• $V$ to a fixed open neighborhood $N_\infty$ of $\infty$ in $R(C)$,

• the univalent vertices of $V$ to their $\infty$-component with respect to $c^0$,

• $V(\Gamma) \setminus V$ to an open subspace $U$ of $R(C) \setminus N_\infty$.

Let $C_V^f(N_\infty, L, c^0)$ denote the space of restrictions to $V$ of configurations of $C(R(C), L; \Gamma)$ mapping univalent vertices of $V$ to $N_\infty$ and to their $\infty$-components (determined by $c^0$), and vertices of $V(\Gamma) \setminus V$ to $U$.

The configuration $c_{V(\Gamma)}^0$ has a neighborhood $N_\Gamma(c_{V(\Gamma)}^0)$ in $C_L^f$ that is a product of $C_V^f(N_\infty, L, c^0)$ by a smooth submanifold $N_2$ of $C_{V(\Gamma)\setminus V}[U]$. In this product decomposition, the $N_2$-part contains the restriction of the configurations $c_{V(\Gamma)}$ to $V(\Gamma) \setminus V$. This space has been studied before (see Proposition 8.6, Section 8.6, and Theorem 8.28), and the manifold $W$ of the statement of Theorem 14.23 will be a product $W_V \times N_2$. This allows us to forget about the $N_2$-part. We focus only on $c^0$ and on a neighborhood $N(c^0)$ of $c^0$ in $C_V(R(C))$.

The configuration $c^0$ is described by

• a $\Delta$-parenthesization $\mathcal{P}^+ = \mathcal{P}^+(c^0) = \mathcal{P}_s^+ \sqcup \mathcal{P}_d$ of $V^+$, where $\mathcal{P}_s = \{V = V(1), V(2), \dots, V(\sigma)\}$ and $V(i+1) \subset V(i)$,

• for any element $A$ of $\mathcal{P}^+$, an element $w_A^0: K(A) \to \mathbb{R}^3$ of the manifold $\tilde{W}_A$ consisting of the injective maps $w_A: K(A) \to \mathbb{R}^3$ up to dilation and translation.

**Notation 14.30.** Let $i \in \bar{\sigma}$. Recall $K_d^s(V(i)) = K(V(i)^+) \cap \mathcal{P}_d$. Set $K^s(V(\sigma)) = K_d^s(V(\sigma))$, and $K^s(V(i)) = K_d^s(V(i)) \sqcup \{V(i+1)\}$ when $i \neq \sigma$. We normalize $w_{V(i)^+}^0$ so that $w_{V(i)^+}^0(v_\infty) = 0$, and the restriction $f_i^0$ of $w_{V(i)^+}^0$ to $V(i)$ is in the manifold consisting of the maps $f_i: K^s(V(i)) \to \mathbb{R}^3$ such that

• $f_i(V(i+1)) = 0$, if $i \neq \sigma$,

• $\sum_{A \in K_d^s(V(i))} \|f_i(A)\|^2 = 1$,

• $\|f_i(A)\| > \eta$ for any $i$ and for any $A \in K_d^s(V(i))$,

• $\|f_i(A_2) - f_i(A_1)\| > \eta$ for any two distinct elements $A_1$ and $A_2$ of $K^s(V(i))$,

for some real number $\eta > 0$,<fnref n="4" /> We fix an open neighborhood $W_i^s$ of $f_i^0$ in the above manifold.

<footnote n="4">⁴Recall Notation 14.26. The map $f_i$ represents an injective configuration $T_0\phi_\infty \circ f_i$ up to dilation of $\tilde{S}(T_\infty R(C), K_d^s(V(i)))$.</footnote>
**Notation 14.31.** We choose univalent basepoints for univalent sets of $\mathcal{P}_d$. As usual, our basepoints also satisfy the conditions that for two elements $A$ and $B$ of $\mathcal{P}_d$, such that $B \subset A$, if $b(A) \in B$, then $b(B) = b(A)$.

For $A \in \mathcal{P}_d$, we normalize the configurations of $\tilde{W}_A$ as follows in a neighborhood $\tilde{N}_A$ of a given $w_A^0 \in \tilde{W}_A$. Choose a kid $k_n(A)$ such that $|p_R(w_A^0(k_n(A))) - p_R(w_A^0(b(A)))|$ or $|p_C(w_A^0(k_n(A))) - p_C(w_A^0(b(A)))|$ is maximal in the set

$$ \left\{ \left| p_{\mathbb{R}}(w_A^0(k)) - p_{\mathbb{R}}(w_A^0(b(A))) \right|, \left| p_{\mathbb{C}}(w_A^0(k)) - p_{\mathbb{C}}(w_A^0(b(A))) \right| : k \in K(A) \right\} $$

and call it the *normalizing kid* of $A$. If

$$ \left| p_{\mathbb{R}}(w_A^0(k_n(A))) - p_{\mathbb{R}}(w_A^0(b(A))) \right| \geq \left| p_{\mathbb{C}}(w_A^0(k_n(A))) - p_{\mathbb{C}}(w_A^0(b(A))) \right|, $$

then we say that $k_n(A)$ is *vertically normalizing* or *v-normalizing* and normalize the configurations $w_A$ in a neighborhood of $w_A^0$ by imposing

$$ w_A(b(A)) = 0 \text{ and } |p_{\mathbb{R}}(w_A(k_n(A)))| = 1. $$

Otherwise, we say that $k_n(A)$ is *horizontally normalizing* or *h-normalizing*, and normalize the configurations $w_A$ in a neighborhood of $w_A^0$ by imposing $w_A(b(A)) = 0$ and $|p_C(w_A(k_n(A)))| = 1$. (These normalizations are compatible with the smooth structure of $C_{V(\Gamma)}(R(\mathcal{C}))$.)

In our neighborhood $\tilde{N}_A$, we also impose that $\|w_A(k) - w_A^0(k)\| < \varepsilon$, for a small $\varepsilon \in ]0, 1[$. So $\tilde{N}_A$ is diffeomorphic to the product $W_A$ of

* the product, over the nonnormalizing kids $k$ of $A$ that do not contain $b(A)$, of the open balls $\tilde{B}(w_A^0(k), \varepsilon)$ of radius $\varepsilon$ centered at $w_A^0(k)$, by

* the set of $w_A(k_n(A))$ in $\tilde{B}(w_A^0(k_n(A)), \varepsilon)$ such that $|p_{\mathbb{R}}(w_A(k_n(A)))| = 1$ (resp. such that $|p_{\mathbb{C}}(w_A(k_n(A)))| = 1$) if $k_n(A)$ is v-normalizing (resp. if $k_n(A)$ is h-normalizing).

We reduce the $\eta$ of Notation 14.30 if necessary, and we choose $\varepsilon$ so that $\|w_A(B_2) - w_A(B_1)\| > \eta$ for any two distinct kids $B_1$ and $B_2$ of $A$ in our normalized neighborhood $W_A$. Note that $\|w_A(k)\| < 3$ for any $k \in K(A)$ in this neighborhood.

All the considered maps $f_i = w_{V(i)^+}|_{V(i)}$, $w_A$ are also considered as maps from $V^+$ to $\mathbb{R}^3$, which are constant on the elements of $K^s(V(i))$ and $K(A)$, respectively, and which respectively map $V^+ \setminus V(i)$ and $V^+ \setminus A$ to 0.
We use a chart $\psi$ of $C_V(R(\mathcal{C}))$ of a neighborhood $N(c^0)$ of $c^0$ in $C_V(R(\mathcal{C}))$ mapping

$$((\mu_A)_{A \in \mathcal{P}^+}, (w_A)_{A \in \mathcal{P}^+}) \in [0, \varepsilon[^{\mathcal{P}^+} \times \prod_{i \in \underline{\sigma}} W_i^s \times \prod_{A \in \mathcal{P}_d} W_A$$

to a configuration $c = \psi((\mu_A)_{A \in \mathcal{P}^+}, (w_A)_{A \in \mathcal{P}^+}) \in C_V(R(\mathcal{C}))$, such that, when the $\mu_A$ do not vanish, $c$ is the injective configuration

$$c = \phi_{\infty} \circ \left( \sum_{A \in \mathcal{P}^+} \left( \prod_{D \in \mathcal{P}^+ : A \subseteq D} \mu_D \right) w_A \right).$$

With this chart induced by Theorems 8.28, 8.32, and 8.35, we have

$$c^0 = \psi((\mu_A^0 = 0), (w_A^0)).$$

**Notation 14.32.** For $k \in \underline{\sigma}$, set $u_k = \mu_{V(k)^+}$, and set $U_k = \prod_{i=1}^k u_i$. For $A \in \mathcal{P}_d$, let $k(A)$ be the maximal integer among the integers $k$ such that $A \subseteq V(k)$. So $\psi$ maps

$$((u_i)_{i \in \underline{\sigma}}, (\mu_A)_{A \in \mathcal{P}_d}, (f_i)_{i \in \underline{\sigma}}, (w_A)_{A \in \mathcal{P}_d}) \in [0, \varepsilon[^{\sigma} \times [0, \varepsilon[^{\mathcal{P}_d}] \times \prod_{i \in \underline{\sigma}} W_i^s \times \prod_{A \in \mathcal{P}_d} W_A$$

to a configuration $c = \psi((u_i), (\mu_A), (f_i), (w_A)) \in C_V(R(\mathcal{C}))$, such that, when the $u_i$ and the $\mu_A$ do not vanish, $c$ is the injective configuration

$$c = \phi_{\infty} \circ \left( \sum_{V(k) \in \mathcal{P}_s} U_k \left( f_k + \sum_{C \in \mathcal{P}_d : k(C) = k} \left( \prod_{D \in \mathcal{P}_d : C \subseteq D} \mu_D \right) w_C \right) \right).$$

We also write

$$c^0 = \psi((u_i^0 = 0), (\mu_A^0 = 0), (f_i^0), (w_A^0)).$$

**Example 14.33.** In the special case of Example 14.25, with the graph

$$\Gamma = \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
\node (v1) at (0, 0) {$v_1$};
\node (v2) at (2, 0) {$v_2$};
\draw[->] (v1) -- (v2);
\end{tikzpicture},$$

consider configurations mapping $v_1$ to $\{z_1\} \times [1, \infty]$ and $v_2$ to $\{z_2\} \times [1, \infty]$. When $V = \{v_1, v_2\}$ and $\mathcal{P}^+ = \{V^+, V\}$, we have $\mathcal{P}_s = \mathcal{P}_d = \{V\}$. Set $f = f_1$, $u = u_1$, $\mu = \mu_V$, $b(V) = v_1$, $w = w_V$. We have $w(v_1) = 0$, $\|f(v_1)\| = 1$, $f^0(v_1) = \vec{N}$, $c = \phi_{\infty} \circ (u(f + \mu w))$, $c(v_1) = \phi_{\infty}(uf(v_1)) = \frac{1}{u}f(v_1)$, and $c(v_2) = \phi_{\infty}(u(f(v_1) + \mu w(v_2)))$.
Back to the general proof of Theorem 14.23 and Propositions 14.27 and 14.28, we will often reduce $\epsilon$ and reduce the spaces $W_k^s$ and $W_A$ to smaller manifolds, which are neighborhoods of $f_k^0$ and $w_A^0$ in the initial manifolds $W_k^s$ and $W_A$. In particular, we assume that the image $N(c^0)$ of $\psi$ is in $C_V(N_\infty)$, and we set

$$N_L^f(c^0) = N(c^0) \cap C_V^f(N_\infty, L, c^0).$$

The intersection of $N_L^f(c^0)$ with $\tilde{C}_V(R(C))$ is determined by the conditions that univalent vertices belong to their $\infty$-components and that their order on the $\infty$-components is prescribed by the isotopy class of injections from $U(\Gamma)$ to $\mathcal{L}$. These conditions are closed. So they still hold in $N_L^f(c^0)$.

In particular, the basepoints $b(A)$ of the univalent elements $A$ of $K_d^s(V(k))$ must go to their $\infty$-components. We call this condition the *first condition*. We examine what this “first condition” imposes on the $f_k$ and prove the following two lemmas.

**Lemma 14.34.** For any $k \in \underline{\sigma}$, for any univalent element $A$ of $K_d^s(V(k))$, we have $f_k^0(A) = \pm \|f_k^0(A)\|\vec{N}$, where $\|f_k^0(A)\| \ge \eta$, and where the $\pm \text{sign}$ is + if $A$ has a univalent vertex above $C$ and — otherwise.

**PROOF:** For an element $A$ of $K_d^s(V(k))$ and a configuration

$$c = \psi((u_i), (\mu_A), (f_i), (w_A)) \in C_V(R(C))$$

such that $U_k = \prod_{i=1}^k u_i \neq 0$, we have

$$c(b(A)) = \frac{1}{U_k} \frac{f_k(A)}{\|f_k(A)\|^2}.$$

So the condition $p_C(c(b(A))) = y(b(A))$ is equivalent to the closed condition

$$p_C(f_k(A)) = U_k \|f_k(A)\|^2 y(b(A)), \qquad (14.1)$$

where

$$\|f_k(A)\|^2 = \|p_C(f_k(A))\|^2 + \|p_R(f_k(A))\|^2 \le 1,$$

with our normalization of Notation 14.30.

So Equation 14.1 implies $\|p_C(f_k(A))\| = O(U_k)$ (meaning that there exists $C \in \mathbb{R}^*$ such that $\|p_C(f_k(A))\| \le CU_k$). In particular, since $U_k = 0$ for $c^0$, we have $p_C(f_k^0(A)) = 0$ and $f_k^0(A) = \pm \|f_k^0(A)\|\vec{N}$, where $\|f_k^0(A)\| \ge \eta$. Lemma 14.34 follows easily since the sign of $p_R(f_k(A))$ is constant on $N_L^f(c^0)$. $\square$

In particular, if $V(\sigma) \in \mathcal{P}_d$, then $f_\sigma^0(V(\sigma))$ equals $\vec{N}$ if $V(\sigma)$ has a univalent vertex above $C$, and $f_\sigma^0(V(\sigma))$ equals $(-\vec{N})$ if $V(\sigma)$ has a univalent vertex under $C$.
**Lemma 14.35.** Let $v_1$ and $v_2$ be two vertices on some $\infty$-component of $L$. Assume that $v_1$ is closer to $\infty$ than $v_2$ and that $v_1 \in A_1$ and $v_2 \in A_2$, for two different kids $A_1$ and $A_2$ of $V(k)$, where $k \in \underline{\sigma}$. Then we have

$$ \|f_k^0(A_1)\| < \|f_k^0(A_2)\|. \qquad (14.2) $$

**PROOF:** The configuration $c^0$ is a limit at 0 of a family $c(t)$ indexed by $t \in ]0, \varepsilon[$ of injective configurations for which $\|c(t)(v_1)\| > \|c(t)(v_2)\|$, with

$$ c(t) = c = \psi((u_i), (\mu_A), (f_i), (w_A)). $$

Therefore, we have $\|f_k^0(A_1)\| \le \|f_k^0(A_2)\|$. Since $f_k^0(A_1) \ne f_k^0(A_2)$, the result follows. $\square$

In particular, we have $|p_R(f_k^0(A_1))| \le |p_R(f_k^0(A_2))| - \eta$. We possibly reduce $W_k^s$ by imposing $|p_R(f_k(A)) - p_R(f_k^0(A))| < \varepsilon$ for some positive $\varepsilon$ such that $\varepsilon < \frac{\eta}{2}$. This condition ensures that the univalent vertices $b(A)$, for the elements $A$ of $K_d^s(V(k))$, are well-ordered on any $\infty$-component.

**Lemma 14.36.** Let $k \in \underline{\sigma}$ be such that $V(k) \notin \mathcal{P}_d$. Recall $U_k = \prod_{i=1}^k u_i$. Let $A_{k,1}, A_{k,2}, \dots, A_{k,\ell(k)}$ denote the univalent elements of $K_d^s(V(k))$, where $\ell(k) \in \mathbb{N}$. Let $K_{td}^s(V(k))$ be the set of nonunivalent elements of $K_d^s(V(k))$. For $D \in K_{td}^s(V(k))$, let $\mathring{B}_D = \mathring{B}(f_k^0(D), \varepsilon)$ be the open ball of center $f_k^0(D)$ and of radius $\varepsilon$ in $\mathbb{R}^3$.

* If $\ell(k) \ge 1$, set

$$ W_k^L = \prod_{i=1}^{\ell(k)-1} [p_R(f_k^0(A_{k,i})) - \varepsilon, p_R(f_k^0(A_{k,i})) + \varepsilon] \times \prod_{D \in K_{td}^s(V(k))} \mathring{B}_D. $$

Up to reducing $\varepsilon$, there is a smooth injective map

$$ \phi_k: [0, \varepsilon^k] \times W_k^L \to W_k^s $$

such that, up to reducing $N_L^f(c^0)$ (to a smaller open neighborhood of $c^0$ in $C_V^f(N_\infty, L, c^0)$), all elements

$$ c = \psi((u_i)_{i \in \underline{\sigma}}, (\mu_A)_{A \in \mathcal{P}_d}, (f_i)_{i \in \underline{\sigma}}, (w_A)_{A \in \mathcal{P}_d}) $$

of $N_L^f(c^0)$ satisfy the condition

$$ f_k = \phi_k \left( \prod_{i=1}^{k} u_i, \left( p_R(f_k(A_{k,i})) \right)_{i \in \underline{\ell(k)-1}}, \left( f_k(D) \right)_{D \in K_{td}^s(V(k))} \right). $$
This condition is equivalent to

$$p_{\mathbb{C}}(c(b(A_{k,i}))) = y(b(A_{k,i}))$$

for any $i \in \ell(k)$ when $c \in \check{C}_V(R(\mathcal{C}))$, and it implies Equation 14.1 for any $c$ in $N(c^0)$.

* If $\ell(k) = 0$, set $W_k^L = W_k^s$.

If $V(\sigma) \in \mathcal{P}_d$ and if $V(\sigma)$ is not univalent, also set $W_{\sigma}^L = W_{\sigma}^s$.

If $V(\sigma) \in \mathcal{P}_d$ and if $V(\sigma)$ is univalent, then all elements $c$ of $N_L^f(c^0)$ satisfy the condition

$$p_{\mathbb{C}}(f_{\sigma}(V(\sigma))) = U_{\sigma}y(b(V(\sigma))).$$

This condition is equivalent to $p_{\mathbb{C}}(c(b(V(\sigma)))) = y(b(V(\sigma)))$ when $c \in \check{C}_V(R(\mathcal{C}))$. These elements $c$ also satisfy

$$p_{\mathbb{R}}(f_{\sigma}(V(\sigma))) = \sqrt{1 - U_{\sigma}^2 |y(b(V(\sigma)))|^2} p_{\mathbb{R}}(f_{\sigma}^0(V(\sigma))).$$

In this case, set $W_{\sigma}^L = \{*_{\sigma}\}$.

Let $N_1$ denote the subspace of $N(c^0)$, where the first condition (stated before Lemma 14.34) is satisfied. Then $N_1$ is a smooth manifold parametrized by

$$[0, \varepsilon^{\sigma}] \times [0, \varepsilon^{[P_d]} \times \prod_{k \in \underline{\sigma}} W_k^L \times \prod_{A \in \mathcal{P}_d} W_A.$$

PROOF: The proof of Lemma 14.34 shows that for $i \in \ell(k) - 1$, $p_{\mathbb{C}}(f_k(A_{k,i}))$ is an implicit function of $U_k = \prod_{i=1}^k u_i$ and $p_{\mathbb{R}}(f_k(A_{k,i}))$, which is close to $\pm \|f_k^0(A_{k,i})\| \neq 0$ on $N_1$ and $N_L^f(c^0)$. This implicit function is determined by Equation 14.1.

Then the condition that $\sum_{A \in K_d^s(V(k))} \|f_k(A)\|^2 = 1$ in $W_k^s$ determines $\|f_k(A_{k,\ell(k)})\| \neq 0$ as a function of $U_k$, $(p_{\mathbb{R}}(f_k(A_{k,i})))_{i \in \ell(k)-1}$ and of the $f_k(D)$ for $D \in K_{td}^s(V(k))$. Now, Equation 14.1 determines $p_{\mathbb{C}}(f_k(A_{k,\ell(k)}))$, which in turn determines $p_{\mathbb{R}}(f_k(A_{k,\ell(k)}))$. This is how the map $\phi_k$ of the statement is constructed. It is easy to check that $\phi_k$ has the desired properties and that $N_1$ is parametrized naturally, as announced, using the maps $\phi_k$. $\square$

In Example 14.33, we have $p_{\mathbb{C}}(f(v_1)) = uy(v_1)$ and

$$p_{\mathbb{R}}(f(v_1)) = \sqrt{1 - u^2 \|y(v_1)\|^2}.$$
So $f(v_1)$ is just a smooth function of the small parameter $u$.

We now restrict to the submanifold $N_1$ of $N(c^0)$ of Lemma 14.36 and take care of the univalent basepoints of the kids of elements of $\mathcal{P}_d$ in the following lemmas.

**Lemma 14.37.** For $X \in S^2$, let $s(X)$ denote the orthogonal reflection of $\mathbb{R}^3$ with respect to the plane orthogonal to $X$. Let $A$ be an element of $\mathcal{P}_d$. Recall that $k(A)$ is the maximal integer $k$ such that $A \subseteq V(k)$. The restriction of $c^0$ to $A$ maps $A$ to

$$X_A^0 = \frac{f_{k(A)}^0(A)}{\|f_{k(A)}^0(A)\|} \in \partial B(R(\mathcal{C}), \infty).$$

It is represented by $s(X_A^0) \circ w_A^0$ up to translation and dilation, as a configuration of the ambient $\mathbb{R}^3$ outside $\mathcal{C}$. If $p$ and $q$ are univalent vertices in two different kids of $A$, if they belong to an $\infty$-component $K^+$, and if $p$ is closer to $\infty$ than $q$, then there exists $\alpha^0 \in \mathbb{R}$ such that $|\alpha^0| > \eta$ and $w_A^0(q) - w_A^0(p) = \alpha^0 \vec{N}$, where $\alpha^0 > 0$ if $K^+$ is above $\mathcal{C}$, and $\alpha^0 < 0$ otherwise.

We introduce some notation before the proof.

**Notation 14.38.** For an element $A$ of $\mathcal{P}_d$ such that $k(A) = k$ and a configuration $c = \psi((u_i), (\mu_A), (f_i), (w_B)) \in N(c^0)$, set

$$M_A = \prod_{D \in \mathcal{P}_d : A \subseteq D} \mu_D,$$

and, for any element $q$ of $A$, define

$$\tilde{f}_k(q) = f_k(A) + \sum_{C \in \mathcal{P}_d : q \in C} M_C w_C(q),$$

so that we have $c(q) = \frac{1}{U_k} \frac{\tilde{f}_k(q)}{\|\tilde{f}_k(q)\|^2}$ when $U_k \neq 0$. Both $M_A$ and $\tilde{f}_k$ depend on the configuration $c$.

**PROOF OF LEMMA 14.37:** The configuration $c_{V(\Gamma)}^0$ is a limit at 0 of a family $c(t)$ indexed by $t \in ]0, \varepsilon[$ of injective configurations of $N_L^f(c^0)$. We have

$$c = c(t) = \psi((u_i), (\mu_B), (f_i), (w_B)),$$

where the $u_i$ and the $\mu_B$ are positive. Set $k = k(A)$. Let $p = b(A)$ be the basepoint of $A$, and let $q$ be the basepoint of a kid of $A$ that does not contain $p$. Since $\tilde{f}_k(q) - \tilde{f}_k(p) = M_A w_A(q)$, we have

$$\|\tilde{f}_k(q)\|^2 = \|\tilde{f}_k(p)\|^2 + 2M_A \langle w_A(q), \tilde{f}_k(p) \rangle + M_A^2 \|w_A(q)\|^2.$$
We get

$$
\begin{align*}
c(q) - c(p) &= \frac{\|\tilde{f}_k(p)\|^2 \tilde{f}_k(q) - \|\tilde{f}_k(q)\|^2 \tilde{f}_k(p)}{U_k \|\tilde{f}_k(q)\|^2 \|\tilde{f}_k(p)\|^2} \\
&= \frac{M_A \|\tilde{f}_k(p)\|^2 w_A(q)}{U_k \|\tilde{f}_k(q)\|^2 \|\tilde{f}_k(p)\|^2} - \frac{2M_A \langle w_A(q), \tilde{f}_k(p) \rangle + M_A^2 \|w_A(q)\|^2}{U_k \|\tilde{f}_k(q)\|^2 \|\tilde{f}_k(p)\|^2} \tilde{f}_k(p).
\end{align*}
$$

When the $\mu_B$ tend to 0 and when $\tilde{f}_k$ tends to $f_k^0$, $\tilde{f}_k(p)$ and $\tilde{f}_k(q)$ tend to $f_k^0(A)$. Thus,

$$
\frac{U_k \| \tilde{f}_k(q) \|^2}{M_A} (c(q) - c(p))
$$

tends to

$$
w_A(q) - 2\langle w_A(q), X_A^0 \rangle X_A^0 = s(X_A^0)(w_A(q)).
$$

So $w_A^0$ is the limit of the $s(X_A^0) \circ c|_A$ up to dilation and translation.

If $A$ contains a univalent vertex of an $\infty$-component above $C$, then $X_A^0$
equals $\vec{N}$ according to Lemma 14.34. In this case, if $a$ and $q$ are univalent
vertices in two different kids of $A$, if they belong to an $\infty$-component $K^+$,
and if $a$ is closer to $\infty$ than $q$, then we have $c(q) - c(a) = -\alpha(t)\vec{N}$ for some
positive $\alpha(t)$ for any $t > 0$. So $(w_A^0(q) - w_A^0(a))$, which is defined up to
dilation, is a positive multiple of $\vec{N}$. $\square$

**Notation 14.39.** Let $\mathcal{P}_{\vec{X}}(= \mathcal{P}_{\vec{X}}(c^0))$ denote the set of elements of $\mathcal{P}_d(= \mathcal{P}_d(c^0))$ that contain or are equal to an element of $\mathcal{P}_X(= \mathcal{P}_X(c^0))$.

**Lemma 14.40.** Let $A \in \mathcal{P}_d$ be such that $k(A) = k$. Let $c$ be as in Notation 14.38. For any univalent kid $B$ of $A$ such that $y(b(B)) - y(b(A)) = 0$, if $c \in N_L^f(c^0)$, we have

$$
p_{\mathcal{C}}(w_A(B)) = U_k \left( 2 \langle \tilde{f}_k(b(A)), w_A(B) \rangle + M_A \|w_A(B)\|^2 \right) y(b(B)). \quad (14.3)
$$

Furthermore, as soon as $p_{\mathcal{C}}(c(b(A))) = y(b(A))$ and $c \in \check{C}_V(R(C))$, Equation 14.3 implies $p_{\mathcal{C}}(c(b(B))) = y(b(B))$ for such a $B$.

If $c \in N_L^f(c^0)$, then there exists $(\lambda_A = \lambda_A(c))_{A \in \mathcal{P}_{\vec{X}}} \in (\mathbb{R}^+)^{\mathcal{P}_{\vec{X}}}$ such that
the following properties are satisfied.

* For any univalent kid $B$ of $A \in \mathcal{P}_{\vec{X}}$, we have

$$
\begin{align}
p_{\mathcal{C}}(w_A(B)) ={}& \lambda_A \|\tilde{f}_{k(A)}(b(A))\|^2 (y(b(B)) - y(b(A))) \nonumber \\
& + U_{k(A)} (2\langle\tilde{f}_{k(A)}(b(A)), w_A(B)\rangle + M_A \|w_A(B)\|^2) y(b(B)). \tag{14.4}
\end{align}
$$
* For $M \in \mathcal{P}_X$, set $\tilde{\lambda}_M = \lambda_M \mu_M$. For any $A \in \mathcal{P}_{\bar{X}} \setminus \mathcal{P}_X$, there exists $M \in \mathcal{P}_X$ such that $M \subset A$ and we have

$$ \lambda_A = \tilde{\lambda}_M \prod_{D \in \mathcal{P}_d : M \subset D \subset A} \mu_D. $$

* The map $\lambda_A$ is continuous on $N_\Gamma(c^0)$.

* We have $\lambda_A M_A = U_{k(A)}$.

* For any three elements A, B, and D in $\mathcal{P}_d$ such that A and B are in $\mathcal{P}_{\bar{X}}$ and $A \cup B \subseteq D$, we have

$$ \lambda_A \prod_{C \in \mathcal{P}_d : A \subseteq C \subset D} \mu_C = \lambda_B \prod_{C \in \mathcal{P}_d : B \subseteq C \subset D} \mu_C. $$

* For a univalent kid B of $A \in \mathcal{P}_{\bar{X}}$, as soon as $p_{\mathbb{C}}(c(b(A))) = y(b(A))$ holds and $c \in \tilde{C}_V(R(\mathcal{C}))$, Equation 14.4 implies $p_{\mathbb{C}}(c(b(B))) = y(b(B))$.

PROOF: Let $A \in \mathcal{P}_d$ be such that $k(A) = k$. Consider a univalent kid B of A and assume $p_{\mathbb{C}}(c(b(A))) = y(b(A))$. This equality is equivalent to

$$ p_{\mathbb{C}}(\tilde{f}_k(b(A))) = U_k \|\tilde{f}_k(b(A))\|^2 y(b(A)), $$

with $\tilde{f}_k(b(A)) = f_k(A) + \sum_{C \in \mathcal{P}_d : A \subseteq C} M_C w_C(A)$ and $\tilde{f}_k(b(B)) = \tilde{f}_k(b(A)) + M_A w_A(B)$. So the condition $p_{\mathbb{C}}(c(b(B))) = y(b(B))$ may be written as

$$ p_{\mathbb{C}}(\tilde{f}_k(b(A)) + M_A w_A(B)) = U_k \|\tilde{f}_k(b(B))\|^2 y(b(B)), $$

which is equivalent to

$$ M_A p_{\mathbb{C}}(w_A(B)) = U_k \|\tilde{f}_k(b(A))\|^2 (y(b(B)) - y(b(A))) \\ + U_k (\|\tilde{f}_k(b(B))\|^2 - \|\tilde{f}_k(b(A))\|^2) y(b(B)), \quad (14.5) $$

with

$$ \|\tilde{f}_k(b(B))\|^2 - \|\tilde{f}_k(b(A))\|^2 = M_A(2\langle\tilde{f}_k(b(A)), w_A(B)\rangle + M_A\|w_A(B)\|^2). $$

When $y(b(B)) - y(b(A)) = 0$ and $M_A \neq 0$, Equation 14.5 simplifies to Equation 14.3, which also holds in the closure $C_L$. In particular, we have $p_{\mathbb{C}}(w_A^0(b(B))) = 0$. So we also get $|p_{\mathbb{R}}(w_A^0(b(B)))| \geq \eta$.
When $M_A \neq 0$, Equation 14.5 is equivalent to

$$ 
\begin{aligned}
p_{\mathbb{C}}(w_A(B)) - U_k \left( 2\langle \tilde{f}_k(b(A)), w_A(B) \rangle + M_A \|w_A(B)\|^2 \right) y(b(B)) \\
= \frac{U_k}{M_A} \|\tilde{f}_k(b(A))\|^2 (y(b(B)) - y(b(A))).
\end{aligned} 
\quad (14.6) $$

It tells that the left-hand side is colinear to $\|\tilde{f}_k(b(A))\|^2(y(b(B)) - y(b(A)))$, and that the scalar product of these two vectors is nonnegative. This remains true in the closure $C_L$. When $y(b(B)) - y(b(A)) \neq 0$, this uniquely defines $\lambda_A = \lambda_A(B, c)$ such that Equation 14.4 holds for B. Furthermore, $\lambda_A(B, c)$ is continuous on $C_L$, we have

$$ \lim_{U_k \to 0} \lambda_A(B, c) = \frac{\|p_{\mathbb{C}}(w_A(B))\|}{\|\tilde{f}_k(b(A))\|^2 \|y(b(B)) - y(b(A))\|}, $$

and $\lambda_A(B, c) = U_k/M_A$ when $M_A \neq 0$. In particular, if $B'$ is another univalent kid of A such that $y(b(B')) - y(b(A)) \neq 0$ and if $M_A \neq 0$, then $\lambda_A(B, c) = \lambda_A(B', c)$. This remains true in the closure $C_L$ when $M_A = 0$. This allows us to define the continuous $\lambda_A = \lambda_A(B, c)$ for the subset $\mathcal{P}'_{\bar{X}}$ of $\mathcal{P}_{\bar{X}}$ made of the sets A that have kids with basepoints on different components. These $\lambda_A$ satisfy $\lambda_A M_A = U_{k(A)}$ when the parameters $\mu_D$ do no vanish. This also remains true in the closure $C_L$ when $M_A = 0$. Observe $\mathcal{P}_X \subseteq \mathcal{P}'_{\bar{X}}$. For $A \in \mathcal{P}_{\bar{X}} \setminus \mathcal{P}'_{\bar{X}}$, there exists $M \in \mathcal{P}_X$ such that $M \subset A$. Define

$$ \lambda(A, M) = \tilde{\lambda}_M \prod_{D \in \mathcal{P}_d : M \subset D \subset A} \mu_D. $$

So we have $\lambda(A, M)M_A = U_{k(A)}$ and $\lambda(A, M) = \lambda(A, M')$ for any subset $M'$ of A in $\mathcal{P}_X$, when $M_A \neq 0$. So $\lambda(A, M) = \lambda(A, M')$ on $C_L$, and we can set $\lambda_A = \lambda(A, M)$. The other properties of the parameters $\lambda_A$ are obvious when the parameters $\mu_D$ do no vanish. So they hold in $C_L$, and Lemma 14.40 is proved. (Note that the set $\mathcal{P}_X$ of possibly separating sets is the subset of $\mathcal{P}_{\bar{X}}$ consisting of its minimal sets with respect to the inclusion.) $\square$

**Lemma 14.41.** *The configuration*

$$ c^0 = \psi((0)_{k \in \underline{\sigma}}, (0)_{A \in \mathcal{P}_d}, (f_k^0)_{k \in \underline{\sigma}}, (w_A^0)_{A \in \mathcal{P}_d}) $$

of $C_V^f(N_\infty, L, c^0)$ is such that

* we have $p_{\mathbb{C}} \circ w_A^0(B) = 0$ for any univalent kid B of A if $A \in \mathcal{P}_d \setminus \mathcal{P}_X$,
• for any $A \in \mathcal{P}_X$, there exists $\lambda_A^0 \ge 0$ such that

$$p_C \circ w_A^0(B) = \lambda_A^0 \|f_{k(A)}^0(A)\|^2 (y(b(B)) - y(b(A)))$$

for any univalent kid B of A.

PROOF: Lemma 14.40 implies that $\lambda_A^0 = 0$ when $A \in \mathcal{P}_X \setminus \mathcal{P}_X$. $\square$

**Example 14.42.** Let us go back to Example 14.33, where $\|y(v_2) - y(v_1)\|$ is nonzero. When $A = V$ and $B = \{v_2\}$, we have $\tilde{f}(v_2) = f(v_1) + \mu w(v_2)$. When $c$ is injective, Equation 14.5 is equivalent to

$$p_C(w(v_2)) = \lambda \|\tilde{f}(v_1)\|^2 (y(v_2) - y(v_1)) + u(2\langle\tilde{f}(v_1), w(v_2)\rangle + \mu \|w(v_2)\|^2) y(v_2),$$
with $\lambda = \lambda_V = \frac{u}{\mu}$. We get $p_C(w^0(v_2)) = \lambda^0(y(v_2) - y(v_1))$.

Let us now define the oriented tree $\mathcal{T}^0$ (as in Definition 14.1) of the statement of Theorem 14.23.

**Notation 14.43.** The set $E(\mathcal{T}^0)$ of edges of $\mathcal{T}^0$ is in one-to-one correspondence with $\{u_i\}_{i \in \underline{\mathcal{G}}} \cup \{\mu_A\}_{A \in \mathcal{P}_X \setminus \mathcal{P}_x} \cup \{\lambda_A\}_{A \in \mathcal{P}_X \setminus \mathcal{P}_x} \cup \{\tilde{\lambda}_A = \lambda_A \mu_A\}_{A \in \mathcal{P}_x}$, and its edges are labeled by these variables. So $E(\mathcal{T}^0)$ is in one-to-one correspondence with the disjoint union of $\underline{\mathcal{G}}$, $\mathcal{P}_x$, and two disjoint copies of $\mathcal{P}_X \setminus \mathcal{P}_x$. (Recall that $\mathcal{P}_X$ is the set of elements of $\mathcal{P}_d$ that contain or are equal to an element of $\mathcal{P}_X$.)

The set of vertices of $\mathcal{T}(\mathcal{P})$ is in one-to-one correspondence with the disjoint union

$$\mathcal{P}_{\bar{X}} \cup \mathcal{P}_s^+ \cup \{r_1\} \cup \{r_A : A \in \mathcal{P}_{\bar{X}} \setminus \mathcal{P}_x\}.$$

Its elements label the vertices.

The edge labeled by $u_i$ ends at the vertex labeled by $V(i)^+$. It starts at the vertex labeled by $V(i-1)^+$ when $i > 1$, and at the univalent vertex labeled by $r_1$ when $i = 1$.

For $A \in \mathcal{P}_X \setminus \mathcal{P}_x$, the edge labeled by $\lambda_A$ starts from the univalent vertex labeled by $r_A$, and it goes to the vertex labeled by A.

For $A \in \mathcal{P}_X \setminus \mathcal{P}_x$ (resp. for $A \in \mathcal{P}_x$), the edge labeled by $\mu_A$ (resp. by $\tilde{\lambda}_A$) starts at the vertex A, It ends at the mother $m(A)$ of A. See Figure 14.6 for an example of a tree $\mathcal{T}^0$.

According to Lemma 14.40, the variables of

$$\{u_i\}_{i \in \underline{\mathcal{G}}} \cup \{\mu_A\}_{A \in \mathcal{P}_X \setminus \mathcal{P}_x} \cup \{\lambda_A\}_{A \in \mathcal{P}_X \setminus \mathcal{P}_x} \cup \{\tilde{\lambda}_A = \lambda_A \mu_A\}_{A \in \mathcal{P}_x}$$

satisfy the equations that define $X(\mathcal{T}^0)$.
<figure><img src="image_118.png" /><figcaption>Figure 14.6: A tree $\mathcal{T}^0 = \mathcal{T}(c^0)$</figcaption></figure>

**Lemma 14.44.** The dimension of $X(\mathcal{T}^0)$ is $d(\mathcal{T}^0) = |\mathcal{P}_s| + |\mathcal{P}_{\bar{X}}| - |\mathcal{P}_x|$.

**PROOF:** Recall $d(\mathcal{T}^0) = |E(\mathcal{T}^0)| - |L(\mathcal{T}^0)| + 1$ from Lemma 14.9. The set of leaves different from $r_1$ is in one-to-one correspondence with $\mathcal{P}_{\bar{X}}$. $\square$

**Lemma 14.45.** Let $c^0 = \psi((0), (0), (f_i^0), (w_B^0))$ be a configuration of $C_V(R(\mathcal{C}))$ that satisfies the equations of Lemmas 14.36 and 14.40. Then $c^0$ belongs to $C_V^f(N_\infty, L, c^0)$.

Let $A$ be a univalent element of $\mathcal{P}_d$. Let $K_u(A)$ denote the set of univalent kids of $A$ that do not contain $b(A)$, and let $K_t(A)$ denote the set of nonunivalent kids of $A$ that do not contain $b(A)$. Recall Notation 14.31. For any element $D$ of $K_t(A)$, let $B'_D$ denote the open ball $\tilde{B}(w_A^0(D), \varepsilon)$ of radius $\varepsilon$ with center $w_A^0(D)$ in $\mathbb{R}^3$ if $D$ is not normalizing, and let $B'_D$ denote the set of elements $w_A(D)$ of $\tilde{B}(w_A^0(D), \varepsilon)$ such that $|p_{\mathbb{R}}(w_A(D))| = 1$ (resp. such that $|p_{\mathbb{C}}(w_A(D))| = 1$) if $D$ is $v$-normalizing (resp. if $D$ is $h$-normalizing). For any element $D$ of $K_u(A)$, let $J'_D$ denote the interval $[p_{\mathbb{R}}(w_A^0(D)) - \varepsilon, p_{\mathbb{R}}(w_A^0(D)) + \varepsilon]$ if $D$ is not $v$-normalizing, and set $J'_D = \{p_{\mathbb{R}}(w_A^0(D))\}$ if $D$ is $v$-normalizing. Set

$$W_A^L = \prod_{D \in K_u(A)} J'_D \times \prod_{D \in K_t(A)} B'_D.$$

When $A$ is a nonunivalent element of $\mathcal{P}_d$, set $W_A^L = W_A$. Recall that $W_k^L$ has been introduced in Lemma 14.36.

Assume that $2\varepsilon < \lambda_A^0$ for all $A \in \mathcal{P}_x$. Let $\mathcal{P}_{x,hn}$ be the set of elements $A$
of $\mathcal{P}_x$ such that $k_n(A)$ is a univalent horizontally normalizing kid. Set

$$W = [0, \varepsilon^{[\mathcal{P}_d \setminus \mathcal{P}_{\bar{X}}]} \times \left( \prod_{k \in \underline{\sigma}} W_k^L \right) \times \left( \prod_{A \in \mathcal{P}_d} W_A^L \right) \times \prod_{A \in \mathcal{P}_x \setminus \mathcal{P}_{x,hn}} ] \lambda_A^0 - \varepsilon, \lambda_A^0 + \varepsilon [ .$$

*Recall Notation 14.43.* There exists a smooth map from $[0, \varepsilon^{[E(\mathcal{T}^0)]} \times W$ to $C_V(R(\mathcal{C}))$ which restricts to $([0, \varepsilon^{[E(\mathcal{T}^0)]} \cap X(\mathcal{T}^0)] \times W$ as a continuous injective map $\varphi$, whose image is an open neighborhood $N_L^f(c^0)$ of $c^0$ in $C_V^f(N_\infty, L, c^0)$.

**PROOF:** Here and in Notation 14.43, the parameters $u_i$ and $\mu_B$ of $E(\mathcal{T}^0)$ are the initial parameters of $c$ in the chart $\psi$ of $C_V(R(\mathcal{C}))$. The parameters $\lambda_A$ are defined in Lemma 14.40. Lemma 14.40 implies that the parameters $\lambda_A$ satisfy the equations of $X(\mathcal{T}^0)$ for configurations in $C_V^f(N_\infty, L, c^0)$. The factor of $[0, \varepsilon^{[\mathcal{P}_d \setminus \mathcal{P}_{\bar{X}}]}$ of $W$ contains the parameters $\mu_B$ for $B \in \mathcal{P}_d \setminus \mathcal{P}_{\bar{X}}$.

Lemma 14.36 shows how to express the parameter $f_k$ of $\psi^{-1}(c)$, for configurations $c$ in $C_V^f(N_\infty, L, c^0)$, as a smooth function of $W \times [0, \varepsilon^{[\underline{\sigma}]}$, where the factor $[0, \varepsilon^{[\underline{\sigma}]}$ contains the $u_i$.

We now construct the $w_A$ as smooth maps from $[0, \varepsilon^{[E(\mathcal{T}^0)]} \times W$ to $\tilde{S}_{K(A)}(\mathbb{R}^3)$, in order to finish constructing a smooth map

$$\varphi: [0, \varepsilon^{[E(\mathcal{T}^0)]} \times W \to C_{V(\Gamma)}(R(\mathcal{C})).$$

The coordinates of the nonunivalent kids of $A$ and the vertical coordinates of the univalent kids of $A$ are part of $W$, and we do not change them. We only need to determine the horizontal coordinate $p_{\mathcal{C}}(w_A(B))$ for any univalent kid $B$ of $A$ as a smooth map.

Note that $\lambda_A$ is a free parameter in $[0, \varepsilon^{[E(\mathcal{T}^0)]} \times W$ when $A \in \mathcal{P}_{\bar{X}} \setminus \mathcal{P}_{x,hn}$.

Let $A$ be a univalent element of $\mathcal{P}_d$. Set $k = k(A)$. By induction, assume that $w_C$ and $\mu_C$ have already been constructed as smooth functions of $[0, \varepsilon^{[E(\mathcal{T}^0)]} \times W$ for any $C \in \mathcal{P}_d$ that contains $A$, so that $\tilde{f}_k(b(A))$ is a smooth function of $[0, \varepsilon^{[E(\mathcal{T}^0)]} \times W$ defined by the expression from Notation 14.38:

$$\tilde{f}_k(b(A)) = f_k(A) + \sum_{C \in \mathcal{P}_d : A \subset C} M_C w_C(A). \quad (14.7)$$

If $A \notin \mathcal{P}_{\bar{X}}$, then Equation 14.3 determines $p_{\mathcal{C}}(w_A(B))$ as a smooth implicit function of $(p_{\mathbb{R}}(w_A(B)),)$, which is determined by $W$, $\tilde{f}_k(b(A))$, the $u_i$, and the $\mu_C$ for the $C \in \mathcal{P}_d$ such that $A \subseteq C$. If $A \in \mathcal{P}_{\bar{X}}$, then Equation 14.4 determines $p_{\mathcal{C}}(w_A(B))$ as a smooth implicit function of the same parameters
(W, $\tilde{f}_k(b(A))$, the $u_i$, and the $\mu_C$ for the $C \in \mathcal{P}_d$ such that $A \subseteq C$), and $\lambda_A$. The parameter $\mu_A$ is among the parameters unless $A \in \mathcal{P}_x$. If $A \in \mathcal{P}_x \setminus \mathcal{P}_{x,hn}$, then $\mu_A$ is determined as $\tilde{\lambda}_A/\lambda_A$, where $\lambda_A$ is a parameter of W. So $\mu_A$ is this smooth function of our parameters. If $A \in \mathcal{P}_{x,hn}$, then $p_{\mathbb{C}}(w_C(k_n(A)))$ is still a smooth implicit function of W, $\tilde{f}_k(b(A))$, the $u_i$, the $\mu_C$ for the $C \in \mathcal{P}_d$ such that $A \subseteq C$, and $\lambda_A$, and the normalizing condition $|p_{\mathbb{C}}(w_A(k_n(A)))| = 1$ determines $\lambda_A$ as a smooth implicit function of the given parameters.

We have constructed a smooth function $\varphi: [0, \varepsilon^{[E(\mathcal{T}^0)]} \times W \to C_{V(\Gamma)}(R(\mathcal{C}))$. Set

$$ \dot{W} = ]0, \varepsilon^{[\mathcal{P}_d \setminus \mathcal{P}_X]} \times \left( \prod_{k \in \underline{\sigma}} W_k^L \right) \times \left( \prod_{A \in \mathcal{P}_d} W_A^L \right) \times \prod_{A \in \mathcal{P}_x \setminus \mathcal{P}_{x,hn}} ]\lambda_A^0 - \varepsilon, \lambda_A^0 + \varepsilon[. $$

Lemma 14.40 implies that $\varphi$ maps $([0, \varepsilon^{[E(\mathcal{T}^0)]} \cap X(\mathcal{T}^0)] \times \dot{W})$ to $C_V^f(N_\infty, L, c^0)$ (up to reducing $\varepsilon$). Since the closure of $\dot{X}(\mathcal{T}^0)$ in $[0, \infty^{[E(\mathcal{T}^0)]}$ is $X(\mathcal{T}^0)$, according to Lemma 14.13, we have

$$ \varphi(([0, \varepsilon^{[E(\mathcal{T}^0)]} \cap X(\mathcal{T}^0)] \times W) \subseteq C_V^f(N_\infty, L, c^0), $$

too. In particular, any $c^0$ that satisfies the equations of Lemmas 14.36 and 14.40 is in $C_V^f(N_\infty, L, c^0)$.

Lemma 14.40 also implies $N_L^f(c^0) \subseteq \varphi((([0, \varepsilon^{[E(\mathcal{T}^0)]} \cap X(\mathcal{T}^0)] \times W))$ for an open neighborhood $N_L^f(c^0)$ of $c^0$ in $C_V^f(N_\infty, L, c^0)$. The injectivity of the restriction of $\varphi$ to $((([0, \varepsilon^{[E(\mathcal{T}^0)]} \cap X(\mathcal{T}^0)] \times W))$ comes from the fact that Equation 14.4 determine the parameters $\lambda_A$ for $A \in \mathcal{P}_x$ and the equations of Lemma 14.40 determine the others.<fnref n="5" /> $\square$

**PROOF OF PROPOSITION 14.27:** Lemmas 14.34, 14.35, 14.37, and 14.41 show that a configuration $c_{V(\Gamma)}^0$ of $C_L$ must satisfy the conditions of the statement of Proposition 14.27. According to Lemma 14.45, if a configuration $c_{V(\Gamma)}^0$ satisfies the conditions of the statement of Proposition 14.27, its restriction $c^0$ to $V = p_b(c_{V(\Gamma)})^{-1}(\infty)$ is in $C_V^f(N_\infty, L, c^0)$, so $c_{V(\Gamma)}^0$ is in $C_L$. $\square$

**PROOF OF PROPOSITION 14.28:** The strata of Proposition 14.28 are smooth submanifolds of $C_{V(\Gamma)}(R(\mathcal{C}))$ since they correspond to the locus where

<footnote n="5">⁵We can remove these other parameters $\lambda_C$, for $C \in \mathcal{P}_X \setminus \mathcal{P}_X$, from our parametrization for the statement. However, it was more convenient to keep them for the proof since the definition of the $\lambda_A$, for $A \in \mathcal{P}_{x,hn}$, involves the $\lambda_C$ for the $C$ that contain $A$. These $\lambda_C$ are functions of the $\lambda_A$ for which $A \in \mathcal{P}_x$.</footnote>
all the variables associated to $\mathcal{T}^0$ in Notation 14.43 are zero. (The parameters $\lambda_A$ are either zero or not zero on the whole stratum, in the charts of Lemma 14.45.) Let us assume $V = V(\Gamma)$. Then the codimension of the stratum of $c^0$ is $d(\mathcal{T}^0) + |(\mathcal{P}_d \setminus \mathcal{P}_{\bar{X}})|$, which is $|\mathcal{P}_s| + |\mathcal{P}_d| - |\mathcal{P}_x|$, according to Lemma 14.44. $\square$

**Lemma 14.46.** The codimension-one faces of $C^f(R(C), L; \Gamma)$ are those listed in Theorem 14.23. In a neighborhood of these faces, $C^f(R(C), L; \Gamma)$ has the structure of a smooth manifold with boundary.

Let $s = s(\vec{N})$ be the orthogonal reflection of $\mathbb{R}^3$ with respect to the horizontal plane. A configuration

$$c_{V(\Gamma)|V}^0 = c^0 = (T_0\phi_\infty \circ f_1^0, (T_0\phi_{\infty*} \circ w_A^0)_{A \in \mathcal{P}_X}, (\lambda_A^0)_{A \in \mathcal{P}_X})$$

of a T-face is the limit at $t=0$ of a family of injective configurations $c(t)_{t \in [0, \varepsilon]}$ on the vertical parts of the tangle, far above or far below, such that $c(t)|_A = s \circ w_A^0$ up to dilation and translation for any $A \in \mathcal{P}_X(= \mathcal{P}_d = \mathcal{P}_x)$. In particular, for an edge $e = (v_1, v_2)$ whose vertices are in A, we have

$$p_{\tau} \circ p_{e}(c^{0}) = \frac{s \circ w_{A}^{0}(v_{2}) - s \circ w_{A}^{0}(v_{1})}{\|s \circ w_{A}^{0}(v_{2}) - s \circ w_{A}^{0}(v_{1})\|}.$$

For an edge $e = (v_1, v_2)$ whose vertices are in different kids of $V$, we have

$$\begin{aligned} p_{\tau} \circ p_e(c^0) &= \frac{\phi_{\infty} \circ f_1^0(v_2) - \phi_{\infty} \circ f_1^0(v_1)}{\|\phi_{\infty} \circ f_1^0(v_2) - \phi_{\infty} \circ f_1^0(v_1)\|} \\ &= \frac{\|f_1^0(v_1)\|^2 f_1^0(v_2) - \|f_1^0(v_2)\|^2 f_1^0(v_1)}{\|\|f_1^0(v_1)\|^2 f_1^0(v_2) - \|f_1^0(v_2)\|^2 f_1^0(v_1)\|}. \end{aligned}$$

For an edge $e = (v_1, v_2)$ such that $v_1 \in V$ and $v_2 \notin V$, we have

$$p_{\tau} \circ p_{e}(c^{0}) = - \frac{f_{1}^{0}(v_{1})}{\|f_{1}^{0}(v_{1})\|}.$$

For an edge $e = (v_1, v_2)$ such that $v_2 \in V$ and $v_1 \notin V$, we have $p_{\tau} \circ p_e(c^0) = \frac{f_1^0(v_2)}{\|f_1^0(v_2)\|}$.

**PROOF:** Let $\partial_\infty(C^f(R(C), L; \Gamma))$ be the subspace of $C^f(R(C), L; \Gamma)$ consisting of the configurations as above that map at least one univalent vertex to $\infty$. Outside this subspace, the spaces $C^f(R(C), L; \Gamma)$ and $C(R(C), L; \Gamma)$ are the same, and $C^f(R(C), L; \Gamma)$ has the structure of a smooth manifold with
ridges. Recall from Remark 14.29 that the only codimension-one new parts are the T-faces of Theorem 14.23, where $\mathcal{P}_s = \{B \sqcup \sqcup_{j \in I} B_j\}$ and $\mathcal{P}_d = \mathcal{P}_X = \mathcal{P}_x = \{B_j : j \in I\}$. Lemmas 14.45 and 14.14 imply that these strata arise as codimension-one faces of $C^f(R(\mathcal{C}), L; \Gamma)$, along which $C^f(R(\mathcal{C}), L; \Gamma)$ is a smooth manifold with boundary.

Lemma 14.34 implies $f_1^0(b) = \pm \|f_1^0(b)\| \vec{N}$ for any univalent vertex $b$ of $B \sqcup \sqcup_{j \in I} B_j$. Lemma 14.37 implies that the restriction of $c^0$ to any $A$ of $\mathcal{P}_X = \{B_j : j \in I\}$ is represented by $s \circ w_A^0$, up to translation and dilation, as a configuration of $\mathbb{R}^3$. Furthermore, Lemma 14.41 implies

$$p_C(w_A^0(b)) = \lambda_A^0 \|f_1^0(b(A))\|^2 (y(b) - y(b(A)))$$

for any $A \in \mathcal{P}_X$ and for any $b \in A$. Therefore, the configuration $c^0$ is the limit of the following family $c(t)$ of configurations, indexed by $t \in ]0, \varepsilon[$, where we have

$$c(t)(b) = \frac{1}{t} \frac{f_1^0(b)}{\|f_1^0(b)\|^2} \quad \text{for any trivalent vertex of } B,$$

$$c(t)(b) = (y(b), 0) + \frac{f_1^0(b)}{t \|f_1^0(b)\|^2} \quad \text{for any univalent vertex of } B, \text{ and}$$

$$c(t)(b) = (y(b(A)), 0) + \frac{f_1^0(b(A))}{t \|f_1^0(b(A))\|^2} + \frac{s \circ w_A^0(b)}{\lambda_A^0 \|f_1^0(b(A))\|^2} \quad \text{for any vertex } b \text{ of an element} \\ A \text{ of } \mathcal{P}_X = \{B_j : j \in I\}. $$

So $c^0$ is the limit at $t = 0$ of the family of injective configurations $c(t)_{t \in ]0, \varepsilon]}$, and $c(t)|_A = s \circ w_A^0$ up to dilation and translation, for any $A \in \mathcal{P}_X$. $\square$

Theorem 14.23 is now proved. $\square$

<figure><img src="image_119.png" /></figure>

## 14.3 Variations of integrals on configuration spaces of long tangles

In this section, we prove Theorem 12.7.

**Lemma 14.47.** For any two propagating forms $\omega$ and $\omega'$ of $C_2(R)$ (as in Definition 3.11) that coincide on $\partial C_2(R)$, there exists a one-form $\eta$ of $C_2(R)$ that vanishes on $\partial C_2(R)$ such that $\omega' = \omega + d\eta$. In particular, for any two homogeneous propagating forms $\omega$ and $\omega'$ of $C_2(R)$ as in Definition 3.13 that coincide on $UB_R$, there exists a one-form $\eta$ of $C_2(R)$ that vanishes on $\partial C_2(R)$ such that $\omega' = \omega + d\eta$.

**PROOF:** Exercise. See the proof of Lemma 3.17. $\square$

<figure><img src="image_120.png" /></figure>
**Lemma 14.48.** The element $Z_n(C, L, (\omega(i)))$ of $\mathcal{A}_n(\mathcal{L})$ is independent of the chosen homogeneous propagating forms $\omega(i)$ of $(C_2(R(C)), \tau)$, under the assumptions of Theorem 12.7.

More generally, if the $\omega(i)$ are only assumed to be homogeneous propagating forms of $C_2(R(C))$, then $Z_n(C, L, (\omega(i)))$ depends only on $(C, L \cap C, \tau)$ and on the restrictions of the $\omega(i)$ to $UC$.

**PROOF:** By Lemma 14.47, it suffices to prove that $Z$ does not vary when $\omega(i)$ is changed to $\omega(i) + d\eta$ for a one-form $\eta$ on $C_2(R(C))$ that vanishes on $\partial C_2(R(C))$. Let $\Omega_\Gamma = \bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e)))$ and let $\tilde{\Omega}_\Gamma$ be obtained from $\Omega_\Gamma$ by replacing $\omega(i)$ by $\eta$. The variations of the integrals $\int_{(C(R(C),L;\Gamma),o(\Gamma))} \Omega_\Gamma$ are computed with Stokes' theorem, as the sum over the codimension-one faces $F$ of $C(R(C), L; \Gamma)$ of $\int_F \tilde{\Omega}_\Gamma$, as allowed by Lemma 14.24.

These faces are the faces listed in Theorem 14.23. The arguments of Lemmas 9.9, 9.10, 9.11, 9.13, 9.14 allow us to get rid of all the faces, except the faces in which some vertices are at $\infty$, and the faces $F(\tilde{\Gamma}_A, L, \Gamma)$ in which $\tilde{\Gamma}_A$ is a connected diagram on $\mathbb{R}$ (these faces are components of $F(A, L, \Gamma)$ as in the proof of Lemma 10.20) and $\Gamma$ is a diagram that contains $\tilde{\Gamma}_A$ as a subdiagram on a component $\mathcal{L}_j$ of $L$. The contribution of the faces $F(\tilde{\Gamma}_A, L, \Gamma)$ is zero when $i \notin j_E(E(\tilde{\Gamma}_A))$ for dimension reasons. It is zero when $i \in j_E(E(\tilde{\Gamma}_A))$ because $\eta$ vanishes on $\partial C_2(R(C))$. So we are left with the faces for which some vertices are at $\infty$. Let $F$ be such a face. Let $V$ be the set of vertices mapped to $\infty$ in $F$, let $E_\infty$ be the set of edges between elements of $V$, and let $E_m$ denote the set of edges with one end in $V$. When $i \in j_E(E_\infty \cup E_m)$, the contribution vanishes because $\eta$ vanishes on $\partial C_2(R(C))$.

Assume $i \notin j_E(E_\infty \cup E_m)$. The face $F$ is diffeomorphic to a product by

$$ \check{C}_{V(\Gamma) \setminus V}(\check{R}(C), L; \Gamma), $$

whose dimension is

$$ 3|T(\Gamma) \cap (V(\Gamma) \setminus V)| + |U(\Gamma) \cap (V(\Gamma) \setminus V)|, $$

of a space $C_V$ of dimension $3|T(\Gamma) \cap V| + |U(\Gamma) \cap V| - 1$, along which

$$ \bigwedge_{e \in E_\infty \cup E_m} p_e^*(\omega(j_E(e))) $$

has to be integrated. The degree $2|E_\infty \cup E_m|$ of this form is bigger than the dimension of $C_V$ as a count of half-edges shows. So the faces for which some vertices are at $\infty$ (including the $T$-faces) do not contribute either. $\square$
**Proposition 14.49.** Let $L: \mathcal{L} \hookrightarrow \tilde{R}(C)$ denote a long tangle representative in a rational homology cylinder. Let $\tau$ denote a parallelization of $C$ as in Definition 12.4. Let $\omega_0$ and $\omega_1$ be two homogeneous propagating forms of $C_2(R(C))$ (as in Definitions 3.11 and 3.13). Let $\tilde{\omega}$ be a closed 2-form on $[0, 1] \times \partial C_2(R(C))$ whose restriction $\tilde{\omega}(t)$ to $\{t\} \times (\partial C_2(R(C)) \setminus UB_{R(C)})$ is $p_{\tau}^*(\omega_{S^2})$ for any $t \in [0, 1]$, and such that the restriction of $\omega_i$ to $\partial C_2(R(C))$ is $\tilde{\omega}(i)$ for $i \in \{0, 1\}$. For any component $K_j$ of $L = \bigsqcup_{j=1}^k K_j$, define $I_j = \sum_{\Gamma_B \in \mathcal{D}^c(\mathbb{R})} \zeta_{\Gamma_B} I(\Gamma_B, K_j, \tilde{\omega})$, where

$$I(\Gamma_B, K_j, \tilde{\omega}) = \int_{u \in [0,1]} \int_{w \in K_j \cap B_{R(C)}} \int_{\tilde{S}(T_w \tilde{R}(C), \tilde{t}_w; \Gamma_B)} \bigwedge_{e \in E(\Gamma_B)} p_e^*(\tilde{\omega}(u)) [\Gamma_B]$$

and $\tilde{t}_w$ denotes the unit tangent vector to $K_j$ at $w$.

(The notation $\tilde{S}(T_w \tilde{R}(C), \tilde{t}_w; \Gamma_B)$ is introduced before Lemma 8.16, and $\mathcal{D}^c(\mathbb{R})$ is introduced at the beginning of Section 10.5.) Define

$$z(\tilde{\omega}) = \sum_{n \in \mathbb{N}} z_n ([0, 1] \times UB_{R(C)}; \tilde{\omega})$$

as in Corollary 9.4. Then we have

$$Z(C, L, \omega_1) = \left( \prod_{j=1}^{k} \exp(I_j) \right) Z(C, L, \omega_0) \exp(z(\tilde{\omega})).$$

**PROOF:** According to Proposition 10.17, this statement holds when $L$ is a link, and when $L$ is the empty link in particular. Using Notation 7.16, it suffices to prove that

$$\check{Z}(C, L, \omega_1) = \left( \prod_{j=1}^{k} \exp(I_j) \right) \check{Z}(C, L, \omega_0)$$

since $Z(C, L, \omega_1) = Z(\check{R}(C), \emptyset, \omega_1) \check{Z}(C, L, \omega_1)$ as in Lemma 7.27. As in the proof of Lemma 14.48 above, the only faces contributing to the variation of $\check{Z}(C, L, \omega_t)$ are the faces $F(\check{\Gamma}_A, L, \Gamma)$ for which $\check{\Gamma}_A$ is a connected diagram on $\mathbb{R}$ and $\Gamma$ is a diagram that contains $\check{\Gamma}_A$ as a subdiagram on a component $\mathcal{L}_j$ of $L$. Their contribution yields the result as in the proof of Lemma 10.18. $\square$

**Lemma 14.50.** Recall Definition 12.6 of $I_\theta(K, \tau)$ for a long component $K: \mathbb{R} \hookrightarrow \check{R}(C)$ of a tangle in a parallelized $\mathbb{Q}$-cylinder $(C, \tau)$. Let $C$ be a
Q-cylinder. Let $(\tau_t)_{t \in [0,1]}$ be a smooth homotopy of parallelizations of $C$. For any component $K$ of a tangle in $C$, we have

$$I_{\theta}(K, \tau_u) - I_{\theta}(K, \tau_0) = 2 \int_{\cup_{t \in [0, u]} p_{\tau_t}(U^+ + K)} \omega_{S^2}.$$

PROOF: When $K$ is closed, $I_{\theta}$ is defined in Lemma 7.15, and the lemma follows from Proposition 10.13 and Lemma 10.21. Lemma 9.1 implies the existence of a closed 2-form $\omega$ on $[0, 1] \times C_2(R(C))$ that restricts to $\{t\} \times C_2(R(C))$ as a homogeneous propagating form of $(C_2(R(C)), \tau_t)$ for all $t \in [0, 1]$. The integral of this form on $\partial([0, u] \times C(R(C), K; \hat{\zeta}^K))$ is zero, and it is half the difference between the two sides of the equality to be proved when $K$ is a long component. $\square$

PROOF OF THEOREM 12.7: Let $\omega$ be a homogeneous propagating form of $(C_2(R(C)), \tau)$. Let us study the variation of $\tilde{Z}(C, L, \tau) = (\tilde{Z}_n(C, L, \omega))_{n \in \mathbb{N}}$ when $\tau$ varies inside its homotopy class.

Let $(\tau(t))_{t \in [0,1]}$ be a smooth homotopy of parallelizations of $C$. Set $\tilde{Z}(t) = \tilde{Z}(C, L, \tau(t))$. Thanks to Proposition 14.49, we have

$$\frac{\partial}{\partial t} \tilde{Z}(t) = \sum_{j=1}^{k} \left( \frac{\partial}{\partial t} \left( 2 \int_{\cup_{u \in [0,t]} p_{\tau(u)}(U^+ + K_j)} \omega \right) \alpha_{\#j} \right) \tilde{Z}(t)$$

as in Lemma 10.21. Lemma 14.50 implies

$$I_{\theta}(K_j, \tau(t)) - I_{\theta}(K_j, \tau(0)) = 2 \int_{\cup_{u \in [0,t]} p_{\tau(u)}(U^+ + K_j)} \omega$$

for any $j$. As in Corollary 10.22, conclude that

$$\prod_{j=1}^{k} \left( \exp(-I_{\theta}(K_j, \tau(t))\alpha) \right)_{\#j} \tilde{Z}(t)$$

is constant, and note $\tilde{Z}_1(C, K_j, \tau(t)) = \frac{1}{2} I_{\theta}(K_j, \tau(t)) [\hat{\zeta}]$ for an interval component $K_j$.

Proposition 14.49 and Lemma 9.1 imply that changing the trivialization $\tau$ in a ball $B_{\tau}$ that does not meet the tangle does not change $\tilde{Z}$ (where the form $\omega^{\partial}$ of Lemma 9.1 is easily assumed to pull back through the projection of $[0, 1] \times (\partial C_2(R(C)) \setminus U(B_{\tau}))$ onto $\partial C_2(R(C)) \setminus U(B_{\tau})$ on $[0, 1] \times (\partial C_2(R(C)) \setminus U(B_{\tau}))$). Then the proof of Theorem 12.7 can be concluded like the proof of Theorem 7.20 at the end of Section 10.5, with the following additional argument for the strands going from bottom to bottom or from top to top. In
the proof of Theorem 7.20, we assumed that $p_{\tau}(U^{+}K_j) = v$ for some $v \in S^2$, and that $g$ maps $K_j$ to rotations with axis $v$ for any $j \in k$, in order to ensure that $^{\tau}\psi(g^{-1})$ induces a diffeomorphism of $\cup_{w \in K_j} \tilde{S}(T_w \tilde{R}, \tilde{t}_w; \tilde{\Gamma}_B)$. Without loss of generality, we instead assume $v = \vec{N}$ and $p_{\tau}(U^{+}K_j) = \pm \vec{N}$ for all components $K_j$ of $L$, except possibly in a neighborhood of the boundary of $C$, which is mapped to 1 by $g$ (so that $p_{\tau}(U^{+}K_j)$ can move from $\pm \vec{N}$ to $\mp \vec{N}$ in this neighborhood). $\square$
# Chapter 15

## The invariant Z as a holonomy for braids

In this chapter, we interpret the extension of $\mathcal{Z}^f$ to long tangles of the previous chapter as a holonomy for long braids, and we study it as such.

Recall the compactification $S_V(T)$ of the space $\tilde{S}_V(T)$ of injective maps from a finite set $V$ to a vector space $T$ up to translation and dilation, from Theorem 8.11. Let $B$ be a finite set of cardinality at least 2. Let $\Gamma$ be a Jacobi diagram on a disjoint union of lines $\mathbb{R}_b$ indexed by elements $b$ of $B$. Let $p_B: U(\Gamma) \to B$ be the natural map induced by $i_\Gamma$. We assume that $p_B$ is onto. Let $U_b = U_b(\Gamma) = p_B^{-1}(b)$ be the set of univalent vertices of $\Gamma$ sent to $\mathbb{R}_b$ by $i_\Gamma$. Let $\tilde{V}(\Gamma) \subset \tilde{S}_{V(\Gamma)}(\mathbb{R}^3)$ be the quotient by the translations and the dilations of the space of injective maps $c$ from $V(\Gamma)$ to $\mathbb{R}^3 = \mathbb{C} \times \mathbb{R}$ that map $U_b(\Gamma)$ to a vertical line $y(c, b) \times \mathbb{R}$ for each $b \in B$, with respect to the order induced by $i_\Gamma$, so that the planar configuration $y(c, \cdot): B \to \mathbb{C}$ is injective. Let $\mathcal{V}(\Gamma)$ denote the closure of the image of $\tilde{V}(\Gamma)$ in $S_{V(\Gamma)}(\mathbb{R}^3) \times S_B(\mathbb{C})$ under the map $(c \mapsto (c, y(c, \cdot)))$.

### 15.1 On the structure of $\mathcal{V}(\Gamma)$

In this section, we investigate the structure of $\mathcal{V}(\Gamma)$, as we did in Section 14.2 for $C(R(\mathbb{C}), L; \Gamma)$.

**Lemma 15.1.** An element $(c, y)$ of $S_{V(\Gamma)}(\mathbb{R}^3) \times S_B(\mathbb{C})$ is in $\mathcal{V}(\Gamma)$ if and only if

* for any $b \in B$ and for any $(v_1, v_2) \in U_b^2$, the restriction of $c$ to $\{v_1, v_2\}$ is vertical, and its direction is that prescribed by $i_\Gamma$,
* for any pair $(b_1, b_2)$ of distinct elements of $B$ and for any $(v_1, v_2)$ in $U_{b_1} \times U_{b_2}$, there exists $\beta \in \mathbb{R}^+$ such that the restriction $c|_{\{v_1, v_2\}}$ of $c$ to $\{v_1, v_2\}$ satisfies

$$p_C(c|_{\{v_1,v_2\}}(v_2) - c|_{\{v_1,v_2\}}(v_1)) = \beta(y|_{\{b_1,b_2\}}(b_2) - y|_{\{b_1,b_2\}}(b_1))$$

(where $\beta$ depends on chosen representatives of $c$ and $y$), and

* for any triple $(b_1, b_2, b_3)$ of distinct elements of $B$ and for any $(v_1, v_2, v_3) \in U_{b_1} \times U_{b_2} \times U_{b_3}$, there exists $\beta \in \mathbb{R}^+$ such that

$$\left(p_C(c|_{\{v_1,v_2,v_3\}}(v_2) - c|_{\{v_1,v_2,v_3\}}(v_1)), p_C(c|_{\{v_1,v_2,v_3\}}(v_3) - c|_{\{v_1,v_2,v_3\}}(v_1))\right) = \beta\left(y|_{\{b_1,b_2,b_3\}}(b_2) - y|_{\{b_1,b_2,b_3\}}(b_1), y|_{\{b_1,b_2,b_3\}}(b_3) - y|_{\{b_1,b_2,b_3\}}(b_1)\right)$$

in $\mathbb{C}^2$.

It is easy to see that an element of $\mathcal{V}(\Gamma)$ must satisfy these conditions since they are closed and satisfied on $\check{\mathcal{V}}(\Gamma)$. We will prove the converse after Lemma 15.6.

Let $(c^0, y^0)$ satisfy the conditions of Lemma 15.1. We are going to study how a neighborhood $N(c^0, y^0)$ of $(c^0, y^0)$ in $S_{V(\Gamma)}(\mathbb{R}^3) \times S_B(\mathbb{C})$ intersects $\check{\mathcal{V}}(\Gamma)$.

Let us introduce notation in the following long notation paragraph, which ends just before Lemma 15.3.

**Notation 15.2.** As in Theorem 8.28, the configuration $c^0$ in $S_{V(\Gamma)}(\mathbb{R}^3)$ is described by a $\Delta$-parenthesization $\mathcal{P}$ of $V = V(\Gamma)$ (as in Definition 8.24) and

$$(c_Z^0 \in \check{S}_{K(Z)}(\mathbb{R}^3))_{Z \in \mathcal{P}}.$$

The configurations $c$ in a neighborhood of $c^0$ in $S_V(\mathbb{R}^3)$ may be expressed as

$$\begin{aligned} c((\mu_Z), (c_Z)) &= \sum_{Z \in \mathcal{P}} \left( \prod_{Y \in \mathcal{P}: Z \subseteq Y \subseteq V} \mu_Y \right) c_Z \\ &= c_V + \sum_{Z \in D(V)} \mu_Z (c_Z + \sum_{Y \in D(Z)} \mu_Y (c_Y + \dots)) \end{aligned}$$

for $((\mu_Z)_{Z \in \mathcal{P} \setminus \{V\}}, (c_Z)_{Z \in \mathcal{P}}) \in [0, \varepsilon^{\mathcal{P} \setminus \{V\}}] \times \prod_{Z \in \mathcal{P}} W_Z$, as in Lemma 8.27. Similarly, the configuration $y^0$ in $S_B(\mathbb{C})$ may be written as

$$(y_D^0 \in \check{S}_{K(D)}(\mathbb{C}))_{D \in \mathcal{P}_B}$$

for a $\Delta$-parenthesization $\mathcal{P}_B$ of $B$, and the configurations $y$ in a neighborhood of $y^0$ in $S_B(\mathbb{C})$ may be expressed as

$$\begin{aligned} y((u_D), (y_D)) &= \sum_{D \in \mathcal{P}_B} \left( \prod_{E \in \mathcal{P}_B: D \subseteq E \subseteq B} u_E \right) y_D \\ &= y_B + \sum_{D \in D(B)} u_D (y_D + \sum_{E \in D(D)} u_E (y_E + \dots)) \end{aligned}$$
for $((u_D)_{D \in \mathcal{P}_B \setminus \{B\}}, (y_D)_{D \in \mathcal{P}_B}) \in [0, \varepsilon^{\mathcal{P}_B \setminus \{B\}}] \times \prod_{D \in \mathcal{P}_B} N_D$.

We normalize the $y_D$ by choosing basepoints $b(D)$ for the $D \in \mathcal{P}_B$ and imposing $y_D(b(D)) = 0$ and $\sum_{E \in K(D)} \|y_D(E)\|^2 = 1$.

For a set $Y$ of $\mathcal{P}$, we choose a basepoint $b(Y)$, which is univalent if there is a univalent vertex in $Y$. We also choose a kid $k_n(Y)$ such that $|p_{\mathbb{R}}(c_Y^0(k_n(Y))) - p_{\mathbb{R}}(c_Y^0(b(Y)))|$ or $|p_{\mathbb{C}}(c_Y^0(k_n(Y))) - p_{\mathbb{C}}(c_Y^0(b(Y)))|$ is maximal in the set

$$ \left\{ \left| p_{\mathbb{R}}(c_Y^0(k)) - p_{\mathbb{R}}(c_Y^0(b(Y))) \right|, \left| p_{\mathbb{C}}(c_Y^0(k)) - p_{\mathbb{C}}(c_Y^0(b(Y))) \right| : k \in K(Y) \right\} $$

and we call it the *normalizing kid* of $Y$. If

$$ |p_{\mathbb{R}}(c_Y^0(k_n(Y))) - p_{\mathbb{R}}(c_Y^0(b(Y)))| \geq |p_{\mathbb{C}}(c_Y^0(k_n(Y))) - p_{\mathbb{C}}(c_Y^0(b(Y)))|, $$

then we say that $k_n(Y)$ is vertically normalizing or v-normalizing, and we normalize the configurations $c_Y$ in a neighborhood of $c_Y^0$ by imposing $c_Y(b(Y)) = 0$ and $|p_{\mathbb{R}}(c_Y(k_n(Y)))| = 1$. Otherwise, we say that $k_n(Y)$ is horizontally normalizing or h-normalizing, and we first normalize the configurations $c_Y$ in a neighborhood of $c_Y^0$ by imposing

$$ c_Y(b(Y)) = 0 \text{ and } \left| p_{\mathbb{C}}(c_Y(k_n(Y))) \right| = 1. $$

(These normalizations are compatible with the smooth structure of $S_V(\mathbb{R}^3)$. In the case of a horizontally normalizing kid, they will be changed in Notation 15.4.)

In our neighborhood, we also impose $\|c_Y(k) - c_Y^0(k)\| < \varepsilon$ for any kid $k$ of $Y$, for a small $\varepsilon \in ]0, 1[$. So the manifold $W_Y$ is diffeomorphic to the product of the product, over the nonnormalizing kids $k$ of $Y$ that do not contain $b(Y)$, of balls $\hat{B}(c_Y^0(k), \varepsilon)$, by the set of elements $c_Y(k_n(Y))$ of $\hat{B}(c_Y^0(k_n(Y)), \varepsilon)$, such that $|p_{\mathbb{R}}(c_Y(k_n(Y)))| = 1$ (resp. such that $|p_{\mathbb{C}}(c_Y(k_n(Y)))| = 1$) if $k$ is v-normalizing (resp. if $k$ is h-normalizing). Note that $|p_{\mathbb{C}}(c_Y(k))| < 2$ for any $k \in K(Y)$.

For a set $Y$ of $\mathcal{P}$, let $B(Y) \subseteq B$ be the set of (labels of) the components of its univalent vertices, and let $\hat{B}(Y)$ denote the smallest element of $\mathcal{P}_B$ such that $B(Y) \subseteq \hat{B}(Y)$ if $|B(Y)| \geq 2$. If $|B(Y)| = 1$, set $\hat{B}(Y) = B(Y)$. If $B(Y) \neq \emptyset$, the set $Y$ is called *univalent*.

For $D \in \mathcal{P}_B$, define the set $\mathcal{P}_{X,D}$ of elements $Y$ of $\mathcal{P}$ such that $\hat{B}(Y) = D$ and $\hat{B}(Z) \neq D$ for every daughter $Z$ of $Y$. Note that any element $Y$ of $\mathcal{P}_{X,D}$ has at least two kids $Y_a$ and $Y_b$ such that $B(Y_a) \neq D$ and $B(Y_b) \neq D$. Set

$$ \mathcal{P}'_X = \cup_{D \in \mathcal{P}_B} \mathcal{P}_{X,D}. $$
Let $\widehat{\mathcal{P}}'_X$ be the subset of $\mathcal{P}$ consisting of the univalent sets $Z$ of $\mathcal{P}$ such that $|B(Z)| \ge 2$.

**Lemma 15.3.** Let $(c^0, y^0)$ be a configuration parametrized as above, which satisfies the conditions of Lemma 15.1.

For any $Y \in \widehat{\mathcal{P}}'_X$, there exists a unique $\lambda^0(Y) \in \mathbb{R}^+$ such that for any two univalent kids $Y_1$ and $Y_2$ of $Y$ such that $B(Y_1) \subset D_1$ and $B(Y_2) \subset D_2$, where $D_1$ and $D_2$ are kids of $\hat{B}(Y)$, we have

$$p_C(c_Y^0(Y_2) - c_Y^0(Y_1)) = \lambda^0(Y)(y_{\hat{B}(Y)}^0(D_2) - y_{\hat{B}(Y)}^0(D_1)).$$

For any $Y \in \widehat{\mathcal{P}}'_X \setminus \mathcal{P}'_X$, we have $\lambda^0(Y) = 0$. Conversely, if an element $(c^0, y^0)$ satisfies the above properties and the first condition of Lemma 15.1, then it satisfies the conditions of Lemma 15.1.

**PROOF:** Let $Y \in \mathcal{P}$. Assume $|B(Y)| \ge 2$. The first condition of Lemma 15.1 ensures that when $v$ and $v'$ belong to $Y \cap U_b$, we have

$$p_C \circ c_Y^0(v) = p_C \circ c_Y^0(v') = (p_C \circ c_Y^0)(b \in B).$$

So, we view $p_C \circ c_Y^0$ as a map from $B(Y)$ to $\mathbb{C}$.

Assume $Y \in \widehat{\mathcal{P}}'_X$. Then $y_{\hat{B}(Y)}^0|_{B(Y)}$ is not constant. So, the second and third conditions ensure that there exists a unique $\lambda^0(Y) \in \mathbb{R}^+$ such that $p_C \circ c_Y^0$ coincides with $\lambda^0(Y)y_{\hat{B}(Y)|_{B(Y)}}^0$ up to translation. Since $p_C \circ c_Y^0$ is constant on $B(Z)$ for any kid $Z$ of $Y$, if there exists such a kid $Z$ such that $\hat{B}(Z) = \hat{B}(Y)$, then $\lambda^0(Y)$ must be equal to zero.

The last assertion is an easy exercise.

<figure><img src="image_121.png" /></figure>

In order to finish the proof of Lemma 15.1, we are going to prove that the configurations that satisfy the conditions of its statement are in $\mathcal{V}(\Gamma)$. We take a closer look at the structure of $\mathcal{V}(\Gamma)$.

**Notation 15.4.** For a univalent $Y \in \mathcal{P}$, define $d(Y) \in B$ such that $b(Y) \in U_{d(Y)}$. Also assume that $p_B(b(V))$ is the basepoint $b_0$ of $B$. As always, our basepoints satisfy that if $Z \subset Y$ and if $b(Y) \in Z$, then $b(Z) = b(Y)$. For $D \in \mathcal{P}_B$, set $U_D = \prod_{E \in \mathcal{P}_B \setminus \{B\} : D \subseteq E} u_E$. For $d \in D$, set

$$\tilde{y}_D(d) = y_D(d) + \sum_{E \in \mathcal{P}_B : d \in E \subset D} \left( \prod_{F \in \mathcal{P}_B : E \subseteq F \subset D} u_F \right) y_E(d).$$

For $Y \in \mathcal{P}_X$, set $M_Y = \prod_{Z \in \mathcal{P} \setminus \{V\} : Y \subseteq Z} \mu_Z$.

<footnote n="1">¹$\lambda^0(Y)$ depends on the fixed normalizations of $c_Y^0$ and $y_{\hat{B}(Y)}^0$.</footnote>
Let $\mathcal{P}'_{X,hn}$ be the set of elements $Z$ of $\widehat{\mathcal{P}}'_{X}$ such that $k_n(Z)$ is a univalent horizontally normalizing kid. For an element $Z$ of $\mathcal{P}'_{X,hn}$, we have $\lambda^0(Z) \neq 0$, which implies $\mathcal{P}'_{X,hn} \subseteq \mathcal{P}'_{X}$. We change the normalizations of the $c_Z$ for the elements $Z$ of $\mathcal{P}'_{X,hn}$ in our neighborhood $N(c^0, y^0)$ of $(c^0, y^0)$ in $S_{V(\Gamma)}(\mathbb{R}^3) \times S_B(\mathbb{C})$ (which we reduce if needed) in $S_B(\mathbb{C})$ so that $c_Z(b(Z)) = 0$ and

$$p_{\mathbb{C}}(c_Z(k_n(Z))) = \lambda^0(Z) \left( \tilde{y}_{\hat{B}(Z)}(d(k_n(Z))) - \tilde{y}_{\hat{B}(Z)}(d(Z)) \right). \quad (15.1)$$

With this normalization, Lemma 15.3 is still valid, and we have $\lambda(Z) = \lambda^0(Z)$ in our neighborhood.

**Lemma 15.5.** With the above normalizations and notation, there exist continuous maps $\lambda$ and $\lambda(Y)$, for $Y \in \mathcal{P}'_X$, from $N(c^0, y^0) \cap \mathcal{V}(\Gamma)$ to $\mathbb{R}^+$ satisfying the following properties. For any $Y \in \mathcal{P}'_X$, we have $\lambda(Y)(c^0) = \lambda^0(Y)$ with the notation of Lemma 15.3 and, for any configuration $(c, y)$ of $N(c^0, y^0) \cap \mathcal{V}(\Gamma)$,

* we have $p_{\mathbb{C}} \circ c(v) = \lambda y(d)$ for any $d \in B$ and for any $v \in U_d$,

* for $D \in \mathcal{P}_B$ and $Y \in \mathcal{P}_{X,D}$,

- if $Y_a$ and $Y_b$ are two univalent kids of $Y$, then we have

$$p_{\mathbb{C}}(c_Y(Y_b) - c_Y(Y_a)) = \lambda(Y) (\tilde{y}_D(d(Y_b)) - \tilde{y}_D(d(Y_a))),$$

- and the equality

$$*(Y) : \lambda(Y)M_Y = \lambda U_{\hat{B}(Y)}$$

holds.

When $V \in \mathcal{P}'_X$, $*(V)$ is equivalent to $\lambda(V) = \lambda$. When $V \notin \mathcal{P}'_X$, we also let $\lambda(V)$ and $\lambda$ both denote $\lambda$, depending on the context.

**PROOF:** In $N(c^0, y^0) \cap \check{\mathcal{V}}(\Gamma)$, where $\prod_{D \in \mathcal{P}_B \setminus \{B\}} u_D \times \prod_{W \in \mathcal{P} \setminus \{V\}} \mu_W \neq 0$, there exists some $\lambda > 0$ such that $p_{\mathbb{C}} \circ c(v) = \lambda y(b)$ for any $b \in U_b$ and for any $v \in U_b$. This map $\lambda$, starting from $N(c^0, y^0) \cap \check{\mathcal{V}}(\Gamma)$, can be extended continuously on $N(c^0, y^0) \cap \mathcal{V}(\Gamma)$, as follows. Let $b \in B$ be such that $b$ and $b_0$ belong to different kids of $B$, and let $v \in U_b$. Then we have $p_{\mathbb{C}}(c(v)) = \lambda y(b)$ on $N(c^0, y^0) \cap \check{\mathcal{V}}(\Gamma)$, and $y(b)$ does not vanish on $N(c^0, y^0)$. The closed condition that $p_{\mathbb{C}}(c(v))$ and $y(b)$ are colinear and that their scalar product is nonnegative is satisfied on $N(c^0, y^0) \cap \mathcal{V}(\Gamma)$. It allows us to define $\lambda(v)$ such that $p_{\mathbb{C}}(c(v)) = \lambda(v)y(b)$ on $N(c^0, y^0) \cap \mathcal{V}(\Gamma)$ and $\lambda(v)$ is continuous. Now, since $\lambda(v) = \lambda$ is independent of $v$ as above on $N(c^0, y^0) \cap \check{\mathcal{V}}(\Gamma)$, it is
also on $N(c^0, y^0) \cap \mathcal{V}(\Gamma)$. Set $\lambda = \lambda(v)$. Then we have $p_{\mathbb{C}} \circ c(v) = \lambda y(b)$ for any $b \in U_b$ and for any $v \in U_b$ on $N(c^0, y^0) \cap \check{\mathcal{V}}(\Gamma)$. This is also true on $N(c^0, y^0) \cap \mathcal{V}(\Gamma)$.

Let $Y \in \mathcal{P}$. Let $Y_a$ and $Y_b$ be two univalent kids of $Y$. If $c \in N(c^0, y^0) \cap \check{\mathcal{V}}(\Gamma)$, then we have

$$M_Y \left( p_{\mathbb{C}}(c_Y(Y_b) - c_Y(Y_a)) \right) = \lambda U_{\hat{B}(Y)} \left( \tilde{y}_{\hat{B}(Y)}(d(Y_b)) - \tilde{y}_{\hat{B}(Y)}(d(Y_a)) \right).$$

In particular, $p_{\mathbb{C}}(c_Y(Y_b) - c_Y(Y_a))$ and $(\tilde{y}_{\hat{B}(Y)}(d(Y_b)) - \tilde{y}_{\hat{B}(Y)}(d(Y_a)))$ are colinear, and their scalar product is nonnegative on $N(c^0, y^0) \cap \check{\mathcal{V}}(\Gamma)$. As above, as soon as there exist two kids $Y_a$ and $Y_b$ as above, such that $d(Y_a)$ and $d(Y_b)$ are in two distinct kids of $\hat{B}(Y)$, we can define the continuous function $\lambda(Y)$ such that

$$p_{\mathbb{C}}(c_Y(Y_b) - c_Y(Y_a)) = \lambda(Y) \left( \tilde{y}_{\hat{B}(Y)}(d(Y_b)) - \tilde{y}_{\hat{B}(Y)}(d(Y_a)) \right)$$

for any two univalent kids $Y_a$ and $Y_b$ of $Y$, and we have $\lambda(Y)M_Y = \lambda U_{\hat{B}(Y)}$. $\square$

Let $\mathcal{P}'_B$ be the set of elements $D$ of $\mathcal{P}_B$ such that $\mathcal{P}_{X,D} \neq \emptyset$. Let $\widehat{\mathcal{P}}'_B$ be the set of elements $D$ of $\mathcal{P}_B$ that contain, or are equal to, an element of $\mathcal{P}'_B$.

For any collections $(B_i)_{i \in \mathbb{Z}/n\mathbb{Z}}$, $(B_i^+)_{i \in \mathbb{Z}/n\mathbb{Z}}$, $(B'_i)_{i \in \mathbb{Z}/n\mathbb{Z}}$ of (possibly equal) sets of $\widehat{\mathcal{P}}'_B$ such that

$$(B_{i-1}^+ \cup B_i) \subseteq B'_i,$$

for any collections $(Y_i)_{i \in \mathbb{Z}/n\mathbb{Z}}$, $(Y_i^+)_{i \in \mathbb{Z}/n\mathbb{Z}}$, $(Y'_i)_{i \in \mathbb{Z}/n\mathbb{Z}}$ such that

$$Y_i \in \mathcal{P}_{X,B_i}, Y_i^+ \in \mathcal{P}_{X,B_i^+}, Y_i' \in \widehat{\mathcal{P}}'_{X}, \text{ and } (Y_i^+ \cup Y_i) \subseteq Y_i',$$

we have

$$\prod_{i=1}^{n} \left( \lambda(Y_i) \frac{M_{Y_i}}{M_{Y_i'}} \right) \prod_{i=1}^{n} \frac{U_{B_i^+}}{U_{B_{i+1}^+}} = \prod_{i=1}^{n} \frac{U_{B_i}}{U_{B_i'}} \prod_{i=1}^{n} \left( \lambda(Y_i^+) \frac{M_{Y_i^+}}{M_{Y_i'}} \right)$$

for configurations of $N(c^0, y^0) \cap \check{\mathcal{V}}(\Gamma)$. This equation is equivalent to the following equation, which also holds in $N(c^0, y^0) \cap \mathcal{V}(\Gamma)$:

$$\prod_{i=1}^{n} \left( \lambda(Y_i) \prod_{Z \in \mathcal{P}: Y_i \subseteq Z \subseteq Y_i'} \mu_Z \right) \prod_{i=1}^{n} \left( \prod_{E \in \mathcal{P}_B: B_i^+ \subseteq E \subseteq B_{i+1}'} u_E \right) \\ = \prod_{i=1}^{n} \left( \lambda(Y_i^+) \prod_{Z \in \mathcal{P}: Y_i^+ \subseteq Z \subseteq Y_i'} \mu_Z \right) \prod_{i=1}^{n} \left( \prod_{E \in \mathcal{P}_B: B_i \subseteq E \subseteq B_i'} u_E \right).$$
For $Y \in \mathcal{P}$, let $K_u(Y)$ denote the set of univalent kids of $Y$ that do not contain $b(Y)$, and let $K_t(Y)$ denote the set of nonunivalent kids of $Y$ that do not contain $b(Y)$.

We are going to prove the following lemma.

**Lemma 15.6.** Let $(c^0, y^0)$ satisfy the conditions of Lemma 15.1 (and hence of Lemma 15.3). There is a neighborhood of $(c^0, y^0)$ in $\mathcal{V}(\Gamma)$ parametrized by the following variables:

1. $((u_D)_{D \in \mathcal{P}_B \setminus \{B\}}, (y_D)_{D \in \mathcal{P}_B}) \in [0, \varepsilon^{[\mathcal{P}_B \setminus \{B\}]}) \times \prod_{D \in \mathcal{P}_B} N_D$,

2. $(\mu_Z)_{Z \in \mathcal{P} \setminus \{V\}} \in [0, \varepsilon^{[\mathcal{P} \setminus \{V\}]})$,

3. the $c_Z$ for nonunivalent sets $Z$ of $\mathcal{P}$,

4. for univalent sets $Z$ of $\mathcal{P}$, the $c_Z(Y)$ for the kids $Y \in K_t(Z)$ of $Z$, and the $p_R \circ c_Z(Y)$ for the kids $Y \in K_u(Z)$ of $Z$,

5. the parameter $\lambda$, and

6. for every element $Y$ of $\mathcal{P}'_X$, the parameter $\lambda(Y)$ defined in Lemma 15.5.

These variables satisfy the following constraints and determine the $p_C \circ c_Z(Y)$ for the univalent kids $Y \in K_u(Z)$ of a set $Z$ of $\mathcal{P}$ as follows.

1. For a vertically normalizing kid $Y$ of an element $Z$ of $\mathcal{P}$, we have $p_R \circ c_Z(Y) = p_R \circ c_Z^0(Y) = \pm 1$.

2. For a horizontally normalizing nonunivalent kid $Y$ of an element $Z$ of $\mathcal{P}$, we have $|p_C \circ c_Z(Y)| = 1$.

3. For a horizontally normalizing univalent kid $Y$ of an element $Z$ of $\mathcal{P}$, we have

$$p_C(c_Z(Y)) = \lambda^0(Z)(\tilde{y}_{\hat{B}(Z)}(d(Y)) - \tilde{y}_{\hat{B}(Z)}(d(Z))).$$

4. For any element $Y$ of $\mathcal{P}_{X,D}$, we have

$$*(Y) : M_Y \lambda(Y) = \lambda U_D$$

where $M_Y = \prod_{Z \in \mathcal{P} \setminus \{V\}}: Y \subseteq Z \mu_Z$ and $U_D = \prod_{E \in \mathcal{P}_B \setminus \{B\}}: D \subseteq E u_E$.
5. For any collections $(B_i)_{i \in \mathbb{Z}/n\mathbb{Z}}$, $(B_i^+)_{i \in \mathbb{Z}/n\mathbb{Z}}$, $(B'_i)_{i \in \mathbb{Z}/n\mathbb{Z}}$ of (possibly equal) sets of $\widehat{\mathcal{P}}'_B$ such that

$$ (B_{i-1}^{+} \cup B_i) \subseteq B'_i, $$

for any collections $(Y_i)_{i \in \mathbb{Z}/n\mathbb{Z}}$, $(Y_i^+)_{i \in \mathbb{Z}/n\mathbb{Z}}$, $(Y'_i)_{i \in \mathbb{Z}/n\mathbb{Z}}$ such that

$$ Y_i \in \mathcal{P}_{X,B_i}, Y_i^+ \in \mathcal{P}_{X,B_i^+}, Y_i' \in \widehat{\mathcal{P}}'_X, \text{ and } (Y_i^+ \cup Y_i) \subseteq Y_i', $$

we have

$$ \prod_{i=1}^{n} \left( \lambda(Y_i) \prod_{Z \in \mathcal{P}: Y_i \subseteq Z \subset Y_i'} \mu_Z \right) \prod_{i=1}^{n} \left( \prod_{E \in \mathcal{P}_B: B_i^+ \subseteq E \subset B_{i+1}'} u_E \right) \\ = \prod_{i=1}^{n} \left( \lambda(Y_i^+) \prod_{Z \in \mathcal{P}: Y_i^+ \subseteq Z \subset Y_i'} \mu_Z \right) \prod_{i=1}^{n} \left( \prod_{E \in \mathcal{P}_B: B_i \subseteq E \subset B_i'} u_E \right). \quad (15.2) $$

6. For any univalent kid $Z_b$ of an element $Z$ of $\mathcal{P}$ such that $d(Z_b) = d(Z)$, we have

$$ p_{\mathcal{C}} \circ c_Z(Z_b) = p_{\mathcal{C}} \circ c_Z(b(Z)) = 0. $$

7. For any univalent element $Z$ of $\mathcal{P}$ such that $|B(Z)| \ge 2$, for any maximal element $Y$ of $\mathcal{P}'_X$ such that $Y \subseteq Z$ and $\hat{B}(Z) = \hat{B}(Y)^2$, for any univalent kid $Z_b$ of $Z$, $p_{\mathcal{C}} \circ c_Z(Z_b)$ is equal to

$$ p_{\mathcal{C}} \circ c_Z(Z_b) = \frac{M_Y}{M_Z} \lambda(Y) \left( \tilde{y}_{\hat{B}(Z)}(d(Z_b)) - \tilde{y}_{\hat{B}(Z)}(d(Z)) \right). \quad (15.3) $$

The analysis performed before Lemma 15.6 ensures that all the elements in our neighborhood $N(c^0, y^0)$ of $(c^0, y^0)$ in $S_V(\mathbb{R}^3) \times S_B(\mathbb{C})$ that are in $V(\Gamma)$ are described by the parameters described in the statement and that they satisfy all the equations of the statement. (See the proof of Lemma 15.5 for the last one.) In order to prove Lemma 15.6, we are going to prove that, conversely, the elements described in its statement are in $V(\Gamma)$. Note that the elements described in this statement such that neither $\lambda$, nor the $\mu_Z$ nor the $u_D$ vanish, correspond to elements of $\tilde{S}_V(\mathbb{R}^3) \times \tilde{S}_B(\mathbb{C})$ and are in $\check{V}(\Gamma)$.

We define a tree $\mathcal{T}(\mathcal{P}, \mathcal{P}_B)$ with oriented edges from the tree whose vertices are the elements of $\widehat{\mathcal{P}}'_X$ and whose edges start at an element $Z$ of $\widehat{\mathcal{P}}'_X \setminus \{V\}$, end at its mother $m(Z)$, and are labeled by $\mu_Z$,

<footnote n="2">²Note that the minimal univalent elements $Z$ of $\mathcal{P}$ such that $|B(Z)| \ge 2$ are in $\mathcal{P}'_X$.</footnote>
* by gluing an edge, which arrives at the vertex $Y$, starts at a vertex labeled by $O_Y$, and which is labeled by $\lambda(Y)$, at the vertex $Y$, for any $Y$ of $\mathcal{P}'_X$,

* by gluing an edge, which arrives at the vertex $V$, starts at a vertex labeled by $B$, and is labeled by $\lambda$, at the vertex $V$,

* by gluing to the root $B$ of this edge the subtree of $\mathcal{P}_B$ whose vertices are the elements of $\hat{\mathcal{P}}'_B$ and whose edges are labeled by the $u_D$ for $D \in \hat{\mathcal{P}}'_B \setminus \{B\}$, where the edge labeled by $u_D$ starts at $D$ and ends at its mother $m(D)$.

An example is drawn in Figure 15.1.

<figure><img src="image_122.png" /><figcaption>Figure 15.1: The tree $\mathcal{T}(\mathcal{P}, \mathcal{P}_B)$ associated to a configuration $c$</figcaption></figure>

**PROOF OF LEMMA 15.1:** We prove that our configuration $(c^0, y^0)$ of Lemma 15.3 and 15.6 is in $\mathcal{V}(\Gamma)$ by exhibiting a family $(c(t), y(t))$ of configurations of $\tilde{\mathcal{V}}(\Gamma)$, for $t \in ]0, \varepsilon[$ tending to 0, which tends to $(c^0, y^0)$. Our family will be described by nonvanishing $u_D(t)$, $\mu_Z(t)$ tending to 0, nonvanishing $\lambda(Y)(t)$ tending to our given $\lambda^0(Y)$, and nonvanishing $\lambda(t)$ tending to our given $\lambda^0$, such that the equations

$$*(Y)(t): M_Y(t)\lambda(Y)(t) = \lambda(t)U_D(t)$$
are satisfied for any element $Y$ of $\mathcal{P}_{X,D}$. The parameters $(y_D)_{D \in \mathcal{P}_B}$, $c_Z(Y)$ for nonunivalent kids $Y$ of elements $Z$ of $\mathcal{P}$, $p_{\mathbb{R}}(c_Z(Y))$ for univalent kids $Y$ of elements $Z$ of $\mathcal{P}$, of the statement of Lemma 15.6 will be defined to be the same as those of $(c^0, y^0)$.

Set $\mu_Y(t) = t$ for any $Y \in \mathcal{P} \setminus (\mathcal{P}'_X \cup \{V\})$, and $u_D(t) = t$ for any $D \in \mathcal{P}_B \setminus (\mathcal{P}'_B \cup \{B\})$.

Denote $\lambda(t)$ by $u_B(t)$ to make notation homogeneous. See Figure 15.1. Recall that we have $\lambda(V) = \lambda$ when $V \in \mathcal{P}_{X,B}$.

For $D \in \mathcal{P}'_B$, we are going to define some integer $g(D) \ge 1$ and set $u_D(t) = t^{g(D)}$ when $D \ne B$. We are going to set $u_B(t) = \lambda(t) = t^{g(B)}$ if $\lambda^0 = 0$, and $u_B(t) = \lambda(t)$, where $\lambda(t) = \lambda^0$ if $\lambda^0 \ne 0$.

Let $Y \in \mathcal{P}'_X$. If $Y$ is maximal in $\mathcal{P}'_X$, set $\lambda'(Y) = 1$. If $Y$ is not maximal in $\mathcal{P}'_X$, let $s(Y)$ be the minimal element of $\mathcal{P}'_X$ that contains $Y$ strictly. If $\lambda^0(s(Y)) = 0$, set $\lambda'(Y)(t) = 1$. If $\lambda^0(s(Y)) \ne 0$, set $\lambda'(Y)(t) = \lambda(s(Y))(t)$. For $Y \in \mathcal{P}'_X$, we are going to define some integer $g(Y) \ge 1$ and set

* $\lambda(Y)(t) = t$ and $\mu_Y(t) = t^{g(Y)}\lambda'(Y)$ if $\lambda^0(Y) = 0$,

* $\lambda(Y)(t) = \lambda^0(Y)$ and $\mu_Y(t) = \frac{t^{g(Y)+1}}{\lambda^0(Y)}\lambda'(Y)(t)$ if $\lambda^0(Y) \ne 0$.

(If $Y = V$, just forget about $\mu_V$, which is useless.)

Order the elements of $\mathcal{P}'_B$, by calling them $D_1, D_2, \dots D_{|\mathcal{P}'_B|}$ so that $D_1$ is maximal in $\mathcal{P}'_B$ (with respect to the inclusion) and $D_{i+1}$ is maximal in $\mathcal{P}'_B \setminus \{D_1, D_2, \dots, D_i\}$. Note that $B \in \mathcal{P}'_B$, so $D_1 = B$.

Recall $\lambda(t) = t^{g(B)}$ if $\lambda^0 = 0$ and $\lambda(t) \ne 0$ if $\lambda^0 \ne 0$. Set $g(D) = 1$ for any $D \in \widehat{\mathcal{P}}'_B \setminus \mathcal{P}'_B$.

We are going to define the integers $g(D_i)$ and the integers $g(Y)$ for every $Y \in \mathcal{P}_{X,D_i}$, for $i \in |\mathcal{P}'_B|$, inductively, so that the following assertion $(*)$ holds for every $i$.

($*$): for every $Y \in \mathcal{P}_{X,D_i}$, we have $M_Y(t)\lambda(Y)(t) = \lambda(t)U_{D_i}(t) = \lambda(t)t^{f(i)}$ for all $t \in ]0, \varepsilon[$, where $f(i) = \sum_{D \in \widehat{\mathcal{P}}'_B : D_i \subseteq D \subseteq B} g(D)$.

Recall $D_1 = B$ and note $f(1) = 0$. If $V \in \mathcal{P}_{X,B}$, set $g(D_1) = 1$. So $(*)$ is satisfied. (If $\lambda^0 \ne 0$, then $g(D_1)$ is not used, and $(*)$ is satisfied.) If $V \notin \mathcal{P}_{X,B}$, then $\lambda^0 = 0$, according to Lemma 15.3. Define $g(D_1)$ to be the maximum over the elements $Y$ of $\mathcal{P}_{X,B}$ of the number of elements of $\mathcal{P}$ that contain $Y$ strictly, plus one. Then define the $g(Y)$ for the elements $Y$ of $\mathcal{P}_{X,B}$ so that $(*)$ holds.

Let $k \in |\mathcal{P}'_B|$.

Assume that we have defined $g(D_i)$ and the $g(Y)$ for all $Y \in \mathcal{P}_{X,D_i}$, for all $i < k$, so that the assertions $(*)$ hold for any $i < k$. Let us define the $g(Y)$ for all $Y \in \mathcal{P}_{X,D_k}$ and $g(D_k)$ so that $(*)$ holds. For any
$Y \in \mathcal{P}_{X,D_k}$, $M_Y(t)\lambda(Y)(t)$ is equal to $\lambda(t)t^{g(Y)+h(Y)}$ for some integer $h(Y)$, and $\lambda(t)U_{D_k}(t)$ is equal to $\lambda(t)t^{g(D_k)+h(D_k)}$ for some integer $h(D_k)$. Let $H(k)$ be the maximum over $h(D_k)$ and all the integers $h(Y)$ for $Y \in \mathcal{P}_{X,D_k}$. Set $g(D_k) = H(k) + 1 - h(D_k)$ and $g(Y) = H(k) + 1 - h(Y)$, so that $(\ast(k))$ holds with $f(k) = H(k) + 1$.

This process is illustrated in Figure 15.2. Its result does not depend on
the arbitrary order that respects our condition on the $D_i$.

<figure><img src="image_123.png" /><figcaption>Figure 15.2: Interior configurations tending to our limit configuration $c^0$, whose associated tree $\mathcal{T}(\mathcal{P}, \mathcal{P}_B)$ is as in Figure 15.1, where $\lambda^0(Z_2) = \lambda^0(Z_3) = \lambda^0(Z_7) = \lambda^0(Z_9) = 0$ and the other $\lambda^0(Z_i)$ are not zero.</figcaption></figure>

<figure><img src="image_124.png" /></figure>

Let us now begin the longer proof of Lemma 15.6, which ends after Sublemma 15.11. The elements of $\mathcal{V}(\Gamma)$ obviously satisfy the equations of the statement of Lemma 15.6. Let us conversely prove that an element $c$ parametrized as in this statement belongs to $\mathcal{V}(\Gamma)$ by exhibiting a sequence of configurations $c(t)$ of $\check{\mathcal{V}}(\Gamma)$ that approaches it. Again, we index our family of configurations $c(t)$ tending to the given one $c$ by a variable $t$ tending to 0.

For $Y \in \mathcal{P} \setminus (\hat{\mathcal{P}}'_{X} \cup \{V\})$, set $\mu_Y = \mu_Y(c)$ and

$$
\mu_Y(t) = \begin{cases} t & \text{if } \mu_Y = 0 \\ \mu_Y & \text{if } \mu_Y \neq 0. \end{cases}
$$
Similarly, for $D \in \mathcal{P}_B \setminus (\widehat{\mathcal{P}}'_B \cup \{B\})$, set $u_D = u_D(c)$ and

$$u_D(t) = \begin{cases} t & \text{if } u_D = 0 \\ u_D & \text{if } u_D \neq 0. \end{cases}$$

We will again focus on the parameters $\mu_Z$ for $Z \in \widehat{\mathcal{P}}'_X \setminus \{V\}$, $u_D$ for $D \in \widehat{\mathcal{P}}'_B \setminus \{B\}$, $\lambda(Y)$ for $Y \in \mathcal{P}'_X$, and $\lambda$. All of these parameters correspond to edges of our tree $\mathcal{T}(\mathcal{P}, \mathcal{P}_B)$. We denote the parameter corresponding to an edge $e$ by $\lambda(e)$ and the corresponding parameter for the family by $\lambda(e)(t)$. The other parameters of Lemma 15.6 are fixed as in the proof of Lemma 15.1.

When $\lambda(e) \neq 0$, set $\lambda(e)(t) = \lambda(e)$.

Let $\mathcal{P}'_{X,c}$ be the set of elements $Y$ of $\mathcal{P}'_X$ such that $\lambda(Y)M_Y = 0$. If $Y \in \mathcal{P}'_X \setminus \mathcal{P}'_{X,c}$, then $(\lambda(Y)M_Y = \lambda U_{\hat{B}(Y)})$ is not zero. Let $\mathcal{P}'_{B,c}$ be the set of elements $D$ of $\mathcal{P}'_B$ such that $\lambda U_D = 0$.

The main equations

$$*(Y)(t) : \lambda(Y)(t)M_Y(t) = \lambda(t)U_{\hat{B}(Y)}(t)$$

associated to elements of $\mathcal{P}'_X \setminus \mathcal{P}'_{X,c}$ are obviously satisfied.

For $Y \in \mathcal{P}'_X$, recall that $O_Y$ denotes the initial vertex of the edge of $\lambda(Y)$ in the tree $\mathcal{T}(\mathcal{P}, \mathcal{P}_B)$. For $Y \in \mathcal{P}'_{X,c}$, define $e(Y)$ to be the closest edge to $O_Y$ such that $\lambda(e(Y)) = 0$ between $O_Y$ and the vertex $V$ in $\mathcal{T}(\mathcal{P}, \mathcal{P}_B)$. For $D \in \mathcal{P}'_{B,c}$, define $e(D)$ to be the closest edge to $D$ such that $\lambda(e(D)) = 0$ between $D$ and the vertex $V$.

For edges $e$ of $\mathcal{T}(\mathcal{P}, \mathcal{P}_B)$ such that $\lambda(e) = 0$ that are not in $e(\mathcal{P}'_{X,c} \cup \mathcal{P}'_{B,c})$, set $\lambda(e)(t) = t$. For edges $e \in e(\mathcal{P}'_{X,c} \cup \mathcal{P}'_{B,c})$, set $\lambda(e)(t) = r(e)t^{k(e)}$, where $(r(e), k(e)) \in \mathbb{R}^+ \times \mathbb{N}$, $r(e) \neq 0$, and $k(e) \neq 0$. We are going to show how to define pairs $(r(e), k(e))$, so that the equations $(Y)(t)$ are satisfied, for all $t$ and for all $Y$ such that $Y \in \mathcal{P}'_X$. Since they imply the equations 15.2, because all the coefficients $\lambda(e)(t)$ are different from zero, this will conclude the proof.

If $\mathcal{P}'_{B,c} = \emptyset$, then no parameter $\lambda(e)$ vanishes. So there is nothing to prove.

For a set $I$ of edges of $\mathcal{T}(\mathcal{P}, \mathcal{P}_B)$, $\lambda(I)$ denotes the product over the edges $e$ of $I$ of the $\lambda(e)$, while $\lambda_t(I)$ denotes the product over the edges $e$ of $I$ of the $\lambda(e)(t)$. With the notation of Definition 14.1, the main equations $(Y)(t)$ may be written as

$$*(Y)(t) : \lambda_t([O_Y, V]) = \lambda_t([\hat{B}(Y), V]).$$

We also have the obvious sublemma.
**Sublemma 15.7.** Condition 15.2 of Lemma 15.6 may be rewritten as follows. For any collections $(Y_i)_{i \in \mathbb{Z}/n\mathbb{Z}}$, $(Y_i^+)$$_{i \in \mathbb{Z}/n\mathbb{Z}}$, $(Y_i')_{i \in \mathbb{Z}/n\mathbb{Z}}$ such that $Y_i \in \mathcal{P}'_X$, $Y_i^+ \in \mathcal{P}'_X$, $Y_i' \in \widehat{\mathcal{P}}'_X$, and $(Y_i^+ \cup Y_i) \subseteq Y_i'$, and for any collection $(B'_i)_{i \in \mathbb{Z}/n\mathbb{Z}}$ of sets of $\widehat{\mathcal{P}}'_B$ such that

$$ (\hat{B}(Y_{i-1}^+) \cup \hat{B}(Y_i)) \subseteq B'_i, $$

the products of the numbers over the arrows in one direction equals the product of the numbers over the arrows in the opposite direction in the cycle of Figure 15.3.

<figure><img src="image_125.png" /><figcaption>Figure 15.3: The cycle of Condition 15.2 of Lemma 15.6.</figcaption></figure>

We can redraw a typical part of the cycle of Figure 15.3 as in Figure 15.4 to emphasize the different natures of its segments and show how the sets at their ends determine the coefficients over the arrows.

<figure><img src="image_126.png" /><figcaption>Figure 15.4: Another representation of a part of the cycle of Figure 15.3.</figcaption></figure>

For $g \in e(\mathcal{P}'_{B,c})$ and for $f \in e(\mathcal{P}'_{X,c})$, let $\mathcal{P}(f,g)$ be the set of elements $Y \in \mathcal{P}'_X$ such that $e(Y) = f$ and $e(\hat{B}(Y)) = g$. Note the following sublemma.
**Sublemma 15.8.** Recall the notation of Definition 14.1 for sets of edges of a tree. Let $g \in e(\mathcal{P}'_{B,c})$, let $f \in e(\mathcal{P}'_{X,c})$. Assume that there exists an element $Y \in \mathcal{P}(f, g)$. Set

$$c(f, g) = \frac{\lambda([\hat{B}(Y), g])}{\lambda([O_Y, f])}.$$

Then $c(f, g)$ is a positive coefficient independent of the chosen element $Y$ of $\mathcal{P}(f, g)$. The equation

$$*_t(f, g) : \lambda_t([f, V]) = c(f, g)\lambda_t([g, V])$$

must be satisfied for our sequence of configurations $c(t)$ to be in $\check{\mathcal{V}}(\Gamma)$. Furthermore, if Equation $*_t(f, g)$ is satisfied for all $t$, then for any $Y^+ \in \mathcal{P}(f, g)$, Equation $(Y^+)(t)$ is satisfied for all $t$.

**PROOF:** With the given coefficient $c(f, g)$, Equation $*_t(f, g)$ is equivalent to Equation $(Y)(t)$. For any other element $Y^+$ of $\mathcal{P}(f, g)$, according to Condition 15.2 of Lemma 15.6—where $n = 1$, $Y'_1$ is the origin of $f$, and $B'_1$ is the origin of $g$, we have

$$\lambda([O_Y, f])\lambda([\hat{B}(Y^+), g]) = \lambda([O_{Y^+}, f])\lambda([\hat{B}(Y), g]).$$

□

Define the equivalence relation $\sim$ on $e(\mathcal{P}'_{X,c} \cup \mathcal{P}'_{B,c})$ to be the relation generated by the equivalences: Whenever $g \in e(\mathcal{P}'_{B,c})$ and $f \in e(\mathcal{P}'_{X,c})$, if $\mathcal{P}(f, g) \neq \emptyset$, then we have $g \sim f$. When $Y \in \mathcal{P}(f, g)$, such a generating elementary equivalence will also be denoted by $g \sim_Y f$, and its inverse will be denoted by $f \sim_Y g$. In this case, set $c(g, f) = c(f, g)^{-1}$.

**Sublemma 15.9.** For any cycle

$$g = g_1 \sim_{Y_1} f_1 \sim_{Y_1^+} g_2 \sim_{Y_2} f_2 \sim_{Y_2^+} g_3 \dots f_n \sim_{Y_n^+} g_{n+1} = g$$

of elementary equivalences for sequences $(g_i)_{i \in \mathbb{Z}/n\mathbb{Z}}$ of edges of $e(\mathcal{P}'_{B,c})$ and $(f_i)_{i \in \mathbb{Z}/n\mathbb{Z}}$ of edges of $e(\mathcal{P}'_{X,c})$, we have

$$\prod_{i=1}^{n} c(g_i, f_i)c(f_i, g_{i+1}) = 1.$$

**PROOF:** Apply Condition 15.2 of Lemma 15.6, as stated in Sublemma 15.7, to the above sequences $(Y_i)$ and $(Y_i^+)$, where $Y'_i$ is the origin $o(f_i)$ of $f_i$ and $B'_i$ is the origin $o(g_i)$ of $g_i$. Note that Sublemma 15.8 implies that, in each triangle of Figure 15.5, the product of the coefficients over the edges of the
<figure><img src="image_127.png" /><figcaption>Figure 15.5: Definition of $c(g_i, f_i)$ and $c(f_i, g_{i+1})$.</figcaption></figure>

boundary of the triangle that are oriented as part of that boundary is equal to the product of the coefficients over the edges with the opposite orientation.

<figure><img src="image_128.png" /></figure>

The following sublemma is an easy corollary of the previous one.

**Sublemma 15.10.** Let $e$ and $e'$ be two elements of $e(\mathcal{P}'_{X,c} \cup \mathcal{P}'_{B,c})$ such that $e \sim e'$. There exists a sequence $(e_i)_{i \in \underline{m}}$ of edges of $e(\mathcal{P}'_{X,c} \cup \mathcal{P}'_{B,c})$ such that $e_1 = e, e_m = e'$, and $e_i \sim_{Z_i} e_{i+1}$ for any $i \in \underline{m-1}$. For such a sequence, set

$$c(e, e') = \prod_{i=1}^{m-1} c(e_i, e_{i+1}).$$

Then $c(e, e')$ is a positive coefficient independent of the chosen sequences as above.

The equation

$$*_t(e, e') : \lambda_t([e, V]) = c(e, e')\lambda_t([e', V])$$

must be satisfied for our sequence of configurations $c(t)$ to be in $\tilde{\mathcal{V}}(\Gamma)$. Furthermore, for any three elements $e, e_0$, and $e'$ of $e(\mathcal{P}'_{B,c})$ in the same equivalence class under $\sim$, the equations $*_t(e, e')$ and $*_t(e', e)$ are equivalent, and the equations $*_t(e, e_0)$ and $*_t(e', e_0)$ imply $*_t(e, e')$.

<figure><img src="image_129.png" /></figure>

The following sublemma will allow us to define a partial order on the set of equivalence classes under $\sim$.

**Sublemma 15.11.** Let $k \in \mathbb{N}$. It is not possible to find edges $e_1, \dots, e_k, e'_1, \dots, e'_k$ of $e(\mathcal{P}'_{X,c} \cup \mathcal{P}'_{B,c})$ such that $e_j \sim e'_j$ and $e_{j+1} \in [e'_j, V]$ for all $j \in \mathbb{Z}/k\mathbb{Z}$.

**PROOF:** Assume that there exist edges $e_1, \dots, e_k, e'_1, \dots, e'_k$ of $e(\mathcal{P}'_{X,c} \cup \mathcal{P}'_{B,c})$ such that $e_j \sim e'_j$ and $e_{j+1} \in [e'_j, V]$ for all $j \in \mathbb{Z}/k\mathbb{Z}$. Let us picture this hypothesis as follows.
<figure><img src="image_130.png" /></figure>

Let us now construct a cycle as in Figure 15.3 under this hypothesis. An elementary equivalence $g \sim_Y f$ where $Y \in \mathcal{P}(f,g)$ is represented by a path $E(g, f)$

$$ \lambda([\hat{B}(Y), g]) \xleftarrow{o(g)} Y \xrightarrow{\lambda([O_Y, f])} \lambda([\hat{O}(Y), g]) \xrightarrow{o(f)} $$

where $o(g)$ is the set associated to the origin of $g$ and $o(f)$ is associated to the origin of $f$. (It is equal to $O_Y$ if $\lambda(Y) = 0$, and it is the set associated to the initial point of $f$ otherwise.) The inverse equivalence $f \sim_Y g$ is similarly represented by the following path $E(f,g)$

$$ \lambda([O_Y, f]) \xleftarrow{o(f)} Y \xrightarrow{\lambda([\hat{B}(Y), g])} \lambda([\hat{O}(Y), g]) \xrightarrow{o(g)}. $$

Again, the coefficients over the arrows are determined by the labels of the ends. In these cases, they do not vanish, and we picture the paths $E(g, f)$ and $E(f, g)$ as $o(g) \leftarrow Y \hookrightarrow o(f)$ and $o(f) \leftarrow Y \hookrightarrow o(g)$, respectively. Note that according to our assumptions, no $e_j$ can start at some $O_Y$.

If $e_j \neq e'_j$ for all $j \in \mathbb{Z}/k\mathbb{Z}$, then our cycle of arrows as above is obtained by assembling the following paths of arrows $A(e_j, e_{j+1})$ from $o(e_j)$ to $o(e_{j+1})$. The path $A(e_j, e_{j+1})$ is obtained from a sequence $E(e_j, e'_j)$ of paths $E(e, e')$ of arrows associated to elementary equivalences by replacing the last arrow $X_j \hookrightarrow o(e'_j)$, which ends at $o(e'_j)$, by an arrow $X_j \to o(e_{j+1})$ ending at $o(e_{j+1})$. The coefficient of this arrow is obtained from the coefficient of $X_j \hookrightarrow o(e'_j)$ by multiplication by $\lambda([e'_j, e_{j+1}])$, which vanishes in our case because it has a factor $\lambda(e'_j)$. According to the recalled criterion, the product of the coefficients over the arrows in the direction of our cycle must be equal to the product of the coefficients over the arrows in the opposite direction. The first product is zero because of its factors $\lambda(e'_j)$. The second one is nonzero because it only contains nonzero factors associated to equivalences. Therefore, the lemma is proved when $e_j \neq e'_j$ for any $j \in \mathbb{Z}/k\mathbb{Z}$. This case is ruled out.

We cannot have $e_j = e'_j$ for all $j \in \mathbb{Z}/k\mathbb{Z}$.

Up to permuting our indices cyclically, it suffices to rule out the case in which $k \ge 2$ and there exists $r \ge 1$ such that $e_r \neq e'_r$, $e_j = e'_j$ for all $j$ such that $r+1 \le j \le k$, and $e_1 \neq e'_1$. In this case, we define a path $A(e_r, e_1)$ by replacing the last arrow $X_r \hookrightarrow o(e'_r)$ in $E(e_r, e'_r)$ with $X_r \to o(e_1)$, and
multiplying the corresponding coefficient by $\lambda([e'_r, e_1]) = \prod_{j=r}^{k} \lambda([e'_j, e_{j+1}])$. Similarly define paths $A(e_s, e_t)$, for all pair $(s, t)$ of integers such that $s < t < k$, $e_s \neq e'_s$, $e_t \neq e'_t$, and $e_j = e'_j$ for all $j$ such that $s+1 \le j \le t-1$. Define the cycle yielding the contradiction by composing these paths, which include our former paths $A(e_j, e_{j+1})$ for which $e_j \neq e'_j$ and $e_{j+1} \neq e'_{j+1}$. $\square$

END OF THE PROOF OF LEMMA 15.6: Sublemma 15.11 allows us to define the partial order $\succeq$ on the set $E(\sim)$ of equivalence classes of the relation $\sim$ on $e(\mathcal{P}'_{X,c} \cup \mathcal{P}'_{B,c})$ such that two equivalence classes $\bar{e}$ and $\bar{f}$ of $E(\sim)$ satisfy $\bar{f} \succeq \bar{e}$ if and only if there exist a positive integer $k \ge 2$ and two sequences $(e_i)_{i \in \underline{k} \setminus \{1\}}$ and $(e'_i)_{i \in \underline{k-1}}$ such that $e'_1 \in \bar{e}$, $e_k \in \bar{f}$, $e_j \sim e'_j$ for all $j \in \underline{k-1} \setminus \{1\}$, and $e_{j+1} \in [e'_j, V]$ (i.e., $e_{j+1}$ is between the initial point of $e'_j$ and $V$) for all $j \in \underline{k-1}$. For example, if $e_2 \in [e'_1, V]$, then $\bar{e}_2 \succeq \bar{e}'_1$.

Fix an arbitrary total order on $E(\sim)$ compatible with the above partial order by writing

$$E(\sim) = \{\bar{g}_i : i \in \underline{m}\}$$

so that for any $(i, j) \in \underline{m}^2$ such that $\bar{g}_j \succeq \bar{g}_i$, we have $j \le i$.

We will pick one representative $g_i \in e(\mathcal{P}'_{B,c})$ in each equivalence class $\bar{g}_i$ of $E(\sim)$ and define $\lambda(g_i)(t) = r(g_i)t^{k(g_i)}$, for $i = 1, \dots, m$, inductively.

For $e \in \bar{g}_i$, the equation

$$*_t(e, g_i) : \lambda_t([e, V]) = c(e, g_i)\lambda_t([g_i, V]),$$

which must be satisfied, determines $\lambda(e)(t) = r(e)t^{k(e)}$ with $(r(e), k(e)) \in \mathbb{R}^{+*} \times (\mathbb{N} \setminus \{0\})$ as a function of $\lambda(g_i)(t)$ and of the $\lambda(e')(t) = r(e')t^{k(e')}$ for edges $e'$ that belong to $\cup_{j=1}^{i-1} \bar{g}_j$. (Recall that we fixed the coefficients of the other edges.) More precisely, $k(e) - k(g_i)$ is a degree one polynomial in the variables $k(e')$ for edges $e'$ that belong to $\cup_{j=1}^{i-1} \bar{g}_j$, which are already defined by induction. In particular, if we choose $k(g_i)$ to be a sufficiently large integer and fix $r(g_i) = 1$, then the $\lambda(e)(t)$ are uniquely determined so that the equations $*_t(e, g_i)$ are satisfied with $k(e) > 0$ for any $e \in \bar{g}_i$.

Thus, once the induction is achieved, according to Sublemmas 15.8 and 15.10, Equation $*(Y)(t)$ is satisfied for all $t$ for any $Y \in \mathcal{P}$. This shows that the elements described in the statement of Lemma 15.6 are in $\mathcal{V}(\Gamma)$ and finishes the proof of Lemma 15.6. $\square$

Let $\mathcal{P}_u$ denote the set of univalents elements of $\mathcal{P}$, which are the elements of $\mathcal{P}$ that contain at least one univalent vertex.

For a configuration $y \in \mathcal{S}_B(\mathbb{C})$ and a Jacobi diagram $\Gamma$ on a disjoint union of lines $\mathbb{R}_b$ indexed by elements $b$ of $B$, let $\mathcal{V}(y, \Gamma)$ denote the preimage of $y$ under

$$p_{\mathcal{S}_B}: \mathcal{V}(\Gamma) \to \mathcal{S}_B(\mathbb{C}).$$
**Lemma 15.12.** With respect to the notation introduced before Lemma 15.3 and in Lemma 15.3, let $\mathcal{P}'_x$ be the subset of $\mathcal{P}'_X$ consisting of its elements $Y$ such that $\lambda^0(Y) \neq 0$. The dimension of the stratum of $c^0$ in the fiber $\mathcal{V}(y^0, \Gamma)$ over the configuration $y^0 \in \mathcal{S}_B(\mathbb{C})$ is $|U(\Gamma)| + 3|T(\Gamma)| - |\mathcal{P}| + |\mathcal{P}'_x| - 1$.

**PROOF:** For any $Y \in \mathcal{P} \setminus \mathcal{P}_u$, the configuration $c_Y$ is defined up to global translation and dilation. (With our normalizations, the quotient by translations is replaced by the fact that we send basepoints to zero, and our horizontal or vertical normalization condition replaces the quotient by dilation.) For any $Y \in \mathcal{P}_u \setminus \mathcal{P}'_x$, the configuration $c_Y$, whose restriction to the set of univalent vertices is vertical, is defined up to global vertical translation and dilation. For any $Y \in \mathcal{P}'_x$, we still have these two codimension-one normalization conditions on the configuration $c_Y$ when $\lambda(Y)$ is fixed. But varying the parameter $\lambda(Y)$ adds one to the dimension.

Say that a subset of $V(\Gamma)$ is *trivalent* if it contains no univalent vertex. Write the set $K(A)$ of kids of an element $A$ of $\mathcal{P}$ as the union of its set $K_u^0(A)$ of univalent kids and its set $K_t^0(A)$ of trivalent kids (including the kid of the basepoint).

$$K(A) = K_u^0(A) \sqcup K_t^0(A).$$

Then the dimension of the involved space of maps up to dilation and (possibly vertical) translation from $K(A)$ to $\mathbb{R}^3$ is

$$\bullet \ 3|K_t^0(A)| - 4 \text{ if } A \in \mathcal{P} \setminus \mathcal{P}_u,$$

$$\bullet \ |K_u^0(A)| + 3|K_t^0(A)| - 2 \text{ if } A \in \mathcal{P}_u \setminus \mathcal{P}'_x,$$

$$\bullet \ |K_u^0(A)| + 3|K_t^0(A)| - 1 \text{ if } A \in \mathcal{P}'_x.$$

Let $\mathcal{P}_{ext}$ denote the union of $\mathcal{P}$ with the set of singletons of elements of $V$. So $\mathcal{P}_{ext}$ contains all the kids of elements of $\mathcal{P}$. The only element of $\mathcal{P}_{ext}$ that is not a kid is $V$. Let $\mathcal{P}_{ext,u}$ denote the set of univalent sets of $\mathcal{P}_{ext}$, and let $\mathcal{P}_{ext,t}$ denote the set of trivalent sets of $\mathcal{P}_{ext}$. The sum over $A \in \mathcal{P}$ of the above dimensions is equal to

$$3|\mathcal{P}_{ext,t}| + |\mathcal{P}_{ext,u}| - 1 - 3|\mathcal{P} \setminus \mathcal{P}_u| - |\mathcal{P}_u| - |\mathcal{P} \setminus \mathcal{P}'_x|,$$

$$\text{where } |\mathcal{P}_{ext,t}| = |T(\Gamma)| + |\mathcal{P} \setminus \mathcal{P}_u|, \text{ and } |\mathcal{P}_{ext,u}| = |U(\Gamma)| + |\mathcal{P}_u|. \quad \square$$

**Lemma 15.6** simplifies when $y^0 \in \check{\mathcal{S}}_B(\mathbb{C})$. It allows us to describe the structure of $\mathcal{V}(\Gamma)$ over $\check{\mathcal{S}}_B(\mathbb{C})$ in the following lemma.

**Lemma 15.13.** Let $y^0 \in \check{\mathcal{S}}_B(\mathbb{C})$. For any $(c^0, y^0) \in \mathcal{V}(\Gamma)$, there exist

* a manifold $W$ with boundary and ridges,
• an open neighborhood $N(y^0)$ of $y^0$ in $\check{S}_B(\mathbb{C})$,

• a small $\varepsilon > 0$,

• an oriented tree $\mathcal{T}_V(c^0)$, and

• a smooth map $\varphi_1: [0, \varepsilon^{[E(\mathcal{T}_V(c^0))}] \times W \times N(y^0) \to S_{V(\Gamma)}(\mathbb{R}^3)$,

such that the product of $\varphi_1$ by the natural projection $p_{N(y^0)}: [0, \varepsilon^{[E(\mathcal{T}_V(c^0))}] \times W \times N(y^0) \to N(y^0)$ has the following property. The map

$$ \varphi_1 \times p_{N(y^0)}: [0, \varepsilon^{[E(\mathcal{T}_V(c^0))}] \times W \times N(y^0) \to S_{V(\Gamma)}(\mathbb{R}^3) \times N(y^0) $$

restricts to $(X(\mathcal{T}_V(c^0)) \cap [0, \varepsilon^{[E(\mathcal{T}_V(c^0))}] \times W \times N(y^0)$ as a bijection onto an open neighborhood of $(c^0, y^0)$ in $\mathcal{V}(\Gamma)$, where $X(\mathcal{T}_V(c^0))$ was introduced in Definition 14.3.

PROOF: According to Lemma 15.1, $(c^0, y^0)$ has a neighborhood parametrized as in Lemma 15.6. In such a parametrization, no $u_D$ is involved since $y^0 = y_V^0 \in \check{S}_B(\mathbb{C})$. Also note $P'_X = P_{X,B}$. In particular, the elements of $P'_X$ are pairwise disjoint sets. Define the tree $\mathcal{T}_V(c^0)$ associated to $(c^0, y^0)$ to be the tree obtained from $\mathcal{T}(P, P_B)$ by replacing the pair of edges respectively labeled by $\lambda(Y)$ and $\mu_Y$ by a single edge labeled by $\tilde{\lambda}(Y) = \lambda(Y)\mu_Y$ for each $Y \in P'_X$. (In this tree, the right B-part in Figure 15.1 is reduced to one edge labeled by $\lambda$, and the edges labeled by $\tilde{\lambda}(Y)$ for $Y \in P'_X$ start at univalent vertices.) The lemma is an easy consequence of Lemma 15.6. $\square$

**Lemma 15.14.** Let $y \in \check{S}_B(\mathbb{C})$. A codimension-one open face of $\mathcal{V}(y, \Gamma)$ is a stratum $(P, P'_X)$ as in Lemma 15.12 such that

• either we have $V(\Gamma) \in P'_X$, $\lambda \neq 0$, $P'_X = \{V(\Gamma)\}$, and $P = \{V(\Gamma), C\}$ for some $C \subset V(\Gamma)$,

• or we have $V(\Gamma) \notin P'_X$, $\lambda = 0$, and $P = \{V(\Gamma)\} \cup P'_X$ (where $P'_X$ can be empty).

PROOF: According to Lemma 15.12, the faces of the $\partial\mathcal{V}(y, \Gamma)$ with maximal dimension are such that $|P| = |P'_X| + 1$, where $V(\Gamma) \in P$. $\square$

Lemmas 14.14 and 15.13 guarantee that $\mathcal{V}(y, \Gamma)$ behaves as a codimension-one face of a manifold with boundary along a face as in Lemma 15.14.
15.2 A one-form on $\check{S}_B(\mathbb{C})$

For a finite set $B$ of cardinality at least 2, a configuration $y \in S_B(\mathbb{C})$, and a Jacobi diagram $\Gamma$ on a disjoint union of lines $\mathbb{R}_b$ indexed by elements $b$ of $B$ such that $p_B: U(\Gamma) \to B$ is onto, recall that $\mathcal{V}(y, \Gamma)$ is the preimage of $y$ under $p_{S_B}: \mathcal{V}(\Gamma) \to S_B(\mathbb{C})$. When $y \in \check{S}_B(\mathbb{C})$, let $\check{\mathcal{V}}(y, \Gamma)$ denote the quotient by vertical translations of $\check{\mathcal{C}}(\mathbb{R}^3, y^0 \times \mathbb{R}; \Gamma)$ for a representative $y^0 \in \check{C}_B[D_1]$ of $y$. Note that $\check{\mathcal{V}}(y, \Gamma)$ is an open $T$-face of $C(R(\mathcal{C}), L; \Gamma)$ for any tangle $L$ whose top configuration is $y$, as in Theorem 14.23, where the set $B$ of Theorem 14.23 is empty, $I = \{j\}$, and $\mathcal{P}_s = \mathcal{P}_d = \mathcal{P}_x = \{V(\Gamma)\}$ with Notation 14.26. Assume that $\Gamma$ is equipped with a vertex-orientation $o(\Gamma)$ as in Definition 6.13 and with an edge-orientation $o_E(\Gamma)$ of $H(\Gamma)$ as before Lemma 7.1. The space $\check{\mathcal{V}}(y, \Gamma)$ is a smooth manifold of dimension $|U(\Gamma)| + 3|T(\Gamma)| - 1$. It is oriented as the part of the boundary of $C(R(\mathcal{C}), L; \Gamma)$ that is the $T$-face in which all univalent vertices of $\Gamma$ tend to $\infty$ above $\mathcal{C}$, for a tangle $L$ whose top configuration is $y$. (Here, $\Gamma$ is also viewed as a diagram on the domain $\mathcal{L}$ of $L$ by representing the original $[i_\Gamma]$ by a map $i_\Gamma: U(\Gamma) \hookrightarrow (\sqcup_{b \in B} [1, \infty]_{[b] \subset \mathcal{L}})$.) Note that $\check{\mathcal{V}}(y, \Gamma)$ is therefore oriented as the part of the boundary of $(-C(R(\mathcal{C}), L; \Gamma))$ that is (minus) the $T$-face in which all univalent vertices of $\Gamma$ tend to $\infty$ below $\mathcal{C}$, for a tangle $L$ whose bottom configuration is $y$. The orientation of $\check{\mathcal{V}}(y, \Gamma)$ depends on $o(\Gamma)$ and on $o_E(\Gamma)$, but it does not depend on the global orientations of the lines $\mathbb{R}_b$, which are only locally oriented by $o(\Gamma)$ near the images of the univalent vertices of $\Gamma$ as in Definition 6.13.<fnref n="3" />

Below, we define a one-form $\eta_\Gamma = \eta_{\Gamma,o(\Gamma)}$ on $\check{S}_B(\mathbb{C})$ to be the integral of $\bigwedge_{e \in E(\Gamma)} p_{e,S^2}^*(\omega_{S^2})$ along the interiors $\check{\mathcal{V}}(y, \Gamma)$ of the compact fibers $\mathcal{V}(y, \Gamma)$. We agree that the integral along the fiber of $dx \wedge \omega$ for a volume form $\omega$ of the fiber is $(\int_{\text{fiber}} \omega)dx$.

**Proposition 15.15.** The integral of $\bigwedge_{e \in E(\Gamma)} p_{e,S^2}^*(\omega_{S^2})$ along the interior $\check{\mathcal{V}}(y, \Gamma)$ of the fiber $\mathcal{V}(y, \Gamma)$ is absolutely convergent. It defines a smooth one-form $\eta_\Gamma$ on $\check{S}_B(\mathbb{C})$.<fnref n="4" /> The definition of $\eta_\Gamma$ extends naturally to diagrams $\Gamma$ on $\sqcup_{b \in B} \mathbb{R}_b$ such that $p_B: U(\Gamma) \to B$ is not onto.<fnref n="5" />

Let $\gamma: [0, 1] \to \check{S}_B(\mathbb{C})$ be a smooth map. Orient $p_{\check{S}_B}^{-1}(\gamma([0, 1]))$ as the local

<footnote n="3">³The reader who prefers working with oriented strands can assume that the lines $\mathbb{R}_b$ are oriented from bottom to top and consider braids $L$ instead of tangles $L$, above, for the moment. However, since we will later need to allow various orientations for our strands $\mathbb{R}_b$, it is better to work with unoriented strands as much as possible.</footnote>

<footnote n="4">⁴Again, $\eta_\Gamma$ depends on the arbitrary vertex-orientation $o(\Gamma)$ of $\Gamma$, but the product $\eta_\Gamma[\Gamma]$ is independent of $o(\Gamma)$.</footnote>

<footnote n="5">⁵In general, $\eta_\Gamma$ pulls back through $\check{S}_{p_B(U(\Gamma))}(\mathbb{C})$. So $\eta_\Gamma = 0$ if $|p_B(U(\Gamma))| < 2$.</footnote>
product $[0, 1] \times \text{fiber.}^6$ Then the integral

$$ \int_{[0,1]} \gamma^*(\eta_\Gamma) = \int_{p_{\tilde{S}_B}^{-1}(\gamma([0,1]))} \bigwedge_{e \in E(\Gamma)} p_{e,S^2}(\omega_{S^2}) $$

is absolutely convergent. The map $(t \mapsto \int_{[0,t]} \gamma^*(\eta_\Gamma))$ is differentiable, and we have

$$ \frac{\partial}{\partial t} \left( \int_{[0,t]} \gamma^*(\eta_\Gamma) \right) (u) = \eta_\Gamma \left( \gamma(u), \frac{\partial}{\partial t} \gamma_u \right). $$

PROOF: In the proof, we assume that $p_B$ is onto, without loss of generality. Lemma 15.13 implies that the integral $\int_{p_{\tilde{S}_B}^{-1}(\gamma([0,1]))} \bigwedge_{e \in E(\Gamma)} p_{e,S^2}^*(\omega_{S^2})$ is absolutely convergent. See the proof of Theorem 14.16. Let us prove that the integral of $\bigwedge_{e \in E(\Gamma)} p_{e,S^2}^*(\omega_{S^2})$ along the interior $\check{V}(y, \Gamma)$ of the fiber $\mathcal{V}(y, \Gamma)$ is absolutely convergent and that it defines a smooth a one-form $\eta_\Gamma$ on $\check{S}_B(\mathbb{C})$.

Let $y^0 \in \check{S}_B(\mathbb{C})$, let $N(y^0)$ be small neighborhood of $y^0$ in $\check{S}_B(\mathbb{C})$, and let

$$ (\zeta_r : N(y^0) \to \mathbb{R})_{r \in [2|B|-3]} $$

be a system of coordinates on $N(y^0)$. These coordinates give rise to associated smooth one-forms $d\zeta_r = dp_{N(y^0)} \circ \zeta_r$ on $\check{V}(N(y^0), \Gamma) = \check{V}(\Gamma) \cap p_{\tilde{S}_B}^{-1}(N(y^0))$. For a local system $(f_1, \dots, f_k)$ of coordinates of the interior of a fiber $\check{V}(y, \Gamma)$, and a local product structure with the base, we also have associated forms $df_i$, which depend on the product structure. (Changing this product structure adds some combination of the $d\zeta_r$ and $df_j$ to $df_i$). We also have an associated volume form of the fiber $\omega_F = df_1 \wedge \dots \wedge df_k$, which depends on the product structure too. A $(k+1)$-form $\Omega$ on $\check{V}(N(y^0), \Gamma)$ may be expressed as $\Omega = \sum_{r=1}^{2|B|-3} d\zeta_r \wedge (g_r \omega_F) + \sum_{i=1}^k \omega_{i,v}$, where $\omega_{i,v}$ vanishes at the tangent vector $\xi_i$ to a curve of the fiber whose coordinates $f_j$ for $j \in \underline{k} \setminus \{i\}$ are constant ($\omega_{i,v}$ is expressed as a wedge product of coordinates forms that does not involve $df_i$, this decomposition is not canonical). In order to check the convergence of the integral of the pull-back $\Omega$ of the form $\bigwedge_{e \in E(\Gamma)} p_{e,S^2}^*(\omega_{S^2})$ on $\check{V}(N(y^0), \Gamma)$ along the fiber, it suffices to cover the fiber by finitely many neighborhoods as above, to express $\Omega$ as above with respect to the corresponding charts, and to check that the $g_r \omega_F$ (which are well-defined, up to forms whose integrals along the fiber vanish) and their derivatives with respect to the coordinates $\zeta_j$ are bounded in each of these neighborhoods. Lemma 15.13 implies that $\Omega$ is the restriction to

$$ (X(\mathcal{T}_V(c^0)) \cap [0, \varepsilon^{[E(\mathcal{T}_V(c^0))]} \times W \times N(y^0) $$

<footnote n="6">⁶Note that it amounts to say that the $[0, 1]$ factor replaces the upward vertical translation parameter of the quotient $\check{V}(y, \Gamma)$, as far as orientations are concerned.</footnote>
of a smooth form on $[0, \varepsilon^{[E(\mathcal{T}_\nu(c^0))}] \times W \times N(y^0)$, locally. The same holds for the $g_r\omega_F$ and all their iterated partial derivatives with respect to the $\zeta_j$. These forms are bounded in a neighborhood of an arbitrary element $(c^0, y^0)$ of $\mathcal{V}(y^0, \Gamma)$ as in Lemma 15.13. Since the fiber $\mathcal{V}(y^0, \Gamma)$ is compact, the integral of $g_r\omega_F$ along the fiber is absolutely convergent, for any $r \in 2|B| - 3$. So are the integrals of its iterated partial derivatives with respect to the $\zeta_j$ on $\check{\mathcal{V}}(N(y^0), \Gamma)$ for a small neighborhood $N(y^0)$ of $y^0$ in $\check{\mathcal{S}}_B(\mathbb{C})$. This proves that $\eta_\Gamma$ is a well-defined smooth one-form on $\check{\mathcal{S}}_B(\mathbb{C})$. $\square$

**Definition 15.16.** For $k \in \mathbb{N} \setminus \{0\}$, set

$$ \eta_{k,B} = \sum_{\Gamma \in \mathcal{D}_k^e (\sqcup_{b \in B} \mathbb{R}_b)} \frac{(3k - |E(\Gamma)|)!}{(3k)! 2^{|E(\Gamma)|}} \eta_{\Gamma} [\Gamma] \in \Omega^1 (\mathcal{S}_B(\mathbb{C}); \mathcal{A}_k (\sqcup_{b \in B} \mathbb{R}_b)), $$

where $\eta_\Gamma = 0$ if $|p_B(U(\Gamma))| < 2$ or if $\Gamma$ is not connected, and

$$ \eta_B = \sum_{k \in \mathbb{N} \setminus \{0\}} \eta_{k,B}. $$

The form $\eta_B$ is a one-form on $\check{\mathcal{S}}_B(\mathbb{C})$ with coefficients in the space $\mathcal{A}(\sqcup_{b \in B} \mathbb{R}_b)$ of Jacobi diagrams on $\sqcup_{b \in B} \mathbb{R}_b$, which is treated as an unoriented manifold as in Definition 6.16. The form $\eta_B$ will be regarded as a *connection*. Let $p_{CS}: \check{\mathcal{C}}_B[D_1] \to \mathcal{S}_B(\mathbb{C})$ denote the natural projection. For a path $\gamma: [a, b] \to \check{\mathcal{C}}_B[D_1]$, define the *holonomy* $\text{hol}_\gamma(\eta_B)$ of $\eta_B$ along $\gamma$ to be

$$ \text{hol}_{\gamma}(\eta_B) = \sum_{r=0}^{\infty} \int_{(t_1, \dots, t_r) \in [a, b]^r : t_1 \le t_2 \le \dots \le t_r} \bigwedge_{i=1}^{r} (p_{CS} \circ \gamma \circ p_i)^*(\eta_B), $$

where $p_i(t_1, \dots, t_r) = t_i$, the wedge product of forms is performed as usual, and the diagrams are multiplied from bottom to top (from left to right) with respect to their order of appearance.

The degree 0 part of $\text{hol}_\gamma(\eta_B)$ is the unit $[\emptyset]$ of $\mathcal{A}_k(\sqcup_{b \in B} \mathbb{R}_b)$, and we have

$$ \text{hol}_{\gamma}(\eta_B) = [\emptyset] + \sum_{r=1}^{\infty} \int_{(t_1, \dots, t_r) \in [a, b]^r : t_1 \le t_2 \le \dots \le t_r} \bigwedge_{i=1}^{r} (p_{CS} \circ \gamma \circ p_i)^*(\eta_B). $$

This holonomy is valued in a space of diagrams on an unoriented domain as in Definition 6.16, Proposition 10.23, and Remark 10.24. It satisfies the following properties.

* For an orientation-preserving diffeomorphism $\psi: [c, d] \to [a, b]$, we have

$$ \text{hol}_{\gamma \circ \psi}(\eta_B) = \text{hol}_{\gamma}(\eta_B). $$
• When $\gamma_1\gamma_2$ is the path composition of $\gamma_1$ and $\gamma_2$, we have

$$ \mathrm{hol}_{\gamma_1\gamma_2}(\eta_B) = \mathrm{hol}_{\gamma_1}(\eta_B) \mathrm{hol}_{\gamma_2}(\eta_B). $$

• We have $\frac{\partial}{\partial t} \mathrm{hol}_{\gamma|_{[a,t]}}(\eta_B) = \mathrm{hol}_{\gamma|_{[a,t]}}(\eta_B)\eta_B(\gamma'(t))$.

• We have $\frac{\partial}{\partial t} \mathrm{hol}_{\gamma|_{[t,b]}}(\eta_B) = -\eta_B(\gamma'(t)) \mathrm{hol}_{\gamma|_{[t,b]}}(\eta_B)$.

The following lemma expresses the variation of the invariant $\mathcal{Z}$ of long tangles under isotopies that do not fix the bottom and top configurations. It uses the above holonomy and the anomaly $\alpha$ of Section 10.3.

**Lemma 15.17.** Let $(h_t)_{t \in [0,1]}$ be an isotopy of $\tilde{R}(\mathcal{C})$ such that $h_t$ is the identity map on $(\mathbb{C} \setminus D_1) \times \mathbb{R}$ for any $t$, $h_t$ may be expressed as $h_t^- \times \mathbf{1}_{]-\infty,0]}$ on $\mathbb{C} \times ]-\infty,0]$ for a planar isotopy $(h_t^-)_{t \in [0,1]}$, and $h_t$ may be expressed as $h_t^+ \times \mathbf{1}_{[1, +\infty[}$ on $\mathbb{C} \times [1, +\infty[$ for a planar isotopy $(h_t^+)_{t \in [0,1]}$. Assume that $h_0 = \mathbf{1}$ and note that $h_t$ preserves $\mathcal{C}$ setwise. Let $L$ be a long tangle representative of $\tilde{R}(\mathcal{C})$ whose bottom (resp. top) configuration is represented by a map $y^-: B^- \to D_1$ (resp. $y^+: B^+ \to D_1$). Let $J_{bb}$ denote the set of components of $L$ going from bottom to bottom, and let $J_{tt}$ denote the set of components of $L$ going from top to top. Set $\varepsilon(K_j) = -$ for $K_j \in J_{bb}$, and $\varepsilon(K_j) = +$ for $K_j \in J_{tt}$. For a component $K_j$ of $J_{bb} \cup J_{tt}$, let $\theta_j: [0,1] \to \mathbb{R}$ be a path such that the difference $(h_t^{\varepsilon(K_j)}(y^{\varepsilon(K_j)}(K_j(1))) - h_t^{\varepsilon(K_j)}(y^{\varepsilon(K_j)}(K_j(0)))))$ is a positive multiple of the complex direction $\exp(i2\pi\theta_j(t))$. With the notation of Theorem 12.7, set $\mathcal{Z}(t) = \mathcal{Z}(\mathcal{C}, h_t(L))$. Then we have

$$ \left( \prod_{K_j \in J_{bb} \cup J_{tt}} \left( \exp(-2\varepsilon(K_j)(\theta_j(t) - \theta_j(0))\alpha) \right) \#_j \right) \mathcal{Z}(t) \\ = \mathrm{hol}_{h^{-}|_{[t,0]} \circ y^{-}}(\eta_{B^{-}}) \mathcal{Z}(0) \mathrm{hol}_{h^{+}|_{[0,t]} \circ y^{+}}(\eta_{B^{+}}). $$

**PROOF:** Let $\tau$ be a parallelization of $\mathcal{C}$. Set $L = (K_j)_{j \in \mathcal{k}}$, and recall

$$ \mathcal{Z}(t) = \exp\left(-\frac{1}{4}p_1(\tau)\beta\right) \left(\prod_{j=1}^{k} \left(\exp\left(-I_{\theta}(K_j(t), \tau)\alpha\right)\right) \#_j\right) Z(\mathcal{C}, h_t(L), \tau). $$

The algebraic boundary of the chain $\cup_{t \in [t_0, t_1]} \mathcal{C}(R(\mathcal{C}), h_t(L); \Gamma)$ is

$$ \mathcal{C}(R(\mathcal{C}), h_{t_1}(L); \Gamma) - \mathcal{C}(R(\mathcal{C}), h_{t_0}(L); \Gamma) - \sum_{t \in [t_0, t_1]} F_t, $$
where the sum runs over the codimension-one faces $F_t$ of $C(R(C), h_t(L); \Gamma)^7$
Faces cancel as in Section 14.3 except for the anomaly $\alpha$ faces and the faces
for which some vertices are at $\infty$.

The variations due to the anomaly $\alpha$ faces contribute to $\frac{\partial}{\partial t}\tilde{Z}(C, h_t(L), \tau)$
as

$$
\left( \sum_{j=1}^{k} \frac{\partial}{\partial t} \left( 2 \int_{[0,t] \times U+K_j} p_{\tau}^{*}(\omega_{S^2}) \right) \alpha_{\#j} \right) \tilde{Z}(\tilde{R}, h_t(L), \tau)
$$

as in Lemma 10.21.

When the bottom and top configurations are not fixed and when $K_j \in J_{bb} \cup J_{tt}$, we have

$$
I_{\theta}(K_j(u), \tau) - I_{\theta}(K_j(0), \tau) = 2 \int_{\cup_{t \in [0, u]} p_{\tau}(U+K_j(t)) \cup S(K_j(t))} \omega_{S^2},
$$

as in the proof of Lemma 14.50, where $S(K_j(t))$ denotes the half-circle from
$\varepsilon(K_j)\vec{N}$ to $-\varepsilon(K_j)\vec{N}$ through $\exp(2i\pi\theta_j(t))$ as in Lemma 12.5. So we get

$$
I_{\theta}(K_j(t), \tau) - I_{\theta}(K_j(0), \tau) = 2 \int_{\cup_{u \in [0,t]} p_{\tau}(U+K_j(u))} \omega_{S^2} - 2\varepsilon(K_j)(\theta_j(t) - \theta_j(0)).
$$

Thus

$$
\tilde{Z}(t) = \left( \prod_{K_j \in J_{bb} \cup J_{tt}} \left( \exp(-2\varepsilon(K_j)(\theta_j(t) - \theta_j(0))\alpha) \right) \#_j \right) Z(t)
$$

gets no variation from the anomaly $\alpha$ faces as in Corollary 10.22.

Let $F_t$ be a codimension-one face of $C(R(C), h_t(L); \Gamma)$ for which a subset $V$ of $V(\Gamma)$ is mapped to $\infty$. Such a face is either a $T$-face as in Theorem 14.23 and Lemma 14.46, or a face $F_\infty(V, L, \Gamma)$ as around Notation 8.18. Set $F = \cup_{t \in [t_0, t_1]} F_t$.

In both cases, an element $c^0$ of the face involves an injective configuration
$T_0\phi_\infty \circ f_1^0$ from the kids of $V$ to $(T_\infty R(C) \setminus 0)$ up to dilation. Let $e = (v_1, v_2)$
be an edge. If the vertices $v_1$ and $v_2$ are in different kids of $V$, then we have

$$
\begin{align*}
p_{\tau} \circ p_{e}(c^{0}) &= \frac{\phi_{\infty} \circ f_{1}^{0}(v_{2}) - \phi_{\infty} \circ f_{1}^{0}(v_{1})}{\|\phi_{\infty} \circ f_{1}^{0}(v_{2}) - \phi_{\infty} \circ f_{1}^{0}(v_{1})\|} \\
&= \frac{\|f_{1}^{0}(v_{1})\|^{2} f_{1}^{0}(v_{2}) - \|f_{1}^{0}(v_{2})\|^{2} f_{1}^{0}(v_{1})}{\|f_{1}^{0}(v_{1})\|^{2} f_{1}^{0}(v_{2}) - \|f_{1}^{0}(v_{2})\|^{2} f_{1}^{0}(v_{1})}.
\end{align*}
$$

<footnote>$^7$This chain is locally modelled by open subsets of $C(R(C), h_t(L); \Gamma) \times ]t - \varepsilon, t + \varepsilon[$ unless the isotopies $h_t^{\pm}$ are degenerate. See Theorem 14.23 and Lemma 14.14. Stokes’ theorem applies thanks to Theorem 14.16.</footnote>
When $v_1 \in V$ and $v_2 \notin V$, then we have

$$p_{\tau} \circ p_{e}(c^{0}) = - \frac{f_{1}^{0}(v_{1})}{\|f_{1}^{0}(v_{1})\|}.$$

If $v_2 \in V$ and $v_1 \notin V$, we similarly have

$$p_{\tau} \circ p_{e}(c^{0}) = \frac{f_{1}^{0}(v_{2})}{\|f_{1}^{0}(v_{2})\|}.$$

Let $E_{\infty}$ be the set of edges between elements of the set $V$ of vertices mapped to $\infty$ in $F$, and let $E_m$ denote the set of edges with one end in $V$. The face $F_t$ is diffeomorphic to a product by $\tilde{C}_{V(\Gamma)\setminus V}(\tilde{R}(\mathcal{C}), h_t(L); \Gamma)$, whose dimension is

$$3|T(\Gamma) \cap (V(\Gamma) \setminus V)| + |U(\Gamma) \cap (V(\Gamma) \setminus V)|,$$

of a space $C_{V,t}$ of dimension

$$3|T(\Gamma) \cap V| + |U(\Gamma) \cap V| - 1 = 2|E_{\infty}| + |E_{m}| - 1,$$

and $\bigwedge_{e \in E_{\infty} \cup E_m} p_e^*(\omega(j_E(e)))$ has to be integrated along $\cup_{t \in [0,1]} C_{V,t}$, according to the above expression of $p_{\tau} \circ p_e$ for edges of $E_{\infty} \cup E_m$. The degree of this form is $2|E_{\infty} \cup E_m|$. So the face $F$ cannot contribute unless $E_m = \emptyset$.

Now the expression $p_{\tau} \circ p_e$ for edges of $E_{\infty}$ makes also clear that, if $f_1^0$ is changed to $\phi_{\infty} \circ T \circ \phi_{\infty} \circ f_1^0$, for a vertical translation $T$ such that $0$ is not in the image of $\phi_{\infty} \circ T \circ \phi_{\infty} \circ f_1^0$, then the image under $\prod_{e \in E_{\infty}} p_e$ is unchanged. So we have a one-parameter group acting on our face $F$ such that $\prod_{e \in E_{\infty}} p_e$ factors through this action. Unless $V^+$ has only one kid, this action is not trivial, and the quotient of the face by this action is of dimension strictly less than the face dimension.

Therefore, for the contributing faces, we have $E_m = \emptyset$ and $V^+$ has only one kid. Thus, according to Lemma 14.46, we are left with the $T$-faces of Theorem 14.23 (for which $B = \emptyset$ and $I$ has one element) for which $\mathcal{P}_x = \{V\}$. These faces yield the derivative $\frac{\partial}{\partial t} \tilde{Z} = d\tilde{Z}(\frac{\partial}{\partial t})$ with

$$d\tilde{Z} = -((t \mapsto h_t^- \circ y^-)^*(\eta_{B^-}))\tilde{Z} + \tilde{Z}((t \mapsto h_t^+ \circ y^+)^*(\eta_{B^+})).$$

This proves the equality

$$\tilde{Z}(t) = \mathrm{hol}_{h^{-}_{|[t,0]} \circ y^{-}}(\eta_{B^{-}}) \tilde{Z}(0) \mathrm{hol}_{h^{+}_{|[0,t]} \circ y^{+}}(\eta_{B^{+}})$$

and leads to the formula for $\mathcal{Z}$.

□
**Corollary 15.18.** Under the hypotheses of Lemma 15.17, let $L_\parallel$ be a parallel of L. Let $h_t(L)_\parallel = (h_t(K_j)_\parallel)_j$ be the parallel of $h_t(L)$ such that

$$lk(h_t(K_j), h_t(K_j)_{\parallel}) - lk(K_j, K_{j\parallel}) = -\epsilon(K_j)2(\theta_j(t) - \theta_j(0))$$

for any component $K_j$ of $J_{bb} \cup J_{tt}$, and $lk(h_t(K), h_t(K)_{\parallel}) = lk(K, K_{\parallel})$ for any other component $K$ of L. Use Definition 12.12 to set

$$\mathcal{Z}^f(t) = \mathcal{Z}^f(\mathcal{C}, h_t(L), h_t(L)_{\parallel}).$$

Then we have

$$\mathcal{Z}^f(t) = \mathrm{hol}_{h^{-}_{|[t,0] \circ y^{-}}(\eta_{B^{-}})} \mathcal{Z}^f(0) \mathrm{hol}_{h^{+}_{|[0,t] \circ y^{+}}(\eta_{B^{+}})}.^8$$

**PROOF:** See Definition 13.4. $\square$

A connection is flat if its holonomy along a null-homotopic loop is trivial.

**Proposition 15.19.** The connection $\eta_B$ is flat on $\tilde{S}_B(\mathbb{C})$. When $\gamma: [0, 1] \to \tilde{C}_B[D_1]$ is smooth with vanishing derivatives at 0 and 1, the image $T(\gamma)$ of the graph $\{(\gamma(t), t)\}$ of $\gamma$ in $D_1 \times [0, 1]$ is a tangle in $D_1 \times [0, 1]$, and we have

$$\mathcal{Z}^f(T(\gamma)) = \mathcal{Z}(T(\gamma)) = \mathrm{hol}_{p_{CS} \circ \gamma}(\eta_B),$$

where $p_{CS} \circ \gamma$ is the composition of $\gamma$ by the natural projection $p_{CS}: \tilde{C}_B[D_1] \to S_B(\mathbb{C})$. For two framed tangles $(C_1, L_1)$ and $(C_2, L_2)$ such that the bottom of $L_2$ coincides with the top of $L_1$, if one of them is a braid $T(\gamma)$ as above, we have

$$\mathcal{Z}^f(C_1C_2, L_1L_2) = \mathcal{Z}^f(C_1, L_1)\mathcal{Z}^f(C_2, L_2),$$

with products obtained by stacking above in natural ways on both sides, reading from left to right.

**PROOF:** Applying Lemma 15.17 when $L$ is a trivial braid, $h_t^- = h_0^-$ is constant, and $h_t^+ \circ y^+ = \gamma(t)$ shows $\mathcal{Z}(T(\gamma)) = \mathrm{hol}_{p_{CS} \circ \gamma}(\eta_B)$. Then the isotopy invariance of $\mathcal{Z}$ shows that $\eta_B$ is flat on $\tilde{S}_B(\mathbb{C})$. Applying Corollary 15.18 when $h_t^- = h_0^-$ is constant and $\gamma(t) = h_t^+ \circ y^+$ proves

$$\mathcal{Z}^f(\mathcal{C}, LT(\gamma)) = \mathcal{Z}^f(\mathcal{C}, L)\mathcal{Z}^f(T(\gamma)).$$

So we have $\mathcal{Z}^f(T(\gamma_1)T(\gamma_2)) = \mathcal{Z}^f(T(\gamma_1))\mathcal{Z}^f(T(\gamma_2))$ for braids.

Applying Corollary 15.18 when $h_t^+ = h_0^+$ is constant and $\gamma(t) = h_{1-t}^- \circ y^-$ proves $\mathcal{Z}^f(C_1C_2, L_1L_2) = \mathcal{Z}^f(C_1, L_1)\mathcal{Z}^f(C_2, L_2)$ when $(C_1, L_1)$ is a braid and $(C_2, L_2)$ is a framed tangle, too. $\square$

<footnote n="8">⁸Again, the holonomies are considered as valued in spaces of diagrams on unoriented domains, where the vertex-orientation of Jacobi diagrams includes local orientations of strands, which can be made consistent with a global orientation induced by $L$, as in Definition 6.16.</footnote>
**Note 15.20.** Recall the space $\tilde{C}_B[\mathbb{C}]$ of injective planar configurations $y: B \hookrightarrow \mathbb{C}$. Let $\gamma: [0, 1] \to \tilde{C}_B[\mathbb{C}]$ be a braid. Proposition 15.19 expresses $\mathcal{Z}(\gamma) = \mathcal{Z}(T(\gamma))$ as the holonomy of the flat connection $\eta_B$ with coefficients in $\mathcal{A}(\sqcup_{b \in B} \mathbb{R}_b)$ along $p_{CS} \circ \gamma$. Kontsevich defined his integral $Z^K$ as the holonomy of another flat connection $\eta_{KZ}$ along $\gamma$ for such a braid [BN95a, Section 4]. The involved Knizhnik–Zamolodchikov connection $\eta_{KZ}$ is defined on the space $\tilde{C}_B[\mathbb{C}]$. The coefficients of $\eta_{KZ}$ belong to the tensor product of the same diagram space $\mathcal{A}(\sqcup_{b \in B} \mathbb{R}_b)$ by $\mathbb{C}$.

Assume $B = \{b_1, b_2\}$. Let $\gamma^c: ]0, 1] \to \tilde{C}_B[\mathbb{C}]$ map $t$ to $\gamma^c(t)$ with $\gamma^c(t)(b_1) = 0$ and $\gamma^c(t)(b_2) = t$. For $\varepsilon \in ]0, 1[$, set $\gamma_{\varepsilon}^c = \gamma^c|_{[\varepsilon, 1]}$. Then the holonomy of $\eta_B$ along $\gamma_{\varepsilon}^c$ is trivial. However, the holonomy of $\eta_{KZ}$ along $\gamma_{\varepsilon}^c$ is not. It does not even converge when $\varepsilon$ tends to zero. In particular, the connection $\eta_{KZ}$ does not factor through $p_{CS}$.

Nevertheless, the holonomy of $\eta_{KZ}$ can be regularized. It lead Thang Lê and Jun Murakami to a definition of the Kontsevich integral for combinatorial $q$-braids $\gamma$ in [LM96]. This definition is related to the limit $\mathcal{Z}^f(T(\gamma))$ of the holonomy of $\eta_B$ as in Note 12.30.

In the above proposition, we proved that $\eta_B$ is flat by proving that its holonomy is $\mathbf{1} = [\emptyset]$ on null-homotopic loops. The flatness of a differentiable connection $\eta$ is often established by proving that its curvature $(d\eta + \eta \wedge \eta)$ vanishes, instead. The following lemma shows how the curvature vanishing implies the homotopy invariance of the holonomy.

**Lemma 15.21.** Set $\Delta^{(r)} = \{(t_1, \dots, t_r) \in [0, 1]^r : t_1 \le t_2 \le \dots \le t_r\}$. Let $\eta$ be a one-form on $\tilde{S}_B(\mathbb{C})$ with coefficients in the space $\mathcal{A}(\sqcup_{b \in B} \mathbb{R}_b)$. Let $\gamma: [0, 1] \times [0, 1] \to \tilde{S}_B(\mathbb{C})$ be a homotopy mapping $([0, 1] \times \{0, 1\}) \cup \{0\} \times [0, 1]$ to a point. Set $\gamma_u(t) = \gamma(u, t)$. Then the holonomy of $\eta$ along $\gamma_1$ is

$$ \mathbf{1} + \sum_{r=1}^{\infty} \int_{[0,1] \times \Delta^{(r)}} \left( \sum_{j=1}^{r} (-1)^{j-1} \bigwedge_{i=1}^{r} (\gamma \circ p_i)^*(\eta) \left( \frac{(\gamma \circ p_j)^*(d\eta + \eta \wedge \eta)}{(\gamma \circ p_j)^*(\eta)} \right) \right), $$

where the fraction means that the denominator is replaced by the numerator in the preceding expression.

In particular, if $d\eta + \eta \wedge \eta$ vanishes, then the holonomy of $\eta$ is trivial along any null-homotopic loop.

**PROOF:** Stokes' theorem allows us to compute

$$ \mathrm{hol}_{\gamma_1}(\eta) = \mathbf{1} + \sum_{r=1}^{\infty} \int_{\Delta^{(r)}} \bigwedge_{i=1}^{r} (\gamma_1 \circ p_i)^*(\eta) $$
by integrating $d \bigwedge_{i=1}^{r} (\gamma_u \circ p_i)^*(\eta)$ over $[0, 1] \times \Delta^{(r)}$ as follows. Set $F_0(\Delta^{(r)}) = \{(0, t_2, \dots, t_r) \in \Delta^r\}$, $F_r(\Delta^r) = \{(t_1, t_2, \dots, t_{r-1}, 1) \in \Delta^r\}$, and

$$F_j(\Delta^r) = \{(t_1, \dots, t_r) \in \Delta^r : t_j = t_{j+1}\}.$$

for $j \in \underline{r-1}$. Then we have

$$\partial \Delta^{(r)} = \sum_{j=0}^{r} (-1)^{j+1} F_j(\Delta^{(r)}),$$

and

$$\partial ([0, 1] \times \Delta^{(r)}) = (\partial [0, 1]) \times \Delta^{(r)} \cup ([0, 1] \times \partial (-\Delta^{(r)})).$$

So we get

$$\int_{\Delta^{(r)}} \bigwedge_{i=1}^{r} (\gamma_1 \circ p_i)^*(\eta) - \int_{\Delta^{(r)}} \bigwedge_{i=1}^{r} (\gamma_0 \circ p_i)^*(\eta) \\ + \sum_{j=0}^{r} (-1)^j \int_{[0,1] \times F_j(\Delta^{(r)})} \bigwedge_{i=1}^{r} (\gamma \circ p_i)^*(\eta) = \int_{[0,1] \times \Delta^{(r)}} d \left( \bigwedge_{i=1}^{r} (\gamma \circ p_i)^*(\eta) \right),$$

where the faces $F_0$ and $F_r$ do not contribute since $\gamma$ maps $([0, 1] \times \{0, 1\})$ to a point, and $\int_{\Delta^{(r)}} \bigwedge_{i=1}^{r} (\gamma_0 \circ p_i)^*(\eta)$ similarly vanishes. We obtain

$$\int_{\Delta^{(r)}} \bigwedge_{i=1}^{r} (\gamma_1 \circ p_i)^*(\eta) \\ = \sum_{j=1}^{r-1} (-1)^{j-1} \int_{[0,1] \times \Delta^{(r-1)}} \bigwedge_{i=1}^{r-1} (\gamma \circ p_i)^*(\eta) \left( \frac{(\gamma \circ p_j)^*(\eta \wedge \eta)}{(\gamma \circ p_j)^*(\eta)} \right) \\ + \sum_{j=1}^{r} (-1)^{j-1} \int_{[0,1] \times \Delta^{(r)}} \left( \bigwedge_{i=1}^{r} (\gamma \circ p_i)^*(\eta) \right) \left( \frac{(\gamma \circ p_j)^*(d\eta)}{(\gamma \circ p_j)^*(\eta)} \right),$$

which yields the result. $\square$

**Corollary 15.22.** We have $d\eta_B + \eta_B \wedge \eta_B = 0$.

**PROOF:** Let us prove that the degree $k$ part $(d\eta_B + \eta_B \wedge \eta_B)_k$ of $(d\eta_B + \eta_B \wedge \eta_B)$ vanishes for any $k \in \mathbb{N}$, by induction on the degree $k$. This is obviously true for $k = 0$. Let us assume that $k > 0$ and $(d\eta_B + \eta_B \wedge \eta_B)_i$ vanishes for $i < k$. For any disk $D = \gamma([0, 1] \times \Delta^{(1)})$ as in Lemma 15.21, the degree
k part of the holonomy of $\eta_B$ along $\partial D$, which vanishes, is the integral of $(d\eta_B + \eta_B \wedge \eta_B)_k$ along $D$ according to Lemma 15.21. Therefore, the integral of $(d\eta_B + \eta_B \wedge \eta_B)_k$ vanishes along any disk $D$, and $(d\eta_B + \eta_B \wedge \eta_B)_k$ is zero. $\square$

Below, we compute $d\eta_B$ and sketch an alternative proof for the equality of Corollary 15.22.

**Lemma 15.23.** Along the open codimension-one faces of $\mathcal{V}(y, \Gamma)$, the integral of $(-\bigwedge_{e \in E(\Gamma)} p_{e,S^2}^*(\omega_{S^2}))$ is absolutely convergent. This allows us to define the smooth two-form $(y \mapsto d\eta_\Gamma(y))$ on $\check{\mathcal{S}}_B(\mathbb{C})$ so that $d\eta_\Gamma(y)$ is the sum of these integrals along the open codimension-one faces of $\mathcal{V}(y, \Gamma)$.

**PROOF:** The integral of $\bigwedge_{e \in E(\Gamma)} p_{e,S^2}^*(\omega_{S^2})$ along these codimension-one faces of $\mathcal{V}(y, \Gamma)$ is absolutely convergent and it defines a smooth two-form on $\check{\mathcal{S}}_B(\mathbb{C})$ as in the proof of Proposition 15.15.

To see that this two-form is $d\eta_\Gamma$, use a chart $\psi: \mathbb{R}^s \to N(y)$ of a neighborhood of $y$ in $\check{\mathcal{S}}_B(\mathbb{C})$, with $s = 2|B| - 3$. Let $\zeta_i$ denote the composition $p_i \circ \psi^{-1}$. So we have $\eta_\Gamma = \sum_{i=1}^s \eta_i d\zeta_i$ and

$$d\eta_\Gamma = \sum_{(i,j) \in \mathcal{S}^2 : i<j} \left( \frac{\partial}{\partial \zeta_i} \eta_j - \frac{\partial}{\partial \zeta_j} \eta_i \right) d\zeta_i \wedge d\zeta_j,$$

where

$$\frac{\partial}{\partial \zeta_2} \eta_1 = \lim_{\substack{t_2 \to 0 \\ t_2 \in [0, \infty[}} \frac{1}{t_2} \left( \lim_{\substack{t_1 \to 0 \\ t_1 \in [0, \infty[}} \frac{1}{t_1} \int_{\psi([0, t_1] \times \{(t_2, 0, \dots, 0)\} - \psi([0, t_1] \times \{(0, 0, \dots, 0)\})} \eta_\Gamma \right).$$

We thus have

$$\frac{\partial}{\partial \zeta_1} \eta_2 - \frac{\partial}{\partial \zeta_2} \eta_1 = \lim_{\substack{(t_1, t_2) \to (0, 0) \\ (t_1, t_2) \in [0, \infty]^2}} \frac{1}{t_1 t_2} \int_{\partial N(t_1, t_2)} \eta_\Gamma,$$

where $N(t_1, t_2) = \psi([0, t_1] \times [0, t_2] \times \{0\})$. Furthermore, we have

$$\int_{\partial N(t_1, t_2)} \eta_\Gamma = \int_{\partial p_{S_B}^{-1}(N(t_1, t_2)) \setminus (\cup_{y \in N(t_1, t_2)} \partial \mathcal{V}(y, \Gamma))} \bigwedge_{e \in E(\Gamma)} p_{e, S^2}^*(\omega_{S^2}).$$

Lemma 15.13 and Theorem 14.16 imply that Stokes' theorem applies to the closed form $\bigwedge_{e \in E(\Gamma)} p_{e,S^2}^*(\omega_{S^2})$. So we have $\int_{\partial p_{S_B}^{-1}(N(t_1, t_2))} \bigwedge_{e \in E(\Gamma)} p_{e,S^2}^*(\omega_{S^2}) = 0$ and

$$\int_{N(t_1, t_2)} d\eta_\Gamma = - \int_{y \in N(t_1, t_2)} \int_{\partial \mathcal{V}(y, \Gamma)} \bigwedge_{e \in E(\Gamma)} p_{e, S^2}^*(\omega_{S^2}). \quad \square$$
**Remark 15.24.** The flatness condition $d\eta_B + \eta_B \wedge \eta_B = 0$ can be proved from the definition of $d\eta_B$, which can be extracted from Definition 15.16 and Lemma 15.23, as an exercise along the following lines.

Let us use Lemma 15.23 to compute $d\eta_B$. Let $y \in \check{S}_B(\mathbb{C})$. According to Lemma 15.14, for a connected Jacobi diagram $\Gamma \in \mathcal{D}_k^e(\bigsqcup_{b \in B} \mathbb{R}_b)$, the faces of the $\partial\mathcal{V}(y, \Gamma)$ with maximal dimension are of two types. Either $V(\Gamma) \notin \mathcal{P}_x'$, or $V(\Gamma) \in \mathcal{P}_x'$ and $\mathcal{P} = \{V(\Gamma), C\}$.

If $V(\Gamma) \notin \mathcal{P}_x'$, then we have $\mathcal{P} = \{V(\Gamma)\} \sqcup \mathcal{P}_x'$, and the configuration of the kids of $V(\Gamma)$ maps the univalent kids of $V(\Gamma)$ to the vertical line through the origin. It is defined up to vertical translation and dilation. Since $\Gamma$ is connected, if a univalent daughter $Y$ of $V(\Gamma)$ contains a trivalent vertex, then it contains a trivalent vertex that is bivalent, univalent, or 0-valent in $\Gamma_Y$. This type of face does not contribute to $d\eta_B$ (as in Sections 9.2 and 9.3). (Note that the univalent daughters of $V(\Gamma)$ are in $\mathcal{P}_x'$. So they must have vertices on at least two strands, and $\Gamma_Y$ cannot be an edge between a univalent vertex and a trivalent one.) For the faces in which all the daughters $Y$ of $V(\Gamma)$ contain only univalent vertices, the integrated form is determined by the configuration space of the kids of $V(\Gamma)$ up to dilation and (conjugates of) vertical translations since $\Gamma_Y$ does not contain chords. The dimension of this space is smaller than $3|T(\Gamma)| + |U(\Gamma)| - 2$ if $\mathcal{P}_x'$ is not empty. If $\mathcal{P}_x'$ is empty, then the face is independent of the planar configuration $y$. So it does not contribute to $d\eta_B$ either.

Thus, the only faces of the $\partial\mathcal{V}(y, \Gamma)$ contributing to $d\eta_B$ are such that $\mathcal{P} = \{V(\Gamma), C\}$ and the univalent vertices of $C$ are on one strand. Most of these faces cancel as an analysis similar to that performed in Section 9.3 shows. The only contributing faces are the STU-faces that involve a collapse of an edge containing one univalent vertex such that the other two diagrams involved in the corresponding STU relation are not connected. Then these other two diagrams are disjoint unions of two components $\Gamma_1$ and $\Gamma_2$ on $\bigsqcup_{b \in B} \mathbb{R}_b$. (If the involved diagrams were connected, then the corresponding faces would cancel by STU.) Consider a configuration $c_1$ of $\Gamma_1$ on $\bigsqcup_{b \in B} y(b) \times \mathbb{R}_b$ and a configuration $c_2$ of $\Gamma_2$ on $\bigsqcup_{b \in B} y(b) \times \mathbb{R}_b$ for a planar configuration $y$. View $\Gamma_1$ far below $\Gamma_2$, and slide it vertically until it is far above. During this sliding, there will be heights at which one univalent vertex of $\Gamma_1$ coincides with one univalent vertex of $\Gamma_2$. (For a generic pair $(c_1, c_2)$, there are no heights at which more than one univalent vertex of $\Gamma_1$ coincides with one univalent vertex of $\Gamma_2$.) Each collision corresponds to a configuration contributing to an incomplete STU relation in $d\eta_B$. Furthermore the sum of the corresponding graph classes is $([\Gamma_1][\Gamma_2] - [\Gamma_2][\Gamma_1])$. This roughly shows how $d\eta_B = -\eta_B \wedge \eta_B$.
# Chapter 16

## Discretizable variants of $\mathcal{Z}^f$ and extensions to $q$-tangles

We introduce and study variants of $\mathcal{Z}^f$ involving nonhomogeneous propagating forms in Sections 16.1, 16.2, and 16.3. These variants allow discrete computations from algebraic intersections as in Chapter 11. We will use them in the proofs of important properties of $\mathcal{Z}^f$ in Chapter 17, where we will finish the proof of Theorem 13.12. Section 16.5 is devoted to the extension of $\mathcal{Z}^f$ and its variants to $q$-tangles. This extension relies on the theory of semi-algebraic sets. We review known facts about semi-algebraic structures and extract useful lemmas for our purposes in Section 16.4.

Throughout this chapter, $N$ denotes some (large) fixed integer $N$, $N \ge 2$, and, for $i \in \underline{3N}$, $\tilde{\omega}(i, S^2) = (\tilde{\omega}(i, t, S^2))_{t \in [0,1]}$ is a closed 2-form on $[0, 1] \times S^2$ such that $\tilde{\omega}(i, 0, S^2)$ is a volume-one form of $S^2$. As in Definition 7.6, for a finite set $A$, an $A$-numbered Jacobi diagram is a Jacobi diagram $\Gamma$ with oriented edges equipped with an injection $j_E: E(\Gamma) \hookrightarrow A$, which numbers the edges. Let $\mathcal{D}_{n,A}^e(\mathcal{L})$ denote the set of $A$-numbered degree $n$ Jacobi diagrams with support $\mathcal{L}$ without looped edges.

### 16.1 Discretizable holonomies

Let $B$ be a finite set. Let $\Gamma$ be a $\underline{3N}$-numbered Jacobi diagram on $\sqcup_{b \in B} \mathbb{R}_b$. For an edge $e$ of $\Gamma$ from a vertex $v(e, 1)$ to a vertex $v(e, 2)$, let

$$p_{e,S^2}: [0,1] \times \mathcal{S}_{V(\Gamma)}(\mathbb{R}^3) \to [0,1] \times S^2$$

send $(t, c)$ to $(t, p_{S^2}((c(v(e, 1)), c(v(e, 2)))))$. The maps $p_{e,S^2}$ provide the form

$$\bigwedge_{e \in E(\Gamma)} p_{e,S^2}^*(\tilde{\omega}(j_E(e), S^2))$$
over $[0, 1] \times S_{V(\Gamma)}(\mathbb{R}^3)$. This form pulls back to provide smooth forms on the smooth strata of $[0, 1] \times \mathcal{V}(\Gamma)$. Define the smooth form $\eta_{\Gamma} = \eta_{\Gamma}((\tilde{\omega}(i, S^2))_{i \in \mathcal{N}})$ on $[0, 1] \times \tilde{S}_B(\mathbb{C})$ so that $\eta_{\Gamma}(t, y)$ is the integral of $\bigwedge_{e \in E(\Gamma)} p_{e,S^2}^*(\tilde{\omega}(j_E(e), S^2))$ along the interiors $\{t\} \times \tilde{V}(y, \Gamma)$ of the fibers $\{t\} \times \mathcal{V}(y, \Gamma)$ of $[0, 1] \times \mathcal{V}(\Gamma)$, as in Proposition 15.15.

**Definitions 16.1.** When $A$ is a subset of $\underline{3N}$ with cardinality $3n$, with $n > 0$, set

$$ \eta_{B,A} = \sum_{\Gamma \in \mathcal{D}_{n,A}^e (\bigsqcup_{b \in B} \mathbb{R}_b)} \zeta_{\Gamma} \eta_{\Gamma} [\Gamma] \in \Omega^1 ([0,1] \times \check{S}_B(\mathbb{C}); \mathcal{A}_n (\bigsqcup_{b \in B} \mathbb{R}_b)) $$

where $\zeta_{\Gamma} = \frac{(|A|-|E(\Gamma)|)!}{|A|!2^{|E(\Gamma)|}}$. The form

$$ \eta_{B,A} = \eta_{B,A} \left( (\tilde{\omega}(i, S^2))_{i \in A} \right) $$

pulls back to a one-form on $[0, 1] \times \check{C}_B[D_1]$ still denoted by $\eta_{B,A}$ with coefficients in $\mathcal{A}_n(\bigsqcup_{b \in B} \mathbb{R}_b)$, which is again viewed as a space of diagrams on an un-oriented domain as in Definition 6.16, Proposition 10.23, and Remark 10.24. Set $\eta_{B,\emptyset} = 0$.

Let $\eta_{B,N}$ denote $\eta_{B,\underline{3N}}$. If $\tilde{\omega}(i, 0, S^2) = \omega_{S^2}$, then the restriction of $\eta_{B,N}$ to $\{0\} \times \check{S}_B(\mathbb{C})$ is the form $\eta_{N,B}$ of Definition 15.16.

For an integer $r$, and for a set $A$ of cardinality $3n$, let $P_r(A)$ denote the set of $r$-tuples $(A_1, A_2, \dots, A_r)$, where $A_i$ is a subset of $A$ with a cardinality multiple of 3, the $A_i$ are pairwise disjoint, and their union is $A$.

Let $n$ be a positive integer. Let $A$ be a subset of cardinality $\underline{3N}$. For a path $\gamma: [0, 1] \to [0, 1] \times \check{S}_B(\mathbb{C})$, define the $A$-holonomy $\widetilde{\text{hol}}_{\gamma}(\eta_{B,A})$ of $\eta_{B,A}$ along $\gamma$ to be

$$ \widetilde{\text{hol}}_{\gamma}(\eta_{B,A}) = \sum_{r=0}^{\infty} \sum_{(A_1, \dots, A_r) \in P_r(A)} \frac{\prod_{i=1}^{r} |A_i|!}{|A|!} \int_{(t_1, \dots, t_r) \in [0,1]^r} \prod_{i=1}^{r} (\gamma \circ p_i)^*(\eta_{B,A_i}), $$

with $p_i(t_1, \dots, t_r) = t_i$ and $\widetilde{\text{hol}}_{\gamma}(\eta_{B,\emptyset}) = [\emptyset]$. Then we have

$$ \widetilde{\text{hol}}_{\gamma_1 \gamma_2}(\eta_{B,A}) = \sum_{(A_1, A_2) \in P_2(A)} \frac{|A_1|! |A_2|!}{|A|!} \widetilde{\text{hol}}_{\gamma_1}(\eta_{B,A_1}) \widetilde{\text{hol}}_{\gamma_2}(\eta_{B,A_2}), $$

$$ \frac{\partial}{\partial t} \widetilde{\text{hol}}_{\gamma|_{[0,t]}}(\eta_{B,A}) = \sum_{(A_1, A_2) \in P_2(A)} \frac{|A_1|! |A_2|!}{|A|!} \widetilde{\text{hol}}_{\gamma|_{[0,t]}}(\eta_{B,A_1}) \eta_{B,A_2}(\gamma(t), \gamma'(t)), $$
and

$$ \frac{\partial}{\partial t} \widetilde{\operatorname{hol}}_{\gamma|_{[t,1]}}(\eta_{B,A}) = - \sum_{(A_1, A_2) \in P_2(A)} \frac{|A_1|! |A_2|!}{|A|!} \eta_{B,A_1}(\gamma(t), \gamma'(t)) \widetilde{\operatorname{hol}}_{\gamma|_{[t,1]}}(\eta_{B,A_2}). $$

Note the following lemma about the behavior of the coefficients.

**Lemma 16.2.** Let $\Gamma = \Gamma_1 \sqcup \Gamma_2$ be a Jacobi diagram equipped with an injection

$$ j_E: E(\Gamma) \hookrightarrow A, $$

where $|A| = 3 \deg(\Gamma)$. We have

$$ \zeta_{\Gamma_1 \cup \Gamma_2} = \sum_{\substack{(A_1, A_2) \in P_2(A), \\ j_E(E(\Gamma_1)) \subseteq A_1, j_E(E(\Gamma_2)) \subseteq A_2, \\ |A_1| = 3 \deg(\Gamma_1), |A_2| = 3 \deg(\Gamma_2)}} \frac{|A_1|! |A_2|!}{|A|!} \zeta_{\Gamma_1} \zeta_{\Gamma_2} $$

**PROOF:** We have

$$ \zeta_{\Gamma_1 \cup \Gamma_2} = \frac{(|A| - |E(\Gamma_1)| - |E(\Gamma_2)|)!}{|A|! 2^{|E(\Gamma_1 \cup \Gamma_2)|}}, $$

and the number of pairs $(A_1, A_2)$ in the sum is

$$ \frac{(|A| - |E(\Gamma_1)| - |E(\Gamma_2)|)!}{(|A_1| - |E(\Gamma_1)|)!(|A_2| - |E(\Gamma_2)|)!} $$

□

**Remark 16.3.** We fixed the cardinality of the sets $A_i$ of edge indices to be $f_0(n_i) = 3n_i$ for degree $n_i$ graphs. We could have replaced $f_0$ by any $f: \mathbb{N} \to \mathbb{N}$ such that $f(n) \ge 3n - 2$ (so that $f(n) \ge |j_E(E(\Gamma))|$ for any degree $n$ Jacobi diagram with at least two univalent vertices) and such that $f(n) + f(m) = f(n + m)$, and we would have obtained the same equalities as above. However, we had to fix such a map $f$ to get these equalities.

If $\tilde{\omega}(i, 0, S^2) = \omega_{S^2}$, then for a subset $A$ of $3N$ with cardinality $3n$ and for a path $\gamma: [0, 1] \to \{0\} \times \tilde{S}_B(\mathbb{C})$, $\widetilde{\operatorname{hol}}_{\gamma}(\eta_{B,A})$ is the degree $n$ part of $\operatorname{hol}_{\gamma}(\eta_B)$.

**Notation 16.4.** For a finite set $A$, let $\mathcal{D}_{k,A}(\mathbb{R})$ denote the set of connected $A$-numbered degree $k$ Jacobi diagrams with support $\mathbb{R}$ without looped edges.
For a vertex-oriented graph $\check{\Gamma} \in \mathcal{D}_{k,A}^c(\mathbb{R})$, define the two-form $\omega(\check{\Gamma})$ on $[0, 1] \times S^2$ to be

$$ \omega(\check{\Gamma})(t, v) = \int_{\{t\} \times Q(v; \check{\Gamma})} \bigwedge_{e \in E(\check{\Gamma})} p_{e,S^2}^*(\tilde{\omega}(j_E(e), S^2)), $$

where $p_{e,S^2}$ again abusively denotes $\mathbf{1}_{[0,1]} \times p_{e,S^2}$ and $Q(v; \check{\Gamma})$ is defined in Section 10.3. For a subset $A$ of $\underline{3N}$ of cardinality $3k$, define the two-form $\omega(A)$ on $[0, 1] \times S^2$ with coefficients in $\check{\mathcal{A}}_k(\mathbb{R})$ to be

$$ \omega(A) = \omega\left(A, \left(\tilde{\omega}(i, S^2)\right)_{i \in \underline{3N}}\right) = \sum_{\check{\Gamma} \in \mathcal{D}_{k,A}^c(\mathbb{R})} \zeta_{\check{\Gamma}} \omega(\check{\Gamma}) [\check{\Gamma}]. $$

Here, we view $\check{\mathcal{A}}(\mathbb{R})$ as a space of Jacobi diagrams on the oriented $\mathbb{R}$. So the sets $U(\Gamma)$ of univalent vertices in involved Jacobi diagrams $\Gamma$ are ordered by $i_\Gamma$, and the univalent vertices are oriented by the orientation of $\mathbb{R}$. Let $s_*: \check{\mathcal{A}}(\mathbb{R}) \to \check{\mathcal{A}}(\mathbb{R})$ be the map that sends (the class of) a Jacobi diagram $\Gamma$ on $\mathbb{R}$ to (the class of) the Jacobi diagram $s(\Gamma)$ obtained from $\Gamma$ by multiplying it by $(-1)^{|U(\Gamma)|}$ and by reversing the order of $U(\Gamma).^1$

**Lemma 16.5.** Let $\iota: [0, 1] \times S^2 \to [0, 1] \times S^2$ map $(t, v)$ to $(t, \iota_{S^2}(v))$, where $\iota_{S^2}$ is the antipodal map of $S^2$. With the above notation, the form

$$ \omega(A) = \omega\left(A, \left(\tilde{\omega}(i, S^2)\right)_{i \in \underline{3N}}\right) $$

is a closed form of $[0, 1] \times S^2$ with coefficients in $\check{\mathcal{A}}_k(\mathbb{R})$, and we have $\iota^*(\omega(A)) = -s_*(\omega(A))$ and $\iota^*(\omega(A)) = (-1)^k(\omega(A))$.

**PROOF:** In order to prove that $\omega(A)$ is closed, it suffices to prove that its integral vanishes on the boundary of any 3-ball $B$ of $[0, 1] \times S^2$. View

$$ Q_k(t, v) = \sum_{\check{\Gamma} \in \mathcal{D}_{k,A}^c(\mathbb{R})} \zeta_{\check{\Gamma}} [\check{\Gamma}] \left( \{t\} \times Q(v; \check{\Gamma}) \times (S^2)^{A \setminus j_E(E(\check{\Gamma}))} \right) $$

as a $(6k - 2)$-chain with coefficients in $\check{\mathcal{A}}_k(\mathbb{R})$. So the integral $\int_{\partial B} \omega(A)$ is the integral of the pull-back of the closed form $\bigwedge_{a \in A} \tilde{\omega}(a, S^2)$ under a natural map $P$ from $\cup_{(t,v) \in \partial B} Q_k(t, v)$ to $[0, 1] \times (S^2)^A$.

The analysis of the boundary of $Q(v; \check{\Gamma})$ in the proof of Proposition 10.13 shows that the codimension-one faces of the boundary of $P(Q_k(t, v))$ can be glued. So the boundary of $P(Q_k(t, v))$ vanishes algebraically.

<footnote n="1">¹To my knowledge, the map $s_*$ might be the identity map. If it is, real-valued Vassiliev invariants cannot distinguish an oriented knot from that obtained by reversing its orientation, as Greg Kuperberg proved in [Kup96]. In other words, there would be no odd Vassiliev invariants. Recall the first two questions in Section 1.6.</footnote>
Therefore, since the cycle $\cup_{(t,v) \in \partial B} P(Q_k(t, v))$ bounds $\cup_{(t,v) \in B} P(Q_k(t, v))$ in $[0, 1] \times (S^2)^A$, the integral $\int_{\partial B} \omega(A)$ vanishes, and $\omega(A)$ is closed.

For $\check{\Gamma} \in \mathcal{D}_{k,A}^c(\mathbb{R})$, recall that the class of $s(\check{\Gamma})$ is obtained from the class of $\check{\Gamma}$ by multiplying it by $(-1)^{|U(\check{\Gamma})|}$ and by reversing the order of $U(\check{\Gamma})$. A configuration of $Q(v; \check{\Gamma})$ is naturally a configuration of $Q(-v; s(\check{\Gamma}))$. Below, we consider $\check{\Gamma}$ and $s(\check{\Gamma})$ as diagrams on unoriented domains whose univalent vertices are equipped with matching local orientations of the domain at univalent vertices. The natural identification from $Q(v; \check{\Gamma})$ to $Q(-v; s(\check{\Gamma}))$ reverses the orientation since the local orientations at univalent vertices coincide, the quotients by dilation coincide, but the translations act in opposite directions. Therefore, for any 2-chain $\Delta$ of $[0, 1] \times S^2$, we have $\int_{\Delta} \omega(\check{\Gamma}) = -\int_{\iota(\Delta)} \omega(s(\check{\Gamma}))$, where $\iota(\Delta)$ is equipped with the orientation of $\Delta$. We get

$$ \sum_{\check{\Gamma} \in \mathcal{D}_{k,A}^c(\mathbb{R})} \zeta_{\check{\Gamma}} \int_{\Delta} \omega(\check{\Gamma}) [\check{\Gamma}] = -\sum_{\check{\Gamma} \in \mathcal{D}_{k,A}^c(\mathbb{R})} \zeta_{s(\check{\Gamma})} \int_{\iota(\Delta)} \omega(s(\check{\Gamma})) [\check{\Gamma}] \\ = -\sum_{\check{\Gamma} \in \mathcal{D}_{k,A}^c(\mathbb{R})} \zeta_{\check{\Gamma}} \int_{\iota(\Delta)} \omega(\check{\Gamma}) [s(\check{\Gamma})] $$

and $\omega(A) = -\iota^*(s_*(\omega(A)))$.

In order to prove $\iota^*(\omega(A)) = (-1)^k(\omega(A))$, use the notation and the arguments of the proof that $\alpha_{2n} = 0$ in the proof of Proposition 10.13 to prove that for any 2-chain $\Delta$ of $[0, 1] \times S^2$ and any $\check{\Gamma} \in \mathcal{D}_{k,A}^c(\mathbb{R})$, we have

$$ \int_{\Delta} \omega(\check{\Gamma}) [\check{\Gamma}] = (-1)^k \int_{\iota(\Delta)} \omega(\check{\Gamma}^{eo}) [\check{\Gamma}^{eo}]. $$

□

**Definition 16.6.** For any oriented tangle component $K$, let $U^+K$ denote the fiber space over $K$ consisting of the tangent vectors to the knot $K$ of $\check{R}(C)$ that orient $K$ up to dilation, as in Section 7.3. When the ambient manifold is equipped with a parallelization $\tau$, define the one-form $\eta(A, p_\tau(U^+K)) = \eta(A, (\tilde{\omega}(i, S^2))_{i \in \mathcal{N}}$, $p_\tau(U^+K))$ on $[0, 1]$ valued in $\check{\mathcal{A}}(\mathbb{R})$ to be

$$ \eta(A, p_{\tau}(U^{+}K))(t) = \eta(A, p_{\tau}(U^{+}K), \tau)(t) = \int_{\{t\} \times p_{\tau}(U^{+}K)} \omega(A, (\tilde{\omega}(i, S^2))_{i \in \mathcal{N}}). $$

More precisely, we have

$$ \eta(A, p_{\tau}(U^{+}K))(u, \frac{\partial}{\partial t}) = \frac{\partial}{\partial t} \left( \int_{[0,t] \times p_{\tau}(U^{+}K)} \omega(A) \right) (u) $$
as in Proposition 15.15. Define the A-holonomy of $\eta(\cdot, p_\tau(U^+K))$ along $[a, b] \subseteq [0, 1]$ to be

$$ \widetilde{\mathrm{hol}}_{[a,b]}(\eta(A, p_{\tau}(U^{+}K))) = \sum_{r=0}^{\infty} \sum_{(A_1, \dots, A_r) \in P_r(A)} \frac{\prod_{i=1}^{r} |A_i|!}{|A|!} \int_{(t_1, \dots, t_r) \in [a,b]^r : t_1 \le t_2 \le \dots \le t_r} \bigwedge_{i=1}^{r} p_i^* \left( \eta(A_i, p_{\tau}(U^{+}K)) \right) $$

with $p_i(t_1, \dots, t_r) = t_i$ and $\widetilde{\mathrm{hol}}_{[a,b]}(\eta(\emptyset, p_{\tau}(U^+K))) = \mathbf{1} = [\emptyset]$.

**Lemma 16.7.** For any oriented tangle component $K$, set $U^-K = U^+(K) - K$. We have

$$ \begin{aligned} \widetilde{\mathrm{hol}}_{[a,b]}(\eta(A, -p_{\tau}(U^{-}K))) &= -\eta(A, p_{\tau}(U^{-}K))) \\ &= s_{*}(\widetilde{\mathrm{hol}}_{[a,b]}(\eta(A, p_{\tau}(U^{+}K)))) \end{aligned} $$

**PROOF:** Lemma 16.5 implies $\eta(A, -p_{\tau}(U^{-}K)) = s_{*}(\eta(A, p_{\tau}(U^{+}K)))$. $\square$

Also note the following lemma for later use.

**Lemma 16.8.** Let $K(\mathcal{L})$ be a connected oriented one-dimensional submanifold of a 3-manifold $R(\mathcal{C})$ equipped with a parallelization $\tau$. Assume that $K = K(\mathcal{L})$ is a finite union $K = \cup_{j=1}^k K_j$ of intervals $K_j$ with disjoint integers. Let $A$ be a subset of $3N$ of cardinality multiple of 3. Then we have $\eta(A, p_{\tau}(U^+K)) = \sum_{j=1}^k \eta(A, p_{\tau}(U^+K_j))$ and

$$ \begin{aligned} \widetilde{\mathrm{hol}}_{[0,1]}(\eta(A, p_{\tau}(U^{+}K))) &= \sum_{(B_1, \dots, B_k) \in P_k(A)} \frac{\prod_{j=1}^{k} |B_j|!}{|A|!} \prod_{j=1}^{k} \widetilde{\mathrm{hol}}_{[0,1]}(\eta(B_j, p_{\tau}(U^{+}K_j))) \\ \intertext{in } \check{\mathcal{A}}(\mathcal{L}). \end{aligned} $$

**PROOF:** It suffices to prove the lemma when $k$ equals 2. Recall

$$ \widetilde{\mathrm{hol}}_{[0,1]}\left(\sum_{j=1}^{2} \eta(\cdot, p_{\tau}(U^{+}K_j))\right) = \sum_{r=0}^{\infty} \sum_{(A_1, \dots, A_r) \in P_r(A)} \frac{\prod_{i=1}^{r} |A_i|!}{|A|!} I(A_1, \dots, A_r), $$

with

$$ \begin{aligned} I(A_1, \dots, A_r) \\ &= \int_{(t_1, \dots, t_r) \in [0,1]^r : t_1 \le t_2 \le \dots \le t_r} \bigwedge_{i=1}^{r} p_i^* \left( \eta(A_i, p_{\tau}(U^+K_1)) + \eta(A_i, p_{\tau}(U^+K_2)) \right). \end{aligned} $$
Write

$$
\begin{align*}
\bigwedge_{i=1}^{r} p_i^* \left( \eta(A_i, p_\tau(U^+ K_1)) + \eta(A_i, p_\tau(U^+ K_2)) \right) \\
&= \sum_{f: \{1, \dots, r\} \to \{1, 2\}} \bigwedge_{i=1}^{r} p_i^* \left( \eta(A_i, p_\tau(U^+ K_{f(i)})) \right).
\end{align*}
$$

For $f$ as above, also write

$$
I_j((A_i)_{i \in f^{-1}(j)}) = \int_{(t_i)_{i \in f^{-1}(j)} \in [0,1]^{f^{-1}(j)} : t_i \le t_s} \bigwedge_{i \in f^{-1}(j)} p_i^* \left( \eta(A_i, p_\tau(U^+ K_j)) \right).
$$

Decompose the terms $I_1((A_i)_{i \in f^{-1}(1)}) I_2((A_i)_{i \in f^{-1}(2)})$ of the right-hand side of the equality to be proved with respect to the relative orders of the involved $t_i$ in $I_1$ and $I_2$.

The contribution to $\widetilde{\text{hol}}_{[0,1]}(\sum_{j=1}^2 \eta(\cdot, p_\tau(U^+K_j)))$ of the terms such that

• $\{A_i : i \in \underline{r}\}$ is fixed as an unordered set,

• $f$ is fixed as a map from this unordered set to $\{1, 2\}$, and

• the partial order induced by the numbering in $\underline{r}$ is fixed over the sets $\{A_i : i \in f^{-1}(j)\}$ for $j = 1, 2$,

is also a sum over the possible total orders on the $\{A_i : i \in \underline{r}\}$ that induce
the given orders on the two subsets.

We can easily identify the involved coefficients of the above similar con-
tributions to prove the result when $s$ equals 2. $\square$

16.2 Variants of $\mathcal{Z}^f$ for tangles

We now present alternative definitions of $\mathcal{Z}^f$ involving nonhomogeneous
propagating forms associated to volume forms $\tilde{\omega}(i, 1, S^2) = \omega_{S^2} + d\eta(i, S^2)$,
where $\omega_{S^2}$ is the homogeneous volume-one form of $S^2$.

Let $\mathcal{C}$ be a rational homology cylinder equipped with a parallelization $\tau$.
Let $L: \mathcal{L} \hookrightarrow \mathcal{R}(\mathcal{C})$ denote a long tangle representative. Recall that we fixed
some (large) integer $N$, $N \ge 2$, and that $(\tilde{\omega}(i, S^2) = (\tilde{\omega}(i, t, S^2))_{t \in [0,1]})_{i \in \underline{3N}}$ is
a family of a closed 2-forms on $[0, 1] \times S^2$ such that $\tilde{\omega}(i, 0, S^2)$ is a volume-one form of $S^2$. For $i \in \underline{3N}$, let $\tilde{\omega}(i) = (\tilde{\omega}(i, t))_{t \in [0,1]}$ be a closed 2-form on $[0, 1] \times C_2(R(\mathcal{C}))$ such that $\tilde{\omega}(i) = (\mathbf{1}_{[0,1]} \times p_\tau)^*(\tilde{\omega}(i, S^2))$ on $[0, 1] \times \partial C_2(R(\mathcal{C}))$. For a diagram $\Gamma \in \mathcal{D}_{k, \underline{3N}}^e(\mathcal{L})$, define

$$I(\mathcal{C}, L, \Gamma, o(\Gamma), (\tilde{\omega}(i, t))_{i \in \underline{3N}}) = \int_{(\tilde{C}(\tilde{R}(\mathcal{C}), L; \Gamma), o(\Gamma))} \bigwedge_{e \in E(\Gamma)} p_e^* (\tilde{\omega}(j_E(e), t)),$$

which converges, according to Theorem 12.2.

**Theorem 16.9.** Let $L: \mathcal{L} \hookrightarrow R(\mathcal{C})$ denote the long tangle associated to a tangle in a rational homology cylinder equipped with a parallelization $\tau$. Let $\{K_j\}_{j \in I}$ be the set of components of $L$. Assume that the bottom and top configurations of $L$ are respectively represented by maps $y^-: B^- \to D_1$ and $y^+: B^+ \to D_1$.

Let $N \in \mathbb{N}$. For $i \in \underline{3N}$, let $\tilde{\omega}(i, S^2) = (\tilde{\omega}(i, t, S^2))_{t \in [0,1]}$ be a closed 2-form on $[0, 1] \times S^2$ such that $\tilde{\omega}(i, 0, S^2)$ is a volume-one form of $S^2$,<fnref n="2" /> and let $\tilde{\omega}(i) = (\tilde{\omega}(i, t))_{t \in [0,1]}$ be a closed 2-form on $[0, 1] \times C_2(R(\mathcal{C}))$ such that $\tilde{\omega}(i) = (\mathbf{1}_{[0,1]} \times p_\tau)^*(\tilde{\omega}(i, S^2))$ on $[0, 1] \times \partial C_2(R(\mathcal{C}))$. For a subset $A$ of $\underline{3N}$ with cardinality $3k$, set

$$Z(\mathcal{C}, L, \tau, A, (\tilde{\omega}(i, t))_{i \in A}) = \sum_{\Gamma \in \mathcal{D}_{k,A}^e(\mathcal{L})} \zeta_{\Gamma} I(\mathcal{C}, L, \Gamma, (\tilde{\omega}(i, t))_{i \in A}) [\Gamma] \in \mathcal{A}_k(\mathcal{L})$$

and

$$Z(\mathcal{C}, L, \tau, A)(t) = Z(\mathcal{C}, L, \tau, A, (\tilde{\omega}(i, t))_{i \in A}).$$

Then we have

$$Z(\mathcal{C}, L, \tau, A)(t) = \sum_{\mathfrak{N}=(A_1, A_2, A_3, (A_{K_j})_{j \in I}) \in P_{3+|I|}(A)} \zeta_d(\mathfrak{N}) Z(\mathfrak{N}, t),$$

with

$$\zeta_d(\mathfrak{N}) = \frac{|A_1|! |A_2|! |A_3|! \prod_{j \in I} |A_{K_j}|!}{|A|!}$$

and

$$Z(\mathfrak{N}, t) = \left( \prod_{j \in I} \widetilde{\mathrm{hol}}_{[0,t]} \left( \eta(A_{K_j}, p_\tau(U^+ K_j)) \right) \#_j \right) \widetilde{\mathrm{hol}}_{[t,0] \times y^-} (\eta_{B^-, A_1}) Z(\mathcal{C}, L, \tau, A_2)(0) \widetilde{\mathrm{hol}}_{[0,t] \times y^+} (\eta_{B^+, A_3}).$$

<footnote n="2">²In this chapter, we apply the theorem only when $\tilde{\omega}(i, 0, S^2)$ is the standard homogenous volume-one form $\omega_{S^2}$ on $S^2$. However, we use this general statement in the next chapter.</footnote>
The terms of this formula belong to spaces of diagrams on unoriented one-manifolds as in Definition 6.16, except for the term $\mathrm{hol}_{[0,t]}(\eta(A_{K_j}, p_\tau(U^+K_j)))$ and its action $\#_j$ for which we first pick an orientation of the $K_j$, which we may forget afterwards.<fnref n="3" /> The formula implies that $Z(C, L, \tau, A)(t)$ depends only on $(\tilde{\omega}(i, t, S^2))_{i \in A}$ for any $t$. (It also depends on $\tau$ and on the specific embedding $L$.) It will be denoted by $Z(C, L, \tau, A, (\tilde{\omega}(i, t, S^2))_{i \in A})$.

**PROOF:** Compute $\frac{\partial}{\partial t}Z(A, t) = dZ(A, \cdot)\left(\frac{\partial}{\partial t}\right)$ as in Lemma 10.20 with the help of Proposition 9.2, using the same analysis of faces as in the proof of Lemma 15.17, to find

$$dZ(A, \cdot) = \sum_{(A_1, A_2) \in P_2(A)} \frac{|A_1|! |A_2|!}{|A_1|! |A_1|! |A_2|!} \sum_{j \in I} (\eta(A_1, p_\tau(U^+K_j)) \#_j) Z(A_2, t) \\ + \sum_{(A_1, A_2) \in P_2(A)} \frac{|A_1|! |A_2|!}{|A_1|!} Z(A_1, t) (\{t\} \times y^*)^* (\eta_{B^+, A_2}) \\ - \sum_{(A_1, A_2) \in P_2(A)} \frac{|A_1|! |A_2|!}{|A_1|!} (\{t\} \times y^-)^* (\eta_{B^-, A_1}) Z(A_2, t),$$

with $\eta(\emptyset, p_\tau(U^+K_j)) = 0$ and $\eta_{B^+, \emptyset} = 0$. (Recall Lemma 16.2 for the behavior of the coefficients.) This shows that both sides of the equality to be proved vary in the same way when $t$ varies. Since they take the same value at $t = 0$, the formula is proved. Apply the formula when $\tilde{\omega}(i, 0, S^2)$ is the standard form $\omega_{S^2}$, and use Lemma 9.1 together with the isotopy invariance of $Z$ of Theorem 12.7 to see that $Z(C, L, \tau, A)(t)$ depends only on the $\tilde{\omega}(i, S^2)$ for $i \in A$. So it depends only on the $\tilde{\omega}(i, t, S^2)$. □

Let us introduce some notation in order to rephrase Theorem 16.9. View $Z(C, L, \tau, \cdot)(t)$ as a map from the set $\mathcal{P}_{(3)}(\underline{3N})$ of subsets of $\underline{3N}$ with cardinality multiple of 3 to $\mathcal{A}_{\le N}(\mathcal{L}) = \bigoplus_{k=0}^N \mathcal{A}_k(\mathcal{L})$, which maps $\emptyset$ to the class of the empty diagram. Similarly, consider $\mathrm{hol}_{[0,t]}(\eta(\cdot, p_\tau(U^+K_j))), \mathrm{hol}_{[t,0] \times y^-}(\eta_{B^-, \cdot})$ and $\widetilde{\mathrm{hol}}_{[0,t] \times y^+}(\eta_{B^+, \cdot})$ as maps from $\mathcal{P}_{(3)}(\underline{3N})$ to spaces of diagrams, which map the empty set to the class of the empty diagram. The values of these maps can be multiplied as in the statement of the theorem using the structures of the space of diagrams.

**Definition 16.10.** For such maps $z_1$ and $z_2$ from $P_{(3)}(\underline{3N})$ to spaces of diagrams, define their product $(z_1z_2)_{\sqcup}$ to be the map with domain $\mathcal{P}_{(3)}(\underline{3N})$ such that

$$(z_1 z_2)_{\sqcup}(A) = \sum_{(A_1, A_2) \in P_2(A)} \frac{|A_1|! |A_2|!}{|A|!} z_1(A_1) z_2(A_2)$$

whenever the products $z_1(A_1)z_2(A_2)$ make sense.

<footnote n="3">³Both sides are independent of the domain orientations in the sense of the last sentence of Proposition 10.23 and Remark 10.24, thanks to Lemma 16.7.</footnote>
This product is associative, and $(z_1 z_2 z_3)_{\sqcup}$ denotes

$$ ((z_1 z_2)_{\sqcup} z_3)_{\sqcup} = (z_1 (z_2 z_3)_{\sqcup})_{\sqcup}. $$

The maps that send all nonempty elements of $P_{(3)}(\underline{3N})$ to 0 and the empty element to the class of the empty diagram are neutral for this product. They are denoted by **1**. With the above definition, we may write the equality of Theorem 16.9 as

$$ Z(\cdot)(t) = \left( \left( \left( \prod_{j \in I} \widetilde{\mathrm{hol}}_{[0,t]} (\eta(\cdot, U_j^+)) \right) \#_j \right) \widetilde{\mathrm{hol}}_{[t,0] \times y^-} (\eta_{B^-, \cdot}) Z(\cdot)(0) \widetilde{\mathrm{hol}}_{[0,t] \times y^+} (\eta_{B^+, \cdot}) \right)_{\sqcup} $$

with $Z(\cdot)(t) = Z(C, L, \tau, \cdot)(t)$ and $U_j^+ = p_\tau(U^+ K_j)$.

## 16.3 Straight tangles

**Definition 16.11.** Recall that $\vec{N}$ denotes the vertical unit vector. A tangle $L: \mathcal{L} \hookrightarrow \mathcal{C}$ is straight with respect to $\tau$ if

* $p_\tau(U^+K) \subset \{-\vec{N}, \vec{N}\}$ for closed components $K$ and components $K$ going from bottom to top or from top to bottom,

* for any interval component $\mathcal{L}_j$, the map $p_\tau$ sends the unit tangent vectors to $L(\mathcal{L}_j)$ to the vertical half great circle $S_{WE}$ from $-\vec{N}$ to $\vec{N}$ that contains the west-east direction (to the right), or to the vertical half great circle $S_{EW}$ from $-\vec{N}$ to $\vec{N}$ that contains the east-west direction (to the left). See Figure 16.1.

<figure><img src="image_131.png" /><figcaption>Figure 16.1: The half-circles $S_{WE}$ and $S_{EW}$.</figcaption></figure>

Orient $S_{WE}$ and $S_{EW}$ from $-\vec{N}$ to $\vec{N}$. Straight tangles with respect to $\tau$ get the following framing induced by $\tau$. For any $k \in K$, $p_\tau(U_k^+ K)$ is an
element $v_k$ of the vertical circle $S_{WE} \cup (-S_{EW})$, which intersects the horizontal plane $\mathbb{C}$ in the real line. Let $\rho_{i,\pi/2}(v_k)$ be the image of $v_k$ under the rotation of angle $\pi/2$ about the axis $i$ ($i \in \mathbb{C}$) points toward the sheet). Then $K_\|$ is the parallel of $K$ obtained by pushing $K$ in the direction of the section ($k \mapsto \tau(\rho_{i,\pi/2}(v_k))$) of the unit normal bundle of $K$. (This is consistent with the conventions of Definition 12.8.)

**Remark 16.12.** A boundaryless straight tangle is a straight link in the sense of Definition 7.37. However, the converse is not true. The present definition is more restrictive.

The following proposition generalizes Lemma 7.35 to interval components.

**Proposition 16.13.** Let $K$ be a component of a straight $q$-tangle in a parallelized homology cylinder $(\mathcal{C}, \tau)$. Then we have

$$I_{\theta}(K, \tau) = I_{\theta}(-K, \tau) = lk(K, K_{\|})$$

with the notation of Definitions 12.9 and 13.4 for $lk(K, K_\|$), and of Definition 12.6 and Lemma 7.15 for $I_\theta$.

In order to prove Proposition 16.13, we describe some propagating forms, which will also be useful in the next chapter.

**Notation 16.14.** For an interval $I$ of $\mathbb{R}$ that contains $[0, 1]$ and for a real number $x \in [1, +\infty[, R_{x,I}(\mathcal{C})$ denotes the part that replaces $D_x \times I$ in $R(\mathcal{C})$, and $R_{x,I}^c(\mathcal{C}) = S^3 \setminus (\mathring{D}_x \times \mathring{I})$ denotes the closure of its complement.

Let $\chi_c$ be a smooth map from $\check{R}(\mathcal{C})$ to $[0, 1]$ that sends $\mathcal{C} = R_{1,[0,1]}(\mathcal{C})$ to 1 and $R_{2,[-1,2]}^c(\mathcal{C})$ to 0. Define

$$\pi_{\mathcal{C}}: \check{R}(\mathcal{C}) \to \mathbb{R}^3 \\ x \mapsto (1 - \chi_{\mathcal{C}}(x))x,$$

with $0x = 0$. The map

$$p: \left( \check{R}(\mathcal{C}) \right)^2 \setminus \left( \check{R}_{2,[-1,2]}(\mathcal{C})^2 \cup \operatorname{diag}(\check{R}_{2,[-1,2]}^c(\mathcal{C})^2) \right) \to S^2 \\ (x, y) \mapsto \frac{\pi_{\mathcal{C}}(y) - \pi_{\mathcal{C}}(x)}{\|\pi_{\mathcal{C}}(y) - \pi_{\mathcal{C}}(x)\|}$$

extends to

$$D(p) = C_2(R(\mathcal{C}) \setminus \mathring{C}_2(R_{2,[-1,2]}(\mathcal{C})).$$

When a parallelization $\tau$ of $\mathcal{C}$ is given, the corresponding extension of $p$ to

$$D(p_{\tau}) = D(p) \cup UR_{2,[-1,2]}(\mathcal{C})$$

is denoted by $p_\tau$.
**Lemma 16.15.** For any volume-one 2-form $\omega(S^2)$ on $S^2$ and any parallelization $\tau$ of $C$ (as in Definition 12.4), there exists a propagating form $\omega$ of $(C_2(\tilde{R}(C)), \tau)$ that restricts to $D(p_\tau)$ as $p_\tau^*(\omega(S^2))$. For any $X \in S^2$, there exists a propagating chain $F$ of $(C_2(\tilde{R}(C)), \tau)$ that restricts to $D(p_\tau)$ as $p_\tau^{-1}(X)$.

**PROOF:** We again need an extension to the interior of $C_2(R_{2,[-1,2]}(C))$ of a closed 2-form defined on the boundary. Since this space is a 6-manifold with ridges with the same homology as $S^2$, the form extends as a closed form. $\square$

**PROOF OF PROPOSITION 16.13:** Lemma 7.35 leaves us with the case of interval components $K$. In this case, we have $I_\theta(K, \tau) = 2I(\hat{\zeta}^K, \omega_{S^2})$ and $I_\theta(-K, \tau) = I_\theta(K, \tau)$.

First assume that $K$ goes from bottom to top. In this case, Lemma 12.5 guarantees that $I(\hat{\zeta}^K, \omega(S^2))$ is independent of the chosen volume-one form $\omega(S^2)$ of $S^2$. Choose a volume-one form $\omega_{0,0}$ of $S^2$ $\varepsilon$-dual to $p_\tau^{-1}(i)$ for the complex horizontal direction $i$ and for a small positive number $\varepsilon$. (Recall Definition 11.6.) Set $\omega_0(S^2) = \frac{1}{2}(\omega_{0,0} - i^*(\omega_{0,0}))$, and choose a propagating form $\omega_0$ that restricts to $D(p_\tau)$ as $p_\tau^*(\omega_0(S^2))$ as in Lemma 16.15. Assume that $\varepsilon$ is sufficiently small so that $p_\tau$ maps $(K \times K_\|) \cap D(p_\tau)$ outside the support of $\omega_0(S^2)$. Define knots $C$ and $C_\|$ such that

* the knots $C$ and $C_\|$ respectively coincide with $K$ and $K_\|$ on $R_{9,[-9,9]}(C)$,

* $(C, C_\|)$ is isotopic to the pair $(\hat{K}, \hat{K}_\|)$ of Definition 12.9 of $lk(K, K_\|)$ so that we have $lk(K, K_\|) = lk(C, C_\|)$, and

* $p_\tau$ maps $(C \times C_\|) \cap D(p_\tau)$ outside the support of $\omega_0(S^2)$ so that we have $\int_{C \times C_\|} \omega_0 = \int_{K \times K_\|} \omega_0$.

Our choice of $\omega_0$ also allows us to let $K_\|$ approach and replace $K$ without changing the rational integral $\int_{K \times K_\|} \omega_0$. So we have $\int_{K \times K_\|} \omega_0 = \int_{K \times K \setminus \Delta} \omega_0$. Then we get

$$ 
\begin{aligned}
\mathrm{lk}(K, K_\|) &= \int_{C \times C_\|} \omega_0 = \int_{K \times K_\|} \omega_0 = \int_{K \times K \setminus \Delta} \omega_0 \\
&= I(\hat{\zeta}^K, \omega_0(S^2)) + I(\hat{\zeta}^K, \omega_0(S^2)) \\
&= I(\hat{\zeta}^K, \omega_0(S^2)) + I(\hat{\zeta}^K, -i^*(\omega_0(S^2))) = 2I(\hat{\zeta}^K, \omega_0(S^2)).
\end{aligned}
$$

This proves Proposition 16.13 when $K$ goes from bottom to top. Since we have $I_\theta(-K, \tau) = I_\theta(K, \tau)$, Definition 12.9 allows us to deduce it when $K$ goes from top to bottom.

Let us now assume that $K$ goes from bottom to bottom (resp. from top to top). Lemma 13.5 reduces the proof to the case $p_\tau(U^+K) \subseteq S_{WE}$.
Let $S(K)$ be the half-circle from $-\vec{N}$ to $\vec{N}$ (resp. from $\vec{N}$ to $-\vec{N}$) through the direction of $[K(0), K(1)]$. Lemma 12.5 implies that $I(\hat{\zeta}^K, \omega(S^2))$ depends only on the integrals of the restriction of $\omega(S^2)$ to the components of $S^2 \setminus (p_\tau(U^+K) \cup S(K))$, with the notation of Lemma 12.5. In particular, when $S(K) = \pm S_{WE}$ (i.e., when the segment $[K(0), K(1)]$ is directed and oriented as the real line, as in Figure 12.4), it does not depend on $\omega(S^2)$, and we conclude as above.

Otherwise, $S^2 \setminus (p_\tau(U^+K) \cup S(K))$ has two connected components $A_1$ and $A_2$ such that $\partial A_1 = p_\tau(U^+K) \cup S(K) = -\partial A_2$. For $j \in \underline{2}$, let $I_j = I(\hat{\zeta}^K, \omega_j(S^2))$, where $\omega_j(S^2)$ is a volume-one form supported on $A_j$. Then $I_2$ is a rational number since it is the intersection of a propagating chain with boundary $p_\tau^{-1}(X)$ for $X \in A_2$ with $C(R(C, K; \hat{\zeta}^K))$ in $C_2(R(C))$.<fnref n="4" /> According to Lemma 12.5, for any volume-one form $\omega(S^2)$ of $S^2$, we have

$$ 
\begin{aligned}
I(\hat{\zeta}^K, \omega(S^2)) &= I_1 + \int_{A_1} (\omega(S^2) - \omega_1(S^2)) = I_1 + \int_{A_1} \omega(S^2) - 1 \\
&= I_2 + \int_{A_1} (\omega(S^2) - \omega_2(S^2)) = I_2 + \int_{A_1} \omega(S^2).
\end{aligned}
$$

The rational numbers $I_1$ and $I_2$ are not changed when $(K, \tau)$ moves continuously so that the angle from the real positive half-line to $\overrightarrow{K(0)K(1)}$ varies in $]0, 2\pi[$ and the trivialization varies accordingly so that $K$ remains straight. Therefore, $I(\hat{\zeta}^K, \omega_{S^2})$ varies like the area of $A_1$, which we compute now. Assume that the direction of $[K(0), K(1)]$ coincides with the direction of $\exp(2i\pi\theta)$ for $\theta \in ]0, 1[$.

If $K$ goes from bottom to bottom, then $A_1$ is the set of vectors $v$ of $S^2$ such that $p_C(v) = \lambda \exp(2i\pi\alpha)$ for some $(\lambda, \alpha) \in ]0, +\infty[ \times ]0, \theta[$, and the area of $A_1$ is $\theta$. So we have $I_\theta(K, \tau) = I_1 + I_2 + 2\theta - 1$.

If $K$ goes from top to top, then $A_1$ is the set of vectors $v$ of $S^2$ such that $p_C(v) = \lambda \exp(2i\pi\alpha)$ for some $(\lambda, \alpha) \in ]0, +\infty[ \times ]\theta, 1[$, and the area of $A_1$ is $1 - \theta$. So we have $I_\theta(K, \tau) = I_1 + I_2 + 1 - 2\theta$.

Let us treat the case $\theta = \frac{1}{2}$ when $K$ goes from bottom to bottom. With the notation of Definition 13.4, set $\theta_1 = -\theta_0 = \frac{1}{2}$. So $\hat{K}_{\parallel, \theta_0, \theta_1}$ looks as in Figure 13.4, and it is isotopic to the parallel $-\widehat{(-K)\parallel}$, which looks as in Figure 12.4. In this case, the interior of $A_1$ contains the direction $i$. We can assume that $\omega_1(S^2) = \omega_{0,0}$ and compute $lk(K, K_\parallel) = lk(C, C_\parallel)$ for a pair $(C, C_\parallel)$ of parallel closed curves that coincides with $(K, -(-K)_\parallel)$ in a big neighborhood of $C$ and that lies in a vertical plane orthogonal to $i$ outside that big neighborhood. We get

$$ 
\begin{aligned}
\mathrm{lk}(K, K_\parallel) &= \mathrm{lk}(C, C_\parallel) = \int_{K \times K \setminus \Delta} \omega_{0,0} = I(\hat{\zeta}^K, \omega_{0,0}) + I(\hat{\zeta}^K, \omega_{0,0}) \\
&= I(\hat{\zeta}^K, \omega_{0,0}) + I(\hat{\zeta}^K, -\iota^*(\omega_{0,0})) = I_1 + I_2 = I_\theta(K, \tau).
\end{aligned}
$$

<footnote n="4">⁴When $C = D_1 \times [0, 1]$ and $\tau = \tau_s$, the map $p_\tau$ extends to $C_2(R(C))$ and $I_2$ is the integral local degree of this extended $p_\tau$ at a point of $A_2$.</footnote>
So Proposition 16.13 holds in the case $\theta = \frac{1}{2}$ when $K$ goes from bottom to bottom. It similarly holds in the case $\theta = \frac{1}{2}$ when $K$ goes from top to top.

We now deduce the case in which $\theta \in ]0, 1[$ from the case $\theta = \frac{1}{2}$. Recall that the rational numbers $I_1$ and $I_2$ are unchanged under isotopies that make $\theta$ vary continuously in $]0, 1[$. Define $\theta_1 = \theta$ and $\theta_0 = \theta - 1$, so that the arcs $\alpha_0$ and $\alpha_1$ of Definition 13.4 vary continuously as $\theta$ varies from 0 to 1. Then the isotopy class of the pair $(\tilde{K}(\theta), \tilde{K}(\theta))_{\parallel, \theta_0, \theta_1}$ is unchanged. So when $K$ goes from bottom to bottom, we have $lk(K(\theta), K(\theta))_{\parallel} = lk(K(\frac{1}{2}), K(\frac{1}{2}))_{\parallel} + 2\theta - 1$ and $I_{\theta}(K(\theta), \tau) = I_{\theta}(K(\frac{1}{2}), \tau) + 2\theta - 1$. When $K$ goes from top to top, $lk(K(\theta), K(\theta))_{\parallel} - I_{\theta}(K, \tau)$ is similarly fixed when $\theta$ varies. So Proposition 16.13 holds in any case. $\square$

View the anomaly $\beta$ of Section 10.2 as the map from $P_3(\underline{3N})$ to $\tilde{\mathcal{A}}(\mathbb{R})$ that sends any subset of $\underline{3N}$ with cardinality $3k$ to $\beta_k$.

With the notation of Definition 16.10, we get the following corollary of Theorem 16.9, Theorem 12.7, and Proposition 16.13.

**Theorem 16.16.** Let $L$ be a straight tangle with respect to a parallelization $\tau$. Let $J_{bb} = J_{bb}(L)$ (resp. $J_{tt} = J_{tt}(L)$) denote the set of components of $L$ going from bottom to bottom (resp. from top to top). For $K \in J_{bb} \cup J_{tt}$, the orientation of $K$ induced by $\tau$ is the orientation of $K$ such that $p_{\tau}(U^{+}K) \subseteq S_{WE}$. Under the assumptions of Theorem 16.9,

$$ \left( Z(C, L, \tau, \cdot, (\tilde{\omega}(i, 1))_{i \in \underline{3N}}) \exp\left(-\frac{1}{4} p_1(\tau) \beta(\cdot)\right) \right)_{\sqcup} $$

depends only on the $\tilde{\omega}(i, 1, S^2)$, on the boundary-fixing diffeomorphism class of $(C, L)$, on the orientations of the components of $J_{bb} \cup J_{tt}$ induced by $\tau$, and on the parallel $L_{\parallel}$ of $L$ induced by $\tau$.

It is denoted by $\mathcal{Z}^f(C, L, L_{\parallel}, \cdot, (\tilde{\omega}(i, 1, S^2))_{i \in \underline{3N}})$. Set

$$ \mathcal{Z}_{\le N}^{f}(C, L, L_{\parallel}, \cdot) = \mathcal{Z}^{f}(C, L, L_{\parallel}, \cdot, (\omega_{S^2})_{i \in \underline{3N}}). $$

Then $\mathcal{Z}_{\le N}^{f}(C, L, L_{\parallel}, \cdot)$ maps any subset of $\underline{3N}$ of cardinality $3k$ to the degree $k$ part $\mathcal{Z}_k^{f}(C, L, L_{\parallel})$ of the invariant $\mathcal{Z}^f(C, L, L_{\parallel})$ of Definition 12.12. We will drop the subscript “$\le N$” from $\mathcal{Z}_{\le N}^{f}(C, L, L_{\parallel}, \cdot)$.

**PROOF:** First note that $\mathcal{Z}^f(C, L, L_{\parallel}, \cdot, (\omega_{S^2})_{i \in \underline{3N}})$ maps any subset of $\underline{3N}$ of cardinality $3k$ to $\mathcal{Z}_k^f(C, L, L_{\parallel})$ as stated.

Let $K$ be a component of $L$. If $K$ is a knot, or a component going from bottom to top or from top to bottom, then the form $\eta(A_K, p_{\tau}(U^{+}K))$ of Definition 16.6 vanishes. Since the components of $J_{bb} \cup J_{tt}$ are equipped with the
orientation induced by $\tau$, $\eta(A_K, p_\tau(U^+K))$ is the same for all components $K$ of $J_{tt}$, and it is independent of $L$. It is denoted by $\eta(A_K, S_{WE})$. Similarly, we have $\eta(A_K, p_\tau(U^+K)) = -\eta(A_K, S_{WE})$ for all components $K$ of $J_{bb}$. Therefore, the factor $(\prod_{j \in I} \widetilde{\mathrm{hol}}_{[0,1]}(\eta(A_{K_j}, p_\tau(U^+K_j)))\#_j)$ in Theorem 16.9 is equal to

$$ \left( \prod_{K_j \in J_{bb}} \widetilde{\mathrm{hol}}_{[0,1]}(-\eta(A_{K_j}, S_{WE})) \#_j \right) \left( \prod_{K_j \in J_{tt}} \widetilde{\mathrm{hol}}_{[0,1]}(\eta(A_{K_j}, S_{WE})) \#_j \right), $$

and $\mathcal{Z}^f(C, L, L_\|, \cdot, (\tilde{\omega}(i, 1, S^2))_{i \in \underline{3N}})$ is determined by $\mathcal{Z}^f(C, L, L_\|)$ and by the holonomies, which depend only on the $\tilde{\omega}(i, \cdot, S^2)$. $\square$

**Definition 16.17.** With the notation of Theorem 16.16 and under its assumptions, Theorems 16.9 and 12.7 together with Proposition 16.13 imply that

$$ \left( \prod_{j=1}^{k} \left( \exp(-lk(K_j, K_{j\|})\alpha) \#_j \right) \mathcal{Z}^f(C, L, L_\|, \cdot, (\tilde{\omega}(i, 1, S^2))_{i \in \underline{3N}}) \right) \sqcup $$

is independent of the framing of $L$ with $L = (K_j)_{j \in \underline{k}}$ and $L_\| = (K_{j\|})_{j \in \underline{k}}$. It is denoted by $\mathcal{Z}(C, L, \cdot, (\tilde{\omega}(i, 1, S^2))_{i \in \underline{3N}})$. It a priori depends on the orientations of the components of $J_{bb} \cup J_{tt}$.

The data of an orientation for the components of $J_{bb} \cup J_{tt}$ is called a $J_{bb,tt}$-orientation, and $L$ is said to be $J_{bb,tt}$-oriented when it is equipped with such an orientation.

All the involved products are as in Definition 16.10, and $lk(K_j, K_{j\|})\alpha$ is considered as a function of subsets of $\underline{3N}$ with cardinality multiple of 3, which depends only on the degree. View the invariant of Theorem 12.7 as such a function $\mathcal{Z}(C, L)(\cdot)$. Then, according to Theorem 16.9—applied when $\tilde{\omega}(i, 0, S^2) = \omega_{S^2}$, we get

$$ \mathcal{Z}(C, L, \cdot, (\tilde{\omega}(i, 1, S^2))_{i \in \underline{3N}}) = \left( \left( \prod_{K_j \in J_{bb}} \widetilde{\mathrm{hol}}_{[0,1]}(-\eta(\cdot, S_{WE})) \#_j \right) \left( \prod_{K_j \in J_{tt}} \widetilde{\mathrm{hol}}_{[0,1]}(\eta(\cdot, S_{WE})) \#_j \right) \right) \cdot \widetilde{\mathrm{hol}}_{[1,0] \times y^{-}}(\eta_{B^{-}, \cdot}) \mathcal{Z}(C, L)(\cdot) \widetilde{\mathrm{hol}}_{[0,1] \times y^{+}}(\eta_{B^{+}, \cdot}) \sqcup \tag{16.1} $$

This allows us to extend the definition of $\mathcal{Z}^f(\cdot, \cdot, (\tilde{\omega}(i, 1, S^2))_{i \in \underline{3N}})$ for $J_{bb,tt}$-oriented framed tangles that are not represented by straight tangles so that
the equality

$$ \mathcal{Z}^f(C, L, L_\|, \dots, (\tilde{\omega}(i, 1, S^2))_{i \in \overline{3N}}) = \left( \prod_{j=1}^k (\exp(lk(K_j, K_{j\|})\alpha)\#_j) \mathcal{Z}(C, L, \dots, (\tilde{\omega}(i, 1, S^2))_{i \in \overline{3N}}) \right) \sqcup $$

holds for all these $J_{bb,tt}$-oriented framed tangles, and we have

$$ \mathcal{Z}^f(C, L, L_\|, \dots, (\tilde{\omega}(i, 1, S^2))_{i \in \overline{3N}}) = \left( \left( \prod_{K_j \in J_{bb}} \widetilde{\mathrm{hol}}_{[0,1]}(-\eta(\cdot, S_{WE})) \#_j \right) \left( \prod_{K_j \in J_{tt}} \widetilde{\mathrm{hol}}_{[0,1]}(\eta(\cdot, S_{WE})) \#_j \right) \widetilde{\mathrm{hol}}_{[1,0] \times y^-}(\eta_{B^-, \cdot}) \mathcal{Z}^f(C, L, L_\|)(\cdot) \widetilde{\mathrm{hol}}_{[0,1] \times y^+}(\eta_{B^+, \cdot}) \right) \sqcup \tag{16.2} $$

In order to compute $\mathcal{Z}^f(C, L, L_\|, \dots,)$ from the discretizable definition of $Z(C, L, \tau, \dots)$ in Theorem 16.9, we first represent $L$ as a straight tangle with another induced parallel $L' = (K'_j)_{j \in \overline{k}}$ (but with the same $J_{bb,tt}$-orientation induced by the parallelization), and we correct by setting

$$ \mathcal{Z}^f(C, L, L_\|, \dots, (\tilde{\omega}(i, 1, S^2))_{i \in \overline{3N}}) = \left( \prod_{j=1}^k (\exp(lk(K_j, K_{j\|} - K'_j)\alpha)\#_j) \mathcal{Z}^f(C, L, L', \dots, (\tilde{\omega}(i, 1, S^2))_{i \in \overline{3N}}) \right) \sqcup $$

**Remark 16.18.** Definition 16.17 is not canonical because of the arbitrary choice of $S_{WE}$. The defined invariant may not have the usual natural dependence on the component orientations (as in Proposition 10.23). Indeed, Definition 16.17 involves the $J_{bb,tt}$-orientation. So it is not symmetric under the reversal of a component orientation. See Remark 16.48 for further explanations.

**Lemma 16.19.** Let $(\omega(i))_{i \in \overline{3N}}$ denote a fixed family of propagating forms of $(C_2(R(C)), \tau)$. These propagators may be expressed as $\tilde{\omega}(i, 1)$ for forms $\tilde{\omega}(i)$ as in Theorem 16.9 (thanks to Lemma 9.1). Set

$$ Z(C, L, \tau, A) = Z(C, L, \tau, A)(1) $$

with the notation of Theorem 16.9. Let $h_t$ be an isotopy of $\tilde{R}(C)$ that is the identity on $(C \setminus D_1) \times \mathbb{R}$ for any $t$ and that restricts to $C \times ]-\infty, 0]$ and $C \times [1, +\infty[$ as isotopies $h_t^- \times \mathbf{1}_{]-\infty, 0]}$ and $h_t^+ \times \mathbf{1}_{[1, +\infty[$, for planar isotopies $h_t^-$
and $h_t^+$. Assume $h_0 = 1$. Let $L$ be a long tangle of $\tilde{R}(\mathcal{C})$ whose bottom (resp. top) configuration is represented by a map $y^-: B^- \to D_1$ (resp. $y^+: B^+ \to D_1$). Let $(\tau_t)_{t \in [0,1]}$ be a smooth homotopy of parallelizations of $\mathcal{C}$ such that $p_{\tau_t}|_{U+(L)}$ is constant with respect to $t$. With the notation of Definition 16.10, for $A \in P_3(\underline{3N})$, set $Z(t, A) = Z(\mathcal{C}, h_t(L), \tau_t, A)$. Then we have

$$Z(t, \cdot) = (\widetilde{\mathrm{hol}}_{h_{[t,0]}^{-} \circ y^{-}}(\eta_{B^{-},\cdot}) Z(0, \cdot) \widetilde{\mathrm{hol}}_{h_{[0,t]}^{+} \circ y^{+}}(\eta_{B^{+},\cdot}))_{\sqcup}.$$

**PROOF:** The proof is similar to that of Lemma 15.17. □

<figure><img src="image_132.png" /></figure>

Lemma 16.19 implies that $(\widetilde{\mathrm{hol}}_{h_{[t,0]}^{-} \circ y^{-}}(\eta_{B^{-},\cdot}) \widetilde{\mathrm{hol}}_{h_{[0,t]}^{-} \circ y^{-}}(\eta_{B^{-},\cdot}))_{\sqcup}$ is neutral for the product of Definition 16.10.

The following proposition can be proved as Proposition 15.19.

**Proposition 16.20.** With the notation and assumptions of Theorem 16.9, when

$$\gamma: [0, 1] \to \check{C}_B [D_1]$$

is smooth with vanishing derivatives at 0 and 1, deform the standard parallelization of $\mathbb{R}^3$ to a homotopic parallelization $\tau$ such that $T(\gamma)$ is straight with respect to $\tau$ at any time of the homotopy. For any subset $A$ of $\underline{3N}$ with cardinality $3k$, we have

$$Z(C_0 = D_1 \times [0, 1], T(\gamma), \tau, A, (\tilde{\omega}(i, 1, S^2))_{i \in \underline{3N}}) = \widetilde{\mathrm{hol}}_{\{1\} \times p_{CS} \circ \gamma}(\eta_{B,A}),$$

where $p_{CS}$ is the natural projection $\check{C}_B [D_1] \to S_B(\mathbb{C})$. Thus we also have

$$\begin{aligned} Z(\gamma, A, (\tilde{\omega}(i, 1, S^2))_{i \in \underline{3N}}) &= \mathcal{Z}^f(C_0, T(\gamma), T(\gamma)_{\|}, A, (\tilde{\omega}(i, 1, S^2))_{i \in \underline{3N}}) \\ &= \widetilde{\mathrm{hol}}_{\{1\} \times p_{CS} \circ \gamma}(\eta_{B,A}). \end{aligned}$$

Proposition 15.19 could be fully generalized to this setting, too, but we will prove more general functoriality properties in Section 17.2.

**Lemma 16.21.** Let $u \in [0, 1]$. Let $\gamma: [0, 1] \to \{u\} \times \check{S}_B(\mathbb{C})$ be a smooth path. We have the following properties.

* $\widetilde{\mathrm{hol}}_{\gamma}(\eta_{B,A})$ depends only on $\gamma(0), \gamma(1)$, the $\tilde{\omega}(i, S^2)$ for $i \in A$, and the homotopy class of $\gamma$ relatively to $\partial\gamma$ in $\{u\} \times \check{S}_B(\mathbb{C})$.

* If $(\bar{\gamma}: t \mapsto \gamma(1-t))$ denotes the inverse of $\gamma$ with respect to the path composition, then $(\widetilde{\mathrm{hol}}_{\gamma}(\eta_{B,\cdot}) \widetilde{\mathrm{hol}}_{\bar{\gamma}}(\eta_{B,\cdot}))_{\sqcup}$ is neutral with respect to the product of Definition 16.10.
• Let $w, t$, and $\varepsilon$ be three elements of $[0, 1[$, such that $[w, w + \varepsilon]$ and $[t, t + \varepsilon]$ are subsets of $[0, 1]$. Let $\ell$ be the boundary of the square $[w, w + \varepsilon] \times p_{CS} \circ \gamma$ ($[t, t + \varepsilon]$) of $[0, 1] \times \tilde{S}_B(\mathbb{C})$, then $\mathrm{hol}_{\ell}(\eta_{B,\cdot})$ is trivial.

**PROOF:** The first assertion is a direct consequence of Proposition 16.20. It implies that for any $u \in [0, 1]$, for any homotopically trivial loop $\ell$ of $\{u\} \times \tilde{S}_B(\mathbb{C})$, $\mathrm{hol}_{\ell}(\eta_{B,\cdot})$ is neutral with respect to the product of Definition 16.10. Since $\gamma\bar{\gamma}$ is such a loop, it implies the second assertion. The third assertion is a direct consequence of Lemma 16.19 and Proposition 16.20. $\square$

## 16.4 Semi-algebraic structures on some configuration spaces

We would like to extend the definitions of our connections $\eta$ of Sections 15.2 and 16.1 on $S_B(\mathbb{C})$, in order to extend the definition of $\mathcal{Z}^f$ to $q$-tangles. Unfortunately, I do not know whether the connections $\eta$ extend as differentiable forms on $S_B(\mathbb{C})$. However, we will be able to extend the definitions of their holonomies and prove that these holonomies along paths make sense (as $\sqrt{t^{-1}}$ may be integrated on $[0, 1]$ though $\sqrt{t^{-1}}$ is not defined at 0). In order to do that, we will need to prove that integrals over singular spaces converge absolutely. Our proofs rely on the theory of semi-algebraic sets. We review the results of this theory that we will use below. Our primary reference is [BCR98, Section 1.4 and Chapter 2] by Jacek Bochnak, Michel Coste, and Marie-Françoise Roy.

**Definition 16.22.** [BCR98, Definition 2.1.4] A semi-algebraic subset of $\mathbb{R}^n$ is a subset of the form

$$ \bigcup_{i=1}^{s} \left( \bigcap_{j=1}^{r_i} \{x \in \mathbb{R}^n : f_{i,j}(x) < 0\} \cap \bigcap_{j=r_i+1}^{s_i} \{x \in \mathbb{R}^n : f_{i,j}(x) = 0\} \right) $$

for an integer $s$, $2s$ integers $r_1, \dots, r_s$, $s_1, \dots, s_s$, such that $s_i \ge r_i$ for any $i \in \underline{s}$, and $\sum_{i=1}^s s_i$ real polynomials $f_{i,j}$ in the natural coordinates of $x$. A semi-algebraic set is a semi-algebraic subset of $\mathbb{R}^n$ for some $n \in \mathbb{N}$.

The set of semi-algebraic subsets of $\mathbb{R}^n$ is obviously stable under finite union, finite intersection, and taking complements. The set of semi-algebraic sets is stable under finite products.

**Theorem 16.23.** Semi-algebraic sets also satisfy the following deeper properties, proved in [BCR98].
[BCR98, Theorem 2.2.1] Let $S$ be a semi-algebraic subset of $\mathbb{R}^{n+1}$. Let $\Pi: \mathbb{R}^{n+1} \to \mathbb{R}^n$ be the projection onto the space of the first $n$ coordinates. Then $\Pi(S)$ is a semi-algebraic subset of $\mathbb{R}^n$.

[BCR98, Proposition 2.2.2] The closure and the interior of a semi-algebraic set are semi-algebraic sets.

□

**Definition 16.24.** [BCR98, Definition 2.2.5] A map from a semi-algebraic subset of $\mathbb{R}^n$ to a semi-algebraic subset of $\mathbb{R}^m$ is *semi-algebraic* if its graph is semi-algebraic in $\mathbb{R}^{n+m}$.

The following proposition [BCR98, Proposition 2.2.7] can be deduced from Theorem 16.23 above as an exercise.

**Proposition 16.25.** Let $f$ be a semi-algebraic map from a semi-algebraic set $A$ to a semi-algebraic subset of $\mathbb{R}^n$. For any semi-algebraic subset $S$ of $A$, $f(S)$ is semi-algebraic. For any semi-algebraic subset $S$ of $\mathbb{R}^n$, $f^{-1}(S)$ is semi-algebraic. The composition of two composable semi-algebraic maps is semi-algebraic.

□

As an example, which will be useful very soon, we prove the following proposition.

**Proposition 16.26.** Let $V$ denote a finite set of cardinality at least 2. Let $T$ be a vector space of dimension $\delta$. The manifold $S_V(T)$ of Theorem 8.11 has a canonical structure of a semi-algebraic set. For any subset $A$ of $V$ with $|A| \ge 2$, the restriction map $S_V(T) \to S_A(T)$ is semi-algebraic with respect to the canonical structures.

**PROOF:** The charts of Lemma 8.9 provide canonical semi-algebraic structures on $\bar{S}_V(T)$ and $\tilde{S}_V(T)$. The restriction maps from $\tilde{S}_V(T)$ to $\bar{S}_A(T)$ are semi-algebraic with respect to these structures. The description of $S_V(T)$ as the closure of the image of $\tilde{S}_V(T)$ in $\prod_{A \in \mathcal{P}_{\ge 2}} \bar{S}_A(T)$ of Lemma 8.39 makes clear that $S_A(T)$ has a natural semi-algebraic structure, thanks to Theorem 16.23.

□

We also have the following easy lemma.

**Lemma 16.27.** The space $\tilde{\mathcal{V}}(\Gamma)$ of Chapter 15 and its compactification $\mathcal{V}(\Gamma)$ carry natural structures of semi-algebraic sets. The projection $p_{S_B}$ from $\mathcal{V}(\Gamma)$ to $S_B(\mathbb{C})$ and its projections to the $S_e(\mathbb{R}^3)$ for ordered pairs $e$ of $V(\Gamma)$ are semi-algebraic maps with respect to these structures. For any configuration $y \in S_B(\mathbb{C})$, the spaces $\tilde{\mathcal{V}}(y, \Gamma)$ and $\mathcal{V}(y, \Gamma)$ are semi-algebraic.
□

**Lemma 16.28.** Let $f: ]a, b[^d \to \mathbb{R}$ be a $C^1$ semi-algebraic map. Then its partial derivatives $\frac{\partial f}{\partial x_i}$ are semi-algebraic functions.

**PROOF:** Let $(e_1, \dots, e_d)$ be a basis of $\mathbb{R}^d = \{\sum_{i=1}^d x_i e_i\}$. As in [BCR98, Proposition 2.9.1], note that the set

$$ \left\{ \left( t, x, f(x), \frac{f(x+te_i) - f(x)}{t} \right) : t \in ]0, 1], x \in ]a, b[^d, x+te_i \in ]a, b[^d \right\} $$

is semi-algebraic. So are its closure, the locus ($t=0$) of this closure, and its projection to the graph of the partial derivative of $f$ with respect to $x_i$. □

**Lemma 16.29.** Let $f$ be a semi-algebraic smooth map from an open hypercube $]0, 1[^d$ to $\mathbb{R}^n$. Then the critical set of $f$, which is the subset of $]0, 1[^d$ for which $f$ is not a submersion, is semi-algebraic.

**PROOF:** According to Lemma 16.28, the partial derivatives $\frac{\partial p_j \circ f}{\partial x_i}$ with respect to the factors of $\mathbb{R}^d$ of the $p_j \circ f$ for the projections $p_j$ on the factors of $\mathbb{R}^n$ are semi-algebraic. It is easy to see that the product and the sum of two real-valued semi-algebraic maps are semi-algebraic. Being in the critical set may be written as "For any subset $I$ of $d$ of cardinality $n$, the determinant $\det[\frac{\partial p_j \circ f}{\partial x_i}(x)]_{i \in I, j \in n}$ is equal to zero." □

An essential property of semi-algebraic sets, which we are going to use, is the following decomposition theorem [BCR98, Proposition 2.9.10].

**Theorem 16.30.** Let $S$ be a semi-algebraic subset of $\mathbb{R}^n$. Then, as a set, $S$ is the disjoint union of finitely many smooth semi-algebraic submanifolds, each semi-algebraically diffeomorphic to an open hypercube $]0, 1[^d$.

The dimension of a semi-algebraic set is the maximal dimension of a hypercube in a decomposition as above. It is proved in [BCR98, Section 2.8] that it does not depend on the decomposition. According to [BCR98, Theorem 2.8.8], the dimension of the image of a semi-algebraic set of dimension $d$ under a semi-algebraic map is smaller than or equal to $d$. According to [BCR98, Proposition 2.8.13], if $A$ is a semi-algebraic set of dimension $\dim(A)$, then we have $\dim(\bar{A} \setminus A) < \dim(A)$.

The following lemma is a corollary of Theorem 16.30.

**Lemma 16.31.** Let $f$ be a continuous semi-algebraic map from a compact semi-algebraic set $A$ of dimension $d$ to a semi-algebraic smooth manifold $B$
with boundary. Let $\omega$ be a smooth differential form of degree $d$ on $B$. Assume
that the restriction of $f$ to each piece of a decomposition as in Theorem 16.30
is smooth. Then the integrals $\int_{\Delta} f^{*}(\omega)$ of $f^{*}(\omega)$ over the open pieces $\Delta$ of
dimension $d$ of such a decomposition converge absolutely, and $\int_{A} f^{*}(\omega)$ is
well defined to be the sum of these $\int_{\Delta} f^{*}(\omega)$.

PROOF: It suffices to prove the lemma when $\omega$ is supported on a subset $[-1, 1]^n$ of an open subset of $B$ semi-algebraically diffeomorphic to $B_{k,n} = [-2, 1]^k \times [-2, 2^{n-k}]$. Indeed, using a partition of unity allows us to write $\omega$ as a finite sum of such forms around the compact $f(A)$. This allows us to reduce the proof to the case $B = B_{k,n}$. Now a degree $d$ differential form on $B_{k,n}$ is a sum over the parts $J$ of cardinality $d$ of $\underline{n}$ of pull-backs of degree $d$ forms on $B_J = B_{J,k,n} = [-2, 1]^{\underline{k} \cap J} \times [-2, 2^{(\underline{n}\setminus\underline{k})\cap J}]$ multiplied by smooth functions on $B_{k,n}$, which are bounded on their compact supports. This allows us to reduce the proof to the case in which $\omega$ is such a pull-back of a form $\omega_J$ on $B_J$, under the projection $p_J: B_{k,n} \to B_J$, multiplied by a bounded function $g_J$ on $B_{k,n}$.

Decompose $A$ as in Theorem 16.30. It suffices to prove that the integral
of $\omega$ over each hypercube $H$ of dimension $d$ converges absolutely. Let $f_J$
denote $p_J \circ f$. Consider the closure $\bar{H} \subset A$ of the hypercube $H$ in $A$, set
$\partial H = \bar{H} \setminus H$. Then $\partial H$ and its image $f_J(\partial H)$ in $B_J$ are algebraic sub-
sets of $B_J$ of dimension less than $d$, according to the dimension properties
recalled before the lemma. Therefore, the form $f_J^*(\omega_J)$ vanishes on the di-
mension $d$ pieces of the intersection of $H$ with the semi-algebraic compact
set $f_J^{-1}(f_J(\partial H))$. Let $\Sigma(f_J)$ be the set of critical points of $f_J|_H$. According
to Lemma 16.29, $\Sigma(f_J)$ is semi-algebraic. According to the Morse–Sard the-
orem 1.4, $f_J(\Sigma(f_J))$, which is semi-algebraic, is of zero measure. Therefore,
its dimension is less than $d$. Now, $B_J \setminus (f_J(\Sigma(f_J) \cup \partial H) \cup \partial B_J)$ is an open
semi-algebraic subset of $B_J$, which therefore has a finite number of connected
components according to Theorem 16.30. On each of these connected compo-
nents, the local degree of $f_J$ is finite because $\bar{H}$ is compact and the points in
the preimage of a regular value are isolated. Our assumptions make this local
degree locally constant. Indeed, for a point $y$ in such a component, there ex-
ists a small $d$-dimensional disk $D(y)$ around $y$ whose preimage contains disk
neighborhoods of the points of the preimage, each mapped diffeomorphically
to $D(y)$. The image of $\bar{H}$ minus these open disks is a compact that does not
meet $y$. Therefore, there is a smaller disk around $y$ that is not met by this
compact.

Then $\int_{\bar{H}} f^*(\omega)$ is the integral of $\omega_J$ weighted by this bounded local degree and by a multiplication by $g_J \circ f$. So it is absolutely convergent. $\square$

Recall that an *open simplex* in $\mathbb{R}^n$ is a subset of the form $v_1 \dots v_k =$
$\{\sum_{i=1}^k t_i v_i : t_i \in ]0, 1], \sum_{i=1}^k t_i = 1\}$, where $v_1, \dots, v_k$ are affinely independent points in $\mathbb{R}^n$. The faces of $v_1 \dots v_k$ are the simplices $v_{i_1} \dots v_{i_j}$ for subsets $\{i_1, \dots, i_j\} \subset \underline{k}$. A locally finite simplicial complex in $\mathbb{R}^n$ is a locally finite collection $K$ of disjoint open simplices such that each face of a simplex of $K$ belongs to $K$. For such a complex $K$, $|K|$ denotes the union of the simplices of $K$.

The following Lojasiewicz triangulation theorem [Loj64, Theorem 1, p. 463, §3] ensures that a compact semi-algebraic set may be viewed as a topological chain (as in Subsection 2.1.5).

**Theorem 16.32.** For any locally finite collection $\{B_i\}$ of semi-algebraic subsets of $\mathbb{R}^n$, there exist a locally finite simplicial complex $K$ of $\mathbb{R}^n$ such that $|K| = \mathbb{R}^n$ and a homeomorphism $\tau$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ such that

* for any open simplex $\sigma$ of $K$, $\tau(\sigma)$ is an analytic submanifold of $\mathbb{R}^n$, and $\tau|_{\sigma}$ is an analytic isomorphism from $\sigma$ to $\tau(\sigma)$,

* for any open simplex $\sigma$ of $K$ and any $B_i$ of the collection $\{B_i\}$, we have $\tau(\sigma) \subset B_i$ or $\tau(\sigma) \subset \mathbb{R}^n \setminus B_i$.

## 16.5 Extending $\mathcal{Z}^f$ to $q$-tangles

In Chapter 15, the behavior of $\mathcal{Z}$ on braids, which are paths in $\tilde{S}_B(\mathbb{C})$ for some finite set $B$ was discussed. Recall that $\mathcal{Z}$ and $\mathcal{Z}^f$ coincide for braids. In this section, we extend $\mathcal{Z}^f$ and its variants of Section 16.2 to paths of $S_B(\mathbb{C})$, where $B$ is a finite set. This is already mostly done in [Poi00], where the main ideas come from. However, our presentation is different, and it provides additional statements and explanations.

Our extension to paths of $S_B(\mathbb{C})$ will allow us to define the extension of $\mathcal{Z}^f$ to $q$-tangles in rational homology cylinders so that Proposition 15.19 is still valid in the setting of $q$-tangles.

Recall the semi-algebraic subsets $\mathcal{V}(\Gamma)$ and $\mathcal{V}(y, \Gamma)$ of $S_{V(\Gamma)}(\mathbb{R}^3) \times S_B(\mathbb{C})$, introduced in Chapter 15, for a $3N$-numbered Jacobi diagram $\Gamma$ on $\sqcup_{b \in B} \mathbb{R}_b$. Both $S_{V(\Gamma)}(\mathbb{R}^3)$ and $S_B(\mathbb{C})$ are stratified by $\Delta$-parenthesizations according to Theorem 8.28. Let $\mathcal{P}_B$ be a $\Delta$-parenthesization of $B$. Let $\mathcal{P}$ be a $\Delta$-parenthesization of $\mathcal{V}(\Gamma)$. Set

$$ \mathcal{V}_{\mathcal{P}_B, \mathcal{P}}(\Gamma) = \mathcal{V}(\Gamma) \cap (S_{\mathcal{V}(\Gamma), \mathcal{P}}(\mathbb{R}^3) \times S_{B, \mathcal{P}_B}(\mathbb{C})) . $$

An element of $S_{V(\Gamma), \mathcal{P}}(\mathbb{R}^3)$ is denoted by $(c_Y)_{Y \in \mathcal{P}}$, where $c_Y \in \tilde{S}_{K(Y)}(\mathbb{R}^3)$. An element of $S_{B, \mathcal{P}_B}(\mathbb{C})$ is denoted by $(y_D)_{D \in \mathcal{P}_B}$, where $y_D \in \tilde{S}_{K(D)}(\mathbb{C})$. Fix $\Gamma$ and $\mathcal{P}_B$. Recall the natural map $p_B: U(\Gamma) \to B$ induced by $i_\Gamma$. Let
$c = ((c_Y)_{Y \in \mathcal{P}}, (y_D)_{D \in \mathcal{P}_B}) \in \mathcal{V}_{\mathcal{P}_B, \mathcal{P}}(\Gamma)$, and let $Y$ be in the set $\widehat{\mathcal{P}}'_X$ introduced in Notation 15.2. Then we have

$$p_{\mathcal{C}} \circ c_Y|_{U(\Gamma) \cap Y} = \lambda(Y) \left( y_{\hat{B}(Y)} \circ p_B - y_{\hat{B}(Y)} \circ p_B(b(Y)) \right)$$

for some $\lambda(Y) \ge 0$, with the normalizations of Notation 15.2. Recall the set $\mathcal{P}'_X(c)$ of separating sets $Y$ for $c$ from Lemma 15.12. Such a separating set is an element $Y$ of $\widehat{\mathcal{P}}'_X$ with $\lambda(Y) \ne 0$. For a subset $\mathcal{P}'_X$ of $\widehat{\mathcal{P}}'_X$, set $\mathcal{V}_{\mathcal{P}_B, \mathcal{P}, \mathcal{P}'_X}(\Gamma) = \{c \in \mathcal{V}_{\mathcal{P}_B, \mathcal{P}}(\Gamma) : \mathcal{P}'_X(c) = \mathcal{P}'_X\}$. We use the data $(\mathcal{P}_B, \mathcal{P}, \mathcal{P}'_X)$ to stratify $\mathcal{V}(\Gamma)$ (or $[0, 1] \times \mathcal{V}(\Gamma)$, whose strata will be the products by $[0, 1]$ of the strata of $\mathcal{V}(\Gamma)$ by definition). Recall that for any $D \in \mathcal{P}_B$, the elements $Y$ of $\mathcal{P}'_X$ with $\hat{B}(Y) = D$ are minimal with respect to the inclusion among the elements of $\mathcal{P}$ with $\hat{B}(Y) = D$.

For $y \in \mathcal{S}_{\mathcal{P}_B, \mathcal{P}_B}(\mathcal{C})$, for a $\Delta$-parenthesization $\mathcal{P}$ of $\mathcal{V}(\Gamma)$, and for a subset $\mathcal{P}'_X$ of $\mathcal{P}$, set

$$\mathcal{V}(y, \Gamma, \mathcal{P}, \mathcal{P}'_X) = \mathcal{V}(y, \Gamma) \cap \mathcal{V}_{\mathcal{P}_B, \mathcal{P}, \mathcal{P}'_X}(\Gamma).$$

Recall from Lemma 15.12 that when $\mathcal{V}(y, \Gamma, \mathcal{P}, \mathcal{P}'_X)$ is not empty, its dimension is $|U(\Gamma)| + 3|T(\Gamma)| - 1 - |\mathcal{P} \setminus \mathcal{P}'_X|$. In particular, the dimension of $\mathcal{V}(y, \Gamma)$ is at most $2|E(\Gamma)| - 1$.

Fix the family $(\tilde{\omega}(i, S^2) = (\tilde{\omega}(i, t, S^2))_{t \in [0,1]})_{i \in \underline{3N}}$ of closed 2-forms on $[0, 1] \times S^2$ once for all in this section, and assume that $\tilde{\omega}(i, 0, S^2) = \omega_{S^2}$ for all $i$. For an edge $e$ of $\Gamma$, recall the map

$$p_{e, S^2} : [0, 1] \times \mathcal{S}_{V(\Gamma)}(\mathbb{R}^3) \to [0, 1] \times S^2,$$

sending $(t, c \in \check{\mathcal{S}}_{V(\Gamma)}(\mathbb{R}^3))$ to $(t, p_{S^2}((c(v(e, 1)), c(v(e, 2)))))$. Also recall the $(2|E(\Gamma)|)$-form

$$\Omega_{\Gamma} = \bigwedge_{e \in E(\Gamma)} p_{e, S^2}^* (\tilde{\omega}(j_E(e), S^2))$$

over $[0, 1] \times \mathcal{S}_{V(\Gamma)}(\mathbb{R}^3)$. This form pulls back to provide smooth forms on the smooth strata of $[0, 1] \times \mathcal{V}(\Gamma)$.

Let $A$ denote a subset of $\underline{3N}$ with cardinality $3n$. An ordered $r$-component $A$-numbered Jacobi diagram $\Gamma^{(r)}$ on $\sqcup_{b \in B} \mathbb{R}_b$ is a degree $n$ $A$-numbered Jacobi diagram $\Gamma^{(r)}$ on $\sqcup_{b \in B} \mathbb{R}_b$ that has $r$ connected components $\Gamma_1, \dots, \Gamma_r$ and such that $i_{\Gamma}$ is represented by an injection of $\mathcal{V}(\Gamma)$ that maps all univalent vertices of $\Gamma_i$ before (or below) the univalent vertices of $\Gamma_{i+1}$ for any $i \in \underline{r-1}$. The data of such an ordered $r$-component $A$-numbered Jacobi diagram $\Gamma^{(r)}$ is equivalent to the data of an $r$-tuple $(\Gamma_1, \dots, \Gamma_r)$ of $A$-numbered connected Jacobi diagrams with pairwise disjoint $j_E(E(\Gamma_i))$ such that the sum of the
degrees of the $\Gamma_i$ is $n$. Let $\mathcal{D}_{n,A}^{e,r}(\sqcup_{b \in B} \mathbb{R}_b)$ denote the set of these ordered $r$-component $A$-numbered Jacobi diagrams $\Gamma^{(r)}$ on $\sqcup_{b \in B} \mathbb{R}_b$.<fnref n="5" />

Such a diagram provides the $(2|E(\Gamma^{(r)})|)$-form

$$ \Omega_{\Gamma^{(r)}} = \bigwedge_{i=1}^{r} P_i^*(\Omega_{\Gamma_i}) $$

on (the smooth strata of) $\prod_{i=1}^r ([0, 1] \times \mathcal{V}(\Gamma_i))$, where

$$ P_i: \prod_{i=1}^{r} ([0, 1] \times \mathcal{V}(\Gamma_i)) \to [0, 1] \times \mathcal{V}(\Gamma_i) $$

is the projection onto the $i$th factor. This form is also the pullback of a smooth form on $[0, 1]^r \times (S^2)^{E(\Gamma^{(r)})}$, by a semi-algebraic map. Recall $\Delta^{(r)} = \{(t_1, \dots, t_r) \in [0, 1]^r : 0 \le t_1 \le t_2 \dots \le t_r\}$. A semi-algebraic path $\gamma: [0, 1] \to [0, 1] \times S_B(\mathbb{C})$ induces the semi-algebraic map

$$ \gamma^{(r)}: \begin{array}{ccc} \Delta^{(r)} & \to & ([0, 1] \times S_B(\mathbb{C}))^r \\ (t_1, \dots, t_r) & \mapsto & (\gamma(t_1), \dots, \gamma(t_r)). \end{array} $$

Consider the product $P_{\Gamma^{(r)}}: \prod_{i=1}^r ([0, 1] \times \mathcal{V}(\Gamma_i)) \to ([0, 1] \times S_B(\mathbb{C}))^r$ of natural projections. Assume that $\gamma$ is injective. Set $C(\Gamma^{(r)}, \gamma) = P_{\Gamma^{(r)}}^{-1}(\gamma^{(r)}(\Delta^{(r)}))$. Then $C(\Gamma^{(r)}, \gamma)$ is a semi-algebraic subset of $\prod_{i=1}^r ([0, 1] \times \mathcal{V}(\Gamma_i))$ of dimension at most $2|E(\Gamma^{(r)})|$ whose $2|E(\Gamma^{(r)})|$-dimensional strata are oriented canonically, as soon as the Jacobi diagrams $\Gamma_i$ are: Fix an arbitrary vertex-orientation for the $\Gamma_i$. The set $C(\Gamma^{(r)}, \gamma)$ is locally oriented as the product of the $C(\Gamma_i^{(1)}, \gamma)$ for $i \in \underline{\mathbb{R}}$. The parameter $t_i$ replaces the translation parameter in $\mathcal{V}(\gamma(t_i), \Gamma_i)$.

Define the A-holonomy $\widetilde{\operatorname{hol}}(\eta_{B,A})$ along injective semi-algebraic paths $\gamma$ of $[0, 1] \times S_B(\mathbb{C})$, with respect to our family $(\tilde{\omega}(i, S^2))_{i \in \underline{3N}}$, to be

$$ \widetilde{\operatorname{hol}}_{\gamma}(\eta_{B,A}) = [\emptyset] + \sum_{r=1}^{\infty} \sum_{\Gamma^{(r)} \in \mathcal{D}_{n,A}^{e,r}(\sqcup_{b \in B} \mathbb{R}_b)} \zeta_{\Gamma^{(r)}} \int_{C(\Gamma^{(r)}, \gamma)} \Omega_{\Gamma^{(r)}}[\Gamma^{(r)}], $$

with

$$ \zeta_{\Gamma^{(r)}} = \frac{(|A| - |E(\Gamma^{(r)})|)!}{|A|!2^{|E(\Gamma^{(r)})|}}. $$

<footnote n="5">⁵The notation $\mathcal{D}_{n,A}^{e,r}$ has a redundancy since the cardinality of $A$ is $3n$. We keep the redundancy for consistency here because we will use other spaces of numbered Jacobi diagrams, where the degree is not determined by the cardinality of the set of indices, in Chapter 17.</footnote>
(Again, we fix an arbitrary vertex-orientation for the components $\Gamma_i$ of each $\Gamma^{(r)}$, and $\text{hol}_\gamma(\eta_{B,A})$ is independent of our choices.)

The involved integrals make sense as soon as $\gamma$ is semi-algebraic, thanks to Lemma 16.31, which also justifies the following lemma.

**Lemma 16.33.** For any injective semi-algebraic path $\gamma$ of $[0, 1] \times S_B(\mathbb{C})$,

$$ \lim_{\epsilon \to 0} \widetilde{\text{hol}}_{\gamma|_{[\epsilon, 1-\epsilon]}}(\eta_{B,A}) $$

makes sense, and it is equal to $\widetilde{\text{hol}}_\gamma(\eta_{B,A})$.

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Together with the identification $\mathcal{Z}(T(\gamma)) = \text{hol}_{p_{CS} \circ \gamma}(\eta_B)$ for braids provided by Proposition 15.19, Lemma 16.33 implies the convergence part of Theorem 13.8. The above convergent integrals extend Definition 16.1 of $\widetilde{\text{hol}}_\gamma(\eta_{B,A})$ for injective semi-algebraic paths in $[0, 1] \times \check{S}_B(\mathbb{C})$. Note the following easy lemma.

**Lemma 16.34.** The A-holonomy $\widetilde{\text{hol}}_\gamma(\eta_{B,\cdot})$, which is valued in $\mathcal{A}_n(\bigsqcup_{b \in B} \mathbb{R}_b)$, extends naturally to noninjective semi-algebraic paths. This holonomy is multiplicative under path composition with respect to the product of Definition 16.10.

□

Recall that $\tilde{\omega}(i, 0, S^2)$ is the standard homogeneous volume-one form on $S^2$. When $\gamma$ is valued in $\{0\} \times S_B(\mathbb{C})$, there is no need to number the diagram edges since $\tilde{\omega}(i, 0, S^2) = \omega_{S^2}$ for all $i$, and we simply have

$$ \widetilde{\text{hol}}_{\gamma}(\eta_B) = \sum_{r=0}^{\infty} \sum_{k=0}^{\infty} \sum_{\Gamma^{(r)} \in \mathcal{D}_{k, \frac{3k}{2}}^{e,r}(\bigsqcup_{b \in B} \mathbb{R}_b)} \zeta_{\Gamma^{(r)}} \int_{C(\Gamma^{(r)}, \gamma)} \Omega_{\Gamma^{(r)}}[\Gamma^{(r)}] \in \mathcal{A}(\bigsqcup_{b \in B} \mathbb{R}_b). $$

In this case, $\widetilde{\text{hol}}_\gamma(\eta_B)$ is nothing but the Poirier functor $Z^l$ of [Poi00, Section 1.4] applied to $\gamma$. The projection in $\mathcal{A}_n(\bigsqcup_{b \in B} \mathbb{R}_b)$ of $\widetilde{\text{hol}}_\gamma(\eta_B)$, which coincides with the holonomy $\widetilde{\text{hol}}_\gamma(\eta_B)$ defined in Section 15.2 when $\gamma$ is valued in $\{0\} \times \check{S}_B(\mathbb{C})$, coincides with $\widetilde{\text{hol}}_\gamma(\eta_{B,A})$, when $|A| = 3n$, in this case of homogeneous forms.

Recall that $S_B(\mathbb{C})$ is a smooth manifold with ridges, which can also be equipped with a semi-algebraic structure for which the local charts provided in Theorem 8.28 are semi-algebraic maps. In such a trivialized open simply connected subspace, any two points can be connected by a semi-algebraic