**Lemma 11.4.** Let $(\tilde{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$. For any link $L: \mathcal{L} \hookrightarrow \tilde{R}$, for any integer $n$, there exists a family $(P(i))_{i \in \underline{3n}}$ of propagating chains of $(C_2(R), \tau)$ in general $3n$-position with respect to $L$.

Let $(P(i))_{i \in \underline{3n}}$ be such a family. The codimension-two rational chains $p_e^{-1}(P(j_E(e)))$ are cooriented by the coorientation of $P(j_E(e))$ in $C_2(R)$. Define $I(\Gamma, o(\Gamma), (P(i))_{i \in \underline{3n}})$ to be the algebraic intersection in $(C(R, L; \Gamma), o(\Gamma))$ of the chains $p_e^{-1}(P(j_E(e)))$ over the edges $e$ of $E(\Gamma)$.

For any finite set $V$, equip $C_V(R)$ with a Riemannian metric that is symmetric with respect to permutations of elements of $V$. Let $d$ denote the associated distance. Our choice of distance will not matter thanks to the following easy lemma.

**Lemma 11.5.** All the distances associated to Riemannian metrics are equivalent on a compact smooth manifold.

**PROOF:** Let $g_1$ and $g_2$ be two Riemannian metrics on the compact manifold $M$, let $\|\cdot\|_1$ and $\|\cdot\|_2$ be the two associated norms on tangent vectors, and let $d_1$ and $d_2$ be the two associated distances. View the unit tangent bundle $UM$ of $M$ as the set of unit tangent vectors to $M$ with respect to $\|\cdot\|_1$. Then the image of $UM$ under the continuous map $\|\cdot\|_2$ is a compact interval $[a, b]$ with $a > 0$, and we have

$$a \le \frac{\|x\|_2}{\|x\|_1} \le b$$

for any nonzero tangent vector $x$ of $M$. Let $p$ and $q$ be two distinct points of $M$. For any smooth path $\gamma: [0, 1] \to M$ such that $\gamma(0) = p$ and $\gamma(1) = q$, we have

$$d_2(p, q) \le \int_0^1 \|\gamma'(t)\|_2 dt \le b \int_0^1 \|\gamma'(t)\|_1 dt.$$

Therefore, we get $d_2(p, q) \le bd_1(p, q)$. Similarly, we have $d_1(p, q) \le \frac{d_2(p,q)}{a}$. $\square$

**Definition 11.6.** For a subset $X$ of $C_V(R)$ and for $\varepsilon > 0$, set

$$N_{\varepsilon}(X) = \{x \in C_V(R) : d(x, X) < \varepsilon\}.$$

For a small positive number $\eta$, a closed 2-form $\omega(i)$ on $C_2(R)$ is said to be $\eta$-dual to $P(i)$, if it is supported in $N_{\eta}(P(i))$ and if $\int_D \omega(i) = \langle D, P(i) \rangle_{C_2(R)}$ for any 2-dimensional disk $D$ embedded in $C_2(R)$ transverse to $P(i)$ whose boundary sits outside $N_{\eta}(P(i))$.
We will prove the following lemma in Section 11.4.

**Lemma 11.7.** Assume Lemma 11.4. Under its hypotheses, for any $\eta > 0$, there exist propagating forms $\omega(i)$ of $(C_2(R), \tau)$ $\eta$-dual to the $P(i)$ of Lemma 11.4. If $\eta$ is small enough, then we have

$$I(\Gamma, o(\Gamma), (P(i))_{i \in \underline{3n}}) = I(\Gamma, o(\Gamma), (\omega(i))_{i \in \underline{3n}})$$

for any $\Gamma \in \mathcal{D}_{k, \underline{3n}}^c(\mathcal{L})$, where $k \le n$.

Thus, $I(\Gamma, o(\Gamma), (\omega(i))_{i \in \underline{3n}})$ is rational in this case, and we get the following theorem.

**Theorem 11.8.** The anomaly $\beta$ is rational, i.e., it belongs to $\mathcal{A}(\emptyset; \mathbb{Q})$. Let $(\tilde{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$. For any link $L: \mathcal{L} \to \tilde{R}$, the value $\mathcal{Z}(R, L)$ belongs to $\mathcal{A}(\mathcal{L}; \mathbb{Q})$.

**PROOF OF THE THEOREM ASSUMING LEMMAS 11.4 AND 11.7:** Theorems 7.40 and 7.20, Lemmas 11.4 and 11.7 imply that $Z(S^3, \tau) = Z(\mathbb{R}^3, \emptyset, \tau)$ is rational for any $\tau$. So $z(S^3, \tau) = p^c(Z(S^3, \tau))$ is rational for any $\tau$, too, and, by Definition 10.5, the anomaly $\beta$ is rational. Therefore, Theorem 7.40, Lemmas 11.4, and 11.7 also imply that $Z(\tilde{R}, L, \tau)$ is rational for any pair $((\tilde{R}, \tau), L)$ such that $(\tilde{R}, \tau)$ is an asymptotic rational homology $\mathbb{R}^3$ and $L$ is a straight link with respect to $\tau$. In particular, the integral $I_\theta(K, \tau)$ is rational for any component $K$ of a straight link $L$. Since $p_1(\tau)$ and the anomalies $\alpha$ and $\beta$ are rational, Theorem 7.20 now implies that $\mathcal{Z}(R, L)$ is rational. $\square$

## 11.2 More on general propagating chains

By a theorem of René Thom [Tho54, Théorème II.27, p. 55], any integral codimension 2 homology class in a manifold can be represented as the class of an embedded closed (oriented) submanifold. We prove a relative version of this result below, following Thom's original proof in this particular case of his theorem.

**Theorem 11.9.** Let $A$ be a compact smooth (oriented) manifold with boundary. Let $C$ be a smooth codimension 2 closed (oriented) submanifold of $\partial A$ such that the homology class of $C$ vanishes in $H_{\dim(A)-3}(A; \mathbb{Z})$. Then there exists a compact smooth codimension 2 submanifold $B$ of $A$ transverse to $\partial A$ whose boundary is $C$.
PROOF: Let us first sketch Thom's proof with his notation. The normal bundle to $C$ in $\partial A$ is an oriented disk bundle. It is the pull-back of a universal disk bundle $A_{SO(2)}$ over a compact classifying space $B_{SO(2)}$ via a map $f_C$ from $C$ to $B_{SO(2)}$. See [MS74, p. 145]. Like René Thom [Tho54, p. 28, 29], define the Thom space $M(SO(2))$ of $SO(2)$ to be the space obtained from the total space $A_{SO(2)}$ by identifying its subspace $E_{SO(2)}$ consisting of the points in the boundaries of the fibers $D^2$ of $A_{SO(2)}$ with a single point $a$. Regard $B_{SO(2)}$ as the zero section of $A_{SO(2)}$. So $B_{SO(2)}$ sits inside $M(SO(2))$.

The map $f_C$ extends canonically to $\partial A$. Its extension $f_{\partial A}$ injects the fibers of an open tubular neighborhood of $C$ in $\partial A$ to fibers of $A_{SO(2)}$ and maps the complement of such a neighborhood to $a$. Thus $C$ is the preimage of $B_{SO(2)}$. In order to prove the theorem, it suffices to extend the map $f_{\partial A}$ to a map $f_A$ from $A$ to $M(SO(2))$ so that, in a neighborhood of any point of $f_A^{-1}(B_{SO(2)})$, the differential of a local projection to the fiber of the normal bundle to $B_{SO(2)}$ composed with $f_A$ is well-defined and surjective. (The fiber of the normal bundle to $B_{SO(2)}$ is isomorphic to the tangent space to a fiber of $A_{SO(2)}$.) Indeed, the compact submanifold $B = f_A^{-1}(B_{SO(2)})$ of $A$, with respect to such an extension, has the desired properties.

The map $f_{\partial A}$ can be extended as a continuous map, using the fact that $M(SO(2))$ is a $K(\mathbb{Z}; 2)$ [Tho54, ii), p. 50]. In other words, the only nontrivial homotopy group of $M(SO(2))$ is its $\pi_2$, which is isomorphic to $\mathbb{Z}$.

Let us now give some details about the above sketch and show how the pair $(B_{SO(2)}, M(SO(2)))$ can be replaced by $(\mathbb{CP}^N, \mathbb{CP}^{N+1})$ for some large integer $N$, following [Tho54, ii), p. 50]. View the fiber of a disk bundle as the unit disk of $\mathbb{C}$. The corresponding complex line bundle over $C$ injects into a trivial complex bundle $\mathbb{C}^{N+1} \times C$ as in [MS74, Lemma 5.3, p. 61] for some integer $N$, by some map $(f_{1,C}, \mathbb{1}(C))$. Therefore, it is the pull-back of the tautological complex line bundle $\gamma_N^1$ over $B'_{SO(2)} = \mathbb{CP}^N$ by the map $f'_C: C \to \mathbb{CP}^N$ that sends a point $x$ of $C$ to the image of the fiber over $x$ under $f_{1,C}$. The disk bundle $A'_{SO(2)}$ associated to $\gamma_N^1$ is diffeomorphic to the normal bundle to $\mathbb{CP}^N$ in $\mathbb{CP}^{N+1}$ by the inverse of the following (orientation-reversing) map:

$$ \begin{aligned} \mathbb{CP}^{N+1} \setminus \{[(0, \dots, 0, 1)]\} & \to \gamma_N^1 \\ [z_1, \dots, z_{N+1}, z] & \mapsto \left( \frac{\bar{z}}{\sum_{i=1}^{N+1} |z_i|^2} (z_1, \dots, z_{N+1}), [z_1, \dots, z_{N+1}] \right). \end{aligned} $$

This map also shows that the space $M'(SO(2))$, obtained from $A'_{SO(2)}$ by identifying $E'_{SO(2)} = \partial A'_{SO(2)}$ to a point, is homeomorphic to the whole $\mathbb{CP}^{N+1}$.

The long exact sequence associated to the fibration $S^1 \hookrightarrow S^{2N+3} \to \mathbb{CP}^{N+1}$ implies that $\pi_2(\mathbb{CP}^{N+1}) = \mathbb{Z}[\mathbb{CP}^1]$ and that $\pi_i(\mathbb{CP}^{N+1})$ is trivial
for any $i \in \underline{2N} + \underline{2} \setminus \{2\}$. See Theorem A.14. Assume that $2N + 2$ is bigger than the dimension of $A$ without loss of generality.

It is not hard to see that $\pi_2(M'(SO(2))) = H_2(M'(SO(2)); \mathbb{Z})$ is freely generated by the class of the image $[F]$ of a fiber under the identification of $E'_{SO(2)}$ with the point $a$. (Indeed, since $B'_{SO(2)}$ is connected, the homology class of $[F]$ is well-defined. Since $M'(SO(2)) \setminus B'_{SO(2)}$ is contractible and since any 2-cycle is homologous to a 2-cycle that is transverse to $B'_{SO(2)}$, any homology class of degree 2 is a multiple of $[F]$, which therefore generates $H_2(M'(SO(2)); \mathbb{Z})$.)

Extend $f'_C$ to a map $f_{\partial A}$ valued in $M'(SO(2))$ as before so that $C = f_{\partial A}^{-1}(B'_{SO(2)})$. Recall that any smooth manifold is triangulable [Cai35], [Whi40]. Fix a triangulation for $(A, \partial A)$ transverse to $C$. In particular, $C = f_{\partial A}^{-1}(B'_{SO(2)})$ avoids the 1-skeleton. Extend $f_{\partial A}$ skeleton by skeleton starting with the zero and one-skeleta, for which there is no obstruction to extending $f_{\partial A}$ to a map valued in $M'(SO(2)) \setminus B'_{SO(2)}$, which is connected. There is no obstruction to extending $f_{\partial A}$, as a map valued in the contractible $M'(SO(2)) \setminus B'_{SO(2)}$, to the two-skeleton of $(A, \partial A)$, but such a map would not necessarily extend to the three-skeleton. Let $f_A^{(2)}$ be an arbitrary generic extension valued in $M'(SO(2))$ of $f_{\partial A}$ to the two-skeleton of $(A, \partial A)$. Define the 2-cochain $c(f_A^{(2)})$ with $\mathbb{Z}$-coefficients such that $c(f_A^{(2)})(D)$ is the algebraic intersection of $B'_{SO(2)}$ and $f_A^{(2)}(D)$ in $M'(SO(2))$ for any 2-cell $D$ of $A$. Then $f_A^{(2)}$ extends to the 3-skeleton if and only if this cochain (which is fixed on $\partial A$ and Poincaré dual to $C$ on $\partial A$) is a cocycle. Thus, to prove that $f_{\partial A}$ extends to the 3-skeleton, it suffices to prove that the class of $c(f_A^{(2)})|_{\partial A}$ in $H^2(\partial A; \mathbb{Z})$ is in the natural image of $H^2(A; \mathbb{Z})$, or, equivalently, that its image in $H^3(A, \partial A; \mathbb{Z})$, by the boundary map of the long cohomology exact sequence of $(A, \partial A)$, vanishes. This image is represented by a cochain that maps a 3-cell $\mathcal{B}$ of $(A, \partial A)$ to the algebraic intersection of $\partial \mathcal{B}$ and $C$, which is, up to a fixed sign, the algebraic intersection of $\mathcal{B}$ and $C$ (pushed inside $A$). Therefore, the class in $H^3(A, \partial A; \mathbb{Z})$ of this relative cocycle is Poincaré dual to the class of $C$ in $H_{\dim(A)-3}(A)$, which vanishes. So $f_{\partial A}$ can be extended to the 3-skeleton. Since the next homotopy groups $\pi_i(M'(SO(2)))$, for $3 \le i < \dim(A)$, vanish, there is no obstruction to extending $f_{\partial A}$ to the manifold $A$.

Finally, make $f_A$ smooth, using an approximation theorem [Hir94, Chapter 2, Theorem 2.6] of continuous maps by smooth maps, and make $f_A$ transverse to $B'_{SO(2)}$, with the help of a transversality theorem [Hir94, Chapter 3, Theorem 2.1]. $\square$

**Corollary 11.10.** If $R$ is a $\mathbb{Z}$-sphere, for any asymptotically standard parallelization $\tau$ of $\tilde{R}$, for any $X \in S^2$, there exists a 4-dimensional submanifold
of $C_2(R)$ transverse to the ridges whose boundary is $p_{\tau}^{-1}(X)$.

PROOF: First extend $p_{\tau}$ as a regular map from a regular neighborhood $N(\partial C_2(R))$ of $\partial C_2(R)$, where $N(\partial C_2(R))$ is a smooth cobordism with ridges embedded in $C_2(R)$ from a smooth manifold $\partial C'_2(R)$ without ridges to $\partial C_2(R)$, and $N(\partial C_2(R))$ is homeomorphic to the product $[0, 1] \times \partial C_2(R)$. Then apply Theorem 11.9 to $C'_2(R) = C_2(R) \setminus \text{Int}(N(\partial C_2(R)))$ and to $p_{\tau}|_{\partial C'_2(R)}(X)$. $\square$

When $R$ is a $\mathbb{Q}$-sphere, perform the same first step as in the above proof. Take a collar neighborhood of $\partial C'_2(R)$ in $N(\partial C_2(R))$, which is (diffeomorphic to and) identified with $[0, 8] \times \partial C'_2(R)$ so that $\partial C'_2(R) = \{0\} \times \partial C'_2(R)$. Assume that $p_{\tau}$ factors through the projection to $\partial C'_2(R)$ on $[0, 8] \times \partial C'_2(R)$. There exists a positive integer $k$ such that $kp_{\tau}|_{\partial C'_2(R)}(X)$ is null-homologous in $C'_2(R)$. Let $p_k: S^2 \to S^2$ be a degree $k$ map that does not fix $X$ and such that $X$ is a regular value of $p_k$ with $k$ preimages. Then $(p_k \circ p_{\tau}|_{\partial C'_2(R)})^{-1}(X)$ bounds a 4-manifold $P'$ properly embedded in $C'_2(R)$ according to Theorem 11.9. For $j \in \underline{k}$, let $\{\gamma_j: [0, 4] \to S^2\}_{j \in \underline{k}}$ be a collection of smooth injective paths ending at $X = \gamma_j(4)$ whose images do not meet outside $X$ and such that $p_k^{-1}(X) = \{\gamma_j(0) : j \in \underline{k}\}$. Also assume that all the derivatives of $\gamma_j$ vanish at 0 and 4. Consider

$$p_{[0,8]} \times p_{\tau}: \begin{array}{ccc} [0,8] \times \partial C'_2(R) & \to & [0,8] \times S^2 \\ (t,x) & \mapsto & (t, p_{\tau}(x)). \end{array}$$

Then

$$P = p_{\tau}^{-1}|_{N(\partial C_2(R)) \setminus ([0,4] \times \partial C'_2(R))}(X) + \frac{1}{k}P' \\ + \frac{1}{k}(p_{[0,8]} \times p_{\tau})|_{[0,4] \times \partial C'_2(R)}(\{(t, \gamma_j(t)) : j \in \underline{k}, t \in [0, 4]\})$$

is a propagating chain of $(\tilde{R}, \tau)$. See Figure 11.1.

## 11.3 Existence of transverse propagating chains

In this section, we prove Lemma 11.4.

In order to warm up, we first prove a weak version of this lemma. The proof is a straightforward adaptation of a proof of Thom [Tho54, p. 23, 24, Lemma I.4].

Assume that $R$ is an integer homology 3-sphere. Let $(P(i))_{i \in \underline{3n}}$ be a family of propagating chains of $(C_2(R), \tau)$ for an asymptotically standard parallelization $\tau$ of $\tilde{R}$. Assume that these chains are submanifolds of $C_2(R)$ transverse to $\partial C_2(R)$ as in Corollary 11.10.
<figure><img src="image_89.png" /><figcaption>Figure 11.1: A multiple of a propagating chain of $(\tilde{R}, \tau)$</figcaption></figure>

Let $N(P(i))$ denote the normal bundle to $P(i)$ embedded in $C_2(R)$ as a tubular neighborhood whose fibers $N_x(P(i))$ over a point $x \in P(i)$ are disks embedded in $C_2(R)$. Let $(K_{i,j})_{j \in J(i)}$ be a finite cover of $P(i)$ by compact subspaces $K_{i,j}$ embedded in open subspaces $O_{i,j}$ of $P(i)$ equipped with bundle charts $(\psi_{i,j}: N(P(i))|_{O_{i,j}} \to A_{i,j} \times D^2)_{j \in J}$. We assume that the above $A_{i,j}$ are $\mathbb{R}^4$, $\mathbb{R}^+ \times \mathbb{R}^3$ or $(\mathbb{R}^+)^2 \times \mathbb{R}^2$, and that the $\psi_{i,j}$ are bundle charts over diffeomorphisms $\phi_{i,j}: O_{i,j} \to A_{i,j}$. We also assume that for any $\{j, k\} \subset J(i)$, for any $x \in O_{i,j} \cap O_{i,k}$, the map

$$v \mapsto p_{D^2} \circ \psi_{i,k} \circ \psi_{i,j}^{-1}(\phi_{i,j}(x), v)$$

is a linear map of $SO(2)$, where $p_{D^2}$ denotes the natural projection onto $D^2$. Consider the space $\mathcal{H}_i$ of smooth diffeomorphisms of $N(P(i))$ that are isotopic to the identity map, that fix a neighborhood of $\partial N(P(i))$ pointwise, and that map any fiber of $N(P(i))$ to itself. Equip this space $\mathcal{H}_i$ with the following distance $d$.<fnref n="2" /> Each fiber is equipped with the distance $d_P$ induced by the norm of $\mathbb{R}^2$. This allows us to define a $C^0$ distance $d_0$ between two elements $h$ and $k$ of $\mathcal{H}_i$ by

$$d_0(h, k) = \sup_{x \in N(P(i))} d_P(h(x), k(x)).$$

Since $A_{i,j} \times D^2$ is a subset of $\mathbb{R}^6$, the differential of a map $\psi_{i,j} \circ h \circ \psi_{i,j}^{-1}$ for $h \in \mathcal{H}_i$ maps every element $x$ of $A_{i,j} \times D^2$ to a linear map of $\mathbb{R}^6$. The norm $\|L\|$ of a linear map $L$ of $\mathbb{R}^6$ is defined to be $\|L\| = \sup_{x \in S^5} \|L(x)\|$. For $h$

<footnote n="2">²This distance induces the strong (or weak, which is the same since $N(P(i))$ is compact) $C^1$-topology. See [Hir94, Chapter 2, p.35].</footnote>
and $k$ in $\mathcal{H}_i$, set

$$d_{(1)}(h, k) = \sup_{j \in J(i), x \in \phi_{i,j}(K_{i,j}) \times D^2} (\|T_x(\psi_{i,j} \circ h \circ \psi_{i,j}^{-1}) - T_x(\psi_{i,j} \circ k \circ \psi_{i,j}^{-1})\|)$$

and

$$d(h, k) = \sup (d_0(h, k), d_{(1)}(h, k)).$$

**Lemma 11.11.** Under the above hypotheses, there is a dense open subset of $\prod_{i=1}^{3n} \mathcal{H}_i$ such that for any $(h_i)$ in this subset, the chains obtained from the $P(i)$ by replacing $P(i)$ with $h_i^{-1}(P(i))$ are in general $3n$ position with respect to $L$ in the sense of Definition 11.3.

**PROOF:** We will first list finitely many sufficient conditions on the $(h_i)$, which guarantee the conclusion “For any graph $\Gamma$ of $\mathcal{D}_{3n}^e(\mathcal{L})$, and for any subset $E$ of $E(\Gamma)$, the map

$$p(\Gamma, E) = \prod_{e \in E} p_e: C(R, L; \Gamma) \to (C_2(R))^{j_E(E)}$$

is transverse to $\prod_{e \in E} h_{j_E(e)}^{-1} P(j_E(e))$.”. Next we will prove that each of these conditions is realized in an open dense subset of $\prod_{i=1}^{3n} \mathcal{H}_i$.

Extend the elements of $\mathcal{H}_i$ to diffeomorphisms of $C_2(R)$, by the identity map of $C_2(R) \setminus \mathring{N}(P(i))$. The propagating chains obtained from the $P(i)$ by replacing $P(i)$ with $h_i^{-1}(P(i))$ are in general $3n$ position with respect to $L$ if and only if the following condition $(*)$ $(\Gamma, E, \ell)$ holds for any triple $(\Gamma, E, \ell)$, where $\Gamma \in \mathcal{D}_{3n}^e(\mathcal{L})$, $E \subseteq E(\Gamma)$, and $\ell$ is a map $\ell: E \to \cup_{i \in \mathcal{I}_{3n}} J(i)$ such that $\ell(e) \in J(j_E(e))$.

$(*)$ $(\Gamma, E, \ell)$: The map $p(\Gamma, E)$ is transverse to $\prod_{e \in E} h_{j_E(e)}^{-1}(P(j_E(e)))$ along $p(\Gamma, E)^{-1}(\prod_{e \in E} h_{j_E(e)}^{-1}(K_{j_E(e),\ell(e)}))$ (as in Definition 11.2).

In order to prove our lemma, it suffices to prove that, for any of the finitely many $(\Gamma, E, \ell)$ as above, the set $\mathcal{H}(\Gamma, E, \ell)$ in which the condition $(*)$ $(\Gamma, E, \ell)$ is realized is a dense open subset of $\prod_{i=1}^{3n} \mathcal{H}_i$. This condition is equivalent to

$(*)$ $(\Gamma, E, \ell)$ : the map $(\prod_{e \in E} h_{j_E(e)}) \circ p(\Gamma, E)$ is transverse to $\prod_{e \in E} P(j_E(e))$ along $p(\Gamma, E)^{-1}(\prod_{e \in E} h_{j_E(e)}^{-1}(K_{j_E(e),\ell(e)}))$.

Set

$$C_{E,\ell} = C(R, L; \Gamma) \cap p(\Gamma, E)^{-1} \left( \prod_{e \in E} \psi_{j_E(e),\ell(e)}^{-1} (\phi_{j_E(e),\ell(e)}(K_{j_E(e),\ell(e)}) \times D^2) \right).$$

The condition $(*)$ $(\Gamma, E, \ell)$ can equivalently be written as “ $(0)_{e \in E}$ is a regular value of the map

$$\prod_{e \in E} (p_{D^2} \circ \psi_{j_E(e), \ell(e)} \circ h_{j_E(e)} \circ p_e)$$
on $C_{E,\ell}$.

Note that the set of regular values of this map on the compact domain $C_{E,\ell}$ is open. Therefore, if $(h_i)_{i \in \mathbb{3n}} \in \mathcal{H}(\Gamma, E, \ell)$ and if the $d_0(h_i, h'_i)$ are small enough, the preimage of $(0)_{e \in E}$ under the restriction of $\prod_{e \in E} p_{D^2} \circ \psi_{j_E(e),\ell(e)} \circ h'_{j_E(e)} \circ p(\Gamma, E)$ to $C_{E,\ell}$ consists of regular points of $\prod_{e \in E} p_{D^2} \circ \psi_{j_E(e),\ell(e)} \circ h_{j_E(e)} \circ p(\Gamma, E)$. These points are regular for $\prod_{e \in E} p_{D^2} \circ \psi_{j_E(e),\ell(e)} \circ h'_{j_E(e)} \circ p(\Gamma, E)$ provided that the $d(h_i, h'_i)$ are small enough. Therefore, the set $\mathcal{H}(\Gamma, E, \ell)$ is open.

To prove density, we use explicit deformations of the $h_i \in \mathcal{H}_i$, for a given $(h_i)_{i \in \mathbb{3n}} \in \prod_{i=1}^{3n} \mathcal{H}_i$. Fix a smooth map $\chi: D^2 \to [0, 1]$, which maps the disk of radius $\frac{1}{2}$ to 1 and the complement of the disk of radius $\frac{3}{4}$ to 0. For each compact $K_{i,j}$, such that $j \in J(i)$, fix a smooth map $\chi_{i,j}: A_{i,j} \to [0, 1]$ that maps $\phi_{i,j}(K_{i,j})$ to 1, and that vanishes outside a compact of $A_{i,j}$. For $w \in D^2$, define

$$h_{i,j,w}: \begin{array}{ccc} A_{i,j} \times D^2 & \to & A_{i,j} \times D^2 \\ (x, v) & \mapsto & (x, v + \chi(v)\chi_{i,j}(x)w). \end{array}$$

Note that $h_{i,j,w}$ is a diffeomorphism as soon as $\|w\|$ is smaller than a fixed positive number $\eta < \frac{1}{2}$. Extend $\psi_{i,j}^{-1} \circ h_{i,j,w} \circ \psi_{i,j}$ by the identity map outside $N(P(i))|_{\mathcal{O}_{i,j}}$. Note that there exists a constant $C$ such that $d(\psi_{i,j}^{-1} \circ h_{i,j,w} \circ \psi_{i,j} \circ h_i, h_i) \le C\|w\|$.

Thus, it suffices to prove that, for any $\varepsilon$ such that $0 < \varepsilon < \eta$, there exists $(w_e)_{e \in E}$ with $\|w_e\| < \varepsilon$ such that the restriction of

$$\prod_{e \in E} (\psi_{j_E(e), \ell(e)}^{-1} \circ h_{j_E(e), \ell(e)}, w_e \circ \psi_{j_E(e), \ell(e)} \circ h_{j_E(e)}) \circ p(\Gamma, E)$$

to $(C_{E,\ell})$ is transverse to $\prod_{e \in E} P(j_E(e))$ along $C_{E,\ell}$. Since this happens when $(-w_e)_{e \in E}$ is a regular value of the restriction of

$$\prod_{e \in E} (p_{D^2} \circ \psi_{j_E(e), \ell(e)} \circ h_{j_E(e)}) \circ p(\Gamma, E)$$

to $C_{E,\ell}$, and since such regular values form a dense set according to the Morse–Sard theorem, the lemma is proved. $\square$

The magic in the Thom proof above is that it proves the density of manifolds in general $3n$ position without bothering to construct a single one.

Lemma 11.11 does not quite prove Lemma 11.4 for two reasons. First, the $h_i$ do not fix the boundary of $\partial C_2(R)$ pointwise, so the perturbations $h_i^{-1}(P(i))$ are no longer propagating chains of $(C_2(R), \tau)$. Second, we have to deal with immersed manifolds (multiplied by an element of $\mathbb{Q}$) rather than embedded ones when $R$ is not an integer homology sphere.
To deal with this latter issue, we start with immersions $f_i$ of manifolds $\tilde{P}(i)$ to $C_2(R)$ whose images $f_i(\tilde{P}(i))$ represent chains $kP$ as in the end of Section 11.2, and (extended) immersions $f_i$ of the pull-backs $N(\tilde{P}(i))$ of the normal bundles to their images. Our immersions $f_i$ have the following properties. The restriction to $f_i^{-1}(C_2'(R))$ of each immersion $f_i$ is an embedding. The preimage $f_i^{-1}(N(\partial C_2(R)))$ has $k$ connected components $C_{j,i}$ ($j \in \underline{k}$) in $N(\tilde{P}(i))$, and $f_i$ embeds each of these $k$ connected components into $N(\partial C_2(R))$. We will think of the intersection with a preimage of $f_i(\tilde{P}(i)) \cap N(\partial C_2(R))$ as the sum of the intersections with the preimages of the $f_i(C_{i,j})$, and argue with covers of $\tilde{P}(i)$ rather than covers of its image. So this latter issue is not a big one—if we do not require the boundary conditions. We keep this in mind, and we no longer discuss this issue.

The first issue is more serious. We want the boundaries of our propagating chains to be equal to $p_{\tau}|_{\partial C_2(R)}(X_i)$ for some $X_i \in S^2$. Recall that $p_{\tau}$ also denotes a regular extension of $p_{\tau}$ on $N(\partial C_2(R))$, that a collar $[0, 8] \times \partial C_2'(R)$ of $\partial C_2'(R)$ in $N(\partial C_2(R))$ has been fixed, and that $p_{\tau}$ factors through the natural projection onto $\partial C_2'(R)$ in this collar. For an interval $I$ included in $[0, 8]$, set

$$N_I = I \times \partial C_2'(R).$$

For $a \in [1, 8]$, set

$$N_{[a,9]} = N(\partial C_2(R)) \setminus [0, a] \times \partial C_2'(R).$$

We will actually impose that our propagating chains intersect $N_{[7,9]}$ as $p_{\tau}|_{N_{[7,9]}}(X_i)$, by modifying our immersions $f_i$ provided by the construction of the end of Section 11.2, only on $N_{[4,7]}$.

We first describe appropriate choices for the $X_i$, to allow transversality near the boundaries.

Let $\Gamma$ of $\mathcal{D}_{3n}^e(\mathcal{L})$, let $E$ be a subset of $E(\Gamma)$. A condition on $(X_i)_{i \in \underline{3n}}$ is that $(X_i)_{i \in j_E(E)}$ is a regular (for the restriction to any stratum of $C(R, L; \Gamma)$) value of the map

$$\prod_{e \in E} p_{\tau} \circ p_e$$

from

$$C(\Gamma, E) = C(R, L; \Gamma) \cap \bigcap_{e \in E} p_e^{-1}(N(\partial C_2(R)))$$

to $(S^2)^{j_E(E)}$. According to the Morse–Sard theorem 1.4, this condition holds when $(X_i)_{i \in \underline{3n}}$ is in a dense subset of $(S^2)^{3n}$, which is furthermore open since $C(\Gamma, E)$ is compact. Thus, this condition holds for any of the finitely many
pairs $(\Gamma, E)$ as above, if $(X_i)_{i \in \underline{3n}}$ belongs to the intersection of the corresponding open dense subsets of $(S^2)^{3n}$, which is still open and dense.

Fix $(X_i)_{i \in \underline{3n}}$ in this open dense subset of $(S^2)^{3n}$. Now, we refer to the proof of Lemma 11.11 and adapt it to produce the desired family of propagating chains of Lemma 11.4. We fix a finite cover $(K_{i,j})_{j \in J(i)}$ of $\tilde{P}(i) \cap f_i^{-1}(C_2'(R) \cup N_{[0,7]}$). We assume that this cover contains a special element $K_{i,0} = \tilde{P}(i) \cap f_i^{-1}(N_{[5,7]})$ and that $K_{i,j}$ is a compact subset of $\tilde{P}(i) \cap f_i^{-1}(C_2'(R) \cup N_{[0,5]})$ for any $j \in J'(i) = J(i) \setminus \{0\}$.

When $j \in J'(i)$, $K_{i,j}$ is embedded in an open subspace $O_{i,j}$ of $\tilde{P}(i) \cap f_i^{-1}(C_2'(R) \cup N_{[0,6]})$. These $O_{i,j}$ are diffeomorphic to $\mathbb{R}^4$ via diffeomorphisms $\phi_{i,j}: O_{i,j} \to \mathbb{R}^4$, and we have bundle charts $(\psi_{i,j}: N(\tilde{P}(i))|_{O_{i,j}} \to \mathbb{R}^4 \times D^2)$, for $j \in J'(i)$, as in the proof of Lemma 11.11.

The bundle $N(K_{i,0})$ is trivialized by $p_\tau$ in the following way. Fix a small neighborhood $D_i$ of $X_i$ in $S^2$ diffeomorphic to the standard disk $D^2$ and a diffeomorphism $\psi_{D,i}$ from $D_i$ to $D^2$. Without loss of generality, assume that $N(K_{i,0}) = f_i^{-1}(p_\tau^{-1}(D_i) \cap N_{[5,7]})$, and identify $N(K_{i,0})$ with $K_{i,0} \times D^2$ so that the projection onto $D^2$ may be expressed as $p_{D^2} = \psi_{D,i} \circ p_\tau \circ f_i$.

The space $\mathcal{H}_i$ is now the space of smooth diffeomorphisms of $N(\tilde{P}(i))$ that are isotopic to the identity map, that fix a neighborhood of $\partial N(\tilde{P}(i))$ and a neighborhood of $f_i^{-1}(f_i(N(\tilde{P}(i))) \cap (N_{[7,9]}))$ pointwise, and that map any fiber of $N(\tilde{P}(i))$ to itself. The space $\mathcal{H}_i$ is equipped with a distance similar to that described before Lemma 11.11.

We want to prove that the subspace of $\prod_{i \in \underline{3n}} \mathcal{H}_i$ consisting of the $(h_i)_{i \in \underline{3n}}$ such that the $f_i(h_i^{-1}(\tilde{P}(i)))$ are in general $3n$ position with respect to $L$, in the sense of Definition 11.3, is open and dense.

It is open as in the proof of Lemma 11.11.<fnref n="3" />

Moreover, for any $\Gamma$ of $\mathcal{D}_{\underline{3n}}^e(\mathcal{L})$, for any triple $(E_X, E_N, E_C)$ of pairwise disjoint subsets of $E(\Gamma)$, the subset $\mathcal{H}(\Gamma, E_X, E_N, E_C)$ of $\prod_{i \in \underline{3n}} \mathcal{H}_i$ such that the restriction of $p(\Gamma, E_N \cup E_C)$ to $C(R, L; \Gamma) \cap p(\Gamma, E_X)^{-1} \prod_{e \in E_X} (N_{[5,9]} \cap p_\tau^{-1}(X_{j_E(e)}))$ is transverse to

$$ \prod_{e \in E_N \cup E_C} \left( f_{j_E(e)} \left( h_{j_E(e)}^{-1} \left( \tilde{P}(j_E(e)) \right) \right) \right) $$

$$ p(\Gamma, E) = \prod_{e \in E} p_e : C(R, L; \Gamma) \to (C_2(R))^{j_E(E)} $$

to $\prod_{e \in E} f_{j_E(e)}(h_{j_E(e)}^{-1}(\tilde{P}(j_E(e))))$ along $\prod_{e \in E} p_e^{-1}(C_2'(R) \cup N_{[0,5]})$, then density could also be proved as in Lemma 11.11.

<footnote n="3">³If we ask only for transversality of the</footnote>
along

$$p(\Gamma, E_X)^{-1} \prod_{e \in E_X} (N_{[5,9]} \cap p_{\tau}^{-1}(X_{j_E(e)})) \cap p(\Gamma, E_N)^{-1} \prod_{e \in E_N} (N_{[5,9]}) \\ \cap p(\Gamma, E_C)^{-1} \prod_{e \in E_C} (C'_2(R) \cup N_{[0,5]})$$

is open.<fnref n="4" /> The $\mathcal{H}(\Gamma, E_X, \emptyset, E_C)$ are furthermore dense as in the proof of Lemma 11.11.

In order to prove Lemma 11.4, it suffices to prove that for any $\Gamma$ of $\mathcal{D}_{3n}^e(\mathcal{L})$, for any pair $(E_N, E_C)$ of disjoint subsets of $E(\Gamma)$, the subset $\mathcal{H}(\Gamma, \emptyset, E_N, E_C)$ of $\prod_{i \in \overline{3n}} \mathcal{H}_i$ is dense. To do that, we fix $(h_i)_{i \in \overline{3n}} \in \prod_{i \in \overline{3n}} \mathcal{H}_i$ and $\varepsilon \in ]0, 1[$, and we prove that there exists $(h'_i)_{i \in \overline{3n}} \in \mathcal{H}(\Gamma, \emptyset, E_N, \overline{E_C})$ such that $\max_{i \in \overline{3n}}(d(h_i, h'_i)) < \varepsilon$. There exists $\eta \in ]0, 1[$ such that the restriction of $h_i$ to

$$f_i^{-1}(f_i(N(\tilde{P}(i))) \cap N_{[7-2\eta,9]})$$

is the identity map for any $i \in j_E(E_N)$.

For $i \in J_E(E_N)$, our $h'_i$ will be constructed as some $h_{i,\eta,w} \circ h_i$. Let $\chi_\eta$ be a smooth map from $[4, 9]$ to $[0, 1]$ that maps $[5, 7 - 2\eta]$ to 1 and that maps the complement of $[5 - \eta, 7 - \eta]$ to 0. Recall our smooth map $\chi: D^2 \to [0, 1]$, which maps the disk of radius $\frac{1}{2}$ to 1 and the complement of the disk of radius $\frac{3}{4}$ to 0. For $w \in D^2$ define

$$h_{\eta,w}: \begin{cases} [4, 9] \times D^2 & \to D^2 \\ (t, v) & \mapsto v + \chi(v)\chi_\eta(t)w. \end{cases}$$

Define $h_{i,\eta,w} \in \mathcal{H}_i$, for $w$ sufficiently small, to coincide with the identity map outside

$$f_i^{-1}(f_i(N(\tilde{P}(i))) \cap N_{[5-\eta,7-\eta]}),$$

and with the map that sends $(p, v) \in (\tilde{P}(i) \cap f_i^{-1}(N_{\{t\}})) \times D^2$

$$\left( \left( \left( \tilde{P}(i) \cap f_i^{-1}(N_{[5-\eta, 7-\eta]}) \right) \times D^2 \right) = N(\tilde{P}(i)) \cap f_i^{-1}(N_{[5-\eta, 7-\eta]}) \right)$$

to $(p, h_{\eta,w}(t, v))$, for $t \in [5 - \eta, 7 - \eta]$.<fnref n="5" /> There exists $u \in ]0, 1[$ such that, as soon as $\|w\| < u$, $h_{i,\eta,w}$ is indeed a diffeomorphism and $d(h_{i,\eta,w} \circ h_i, h_i) < \varepsilon$.

<footnote n="4">⁴Our hypotheses on $(X_i)_{i \in \overline{3n}}$ guarantee that $C(R, L; \Gamma) \cap p(\Gamma, E_X)^{-1} \prod_{e \in E_X} (N_{[5,9]} \cap p_{\tau}^{-1}(X_{j_E(e)}))$ is a manifold.</footnote>

<footnote n="5">⁵For any $t \in [0, 9]$, we assume $f_i((\tilde{P}(i) \cap f_i^{-1}(N_{\{t\}})) \times D^2) \subset N_{\{t\}}$, without loss of generality.</footnote>
Fix $(h'_i)_{i \in J_E(E_C)}$ such that $d(h_i, h'_i) < \varepsilon$ and $(h'_i)_{i \in J_E(E_C)} \times (h_i)_{i \notin J_E(E_C)}$ is in the dense open set $\cap_{E_x \subseteq E_N} \mathcal{H}(\Gamma, E_x, \emptyset, E_C)$ (this does not impose anything on $(h_i)_{i \notin J_E(E_C)}$).

After reducing $u$, we may now assume that as soon as $\|w\| < u$, for any $E_x \subseteq E_N$, $p(\Gamma, E_C \cup E_x)$ is transverse to

$$ \prod_{e \in E_C} f_{j_E(e)} \left( \left( (h'_{j_E(e)})^{-1} (\tilde{P}(j_E(e))) \right) \right) \times \prod_{e \in E_x} f_{j_E(e)} \left( \left( (h_{j_E(e), \eta, w_{j_E(e)}} \circ h_{j_E(e)})^{-1} (\tilde{P}(j_E(e))) \right) \right) $$

along

$$ p(\Gamma, E_C)^{-1} \left( \prod_{e \in E_C} (C'_2(R) \cup N_{[0,5]}) \right) \cap p(\Gamma, E_x)^{-1} \left( \prod_{e \in E_x} (N_{[7-2\eta,9]}) \right) $$

since $h_{j_E(e)}^{-1}(\tilde{P}(j_E(e))) = p_{\tau}^{-1}(X_{j_E(e)})$ on $N_{[7-2\eta,9]}$. Furthermore, $p(\Gamma, E_C \cup E_N)$ is transverse to

$$ M \left( (h'_i)_{i \in J_E(E_C)}, (h_{i, \eta, w_i})_{i \in J_E(E_N)} \right) = \prod_{e \in E_C} \left( f_{j_E(e)} \left( \left( (h'_{j_E(e)})^{-1} (\tilde{P}(j_E(e))) \right) \right) \right) \times \prod_{e \in E_N} f_{j_E(e)} \left( \left( (h_{j_E(e), \eta, w_{j_E(e)}} \circ h_{j_E(e)})^{-1} (\tilde{P}(j_E(e))) \right) \right) $$

along

$$ p(\Gamma, E_C)^{-1} \left( \prod_{e \in E_C} (C'_2(R) \cup N_{[0,5]}) \right) \cap p(\Gamma, E_N)^{-1} \left( \prod_{e \in E_N} N_{[5,9]} \right), $$

if and only if, for any subset $E_x$ of $E_N$, the following condition $(*) (E_x)$ holds. $(*) (E_x) : p(\Gamma, E_C \cup E_N)$ is transverse to $M \left( (h'_i)_{i \in J_E(E_C)}, (h_{i, \eta, w_i})_{i \in J_E(E_N)} \right)$ along

$$ p(\Gamma, E_C)^{-1} \left( \prod_{e \in E_C} (C'_2(R) \cup N_{[0,5]}) \right) \cap p(\Gamma, E_x)^{-1} \left( \prod_{e \in E_x} N_{[7-2\eta,9]} \right) \cap p(\Gamma, E_N \setminus E_x)^{-1} \left( \prod_{e \in E_N \setminus E_x} N_{[5,7-2\eta]} \right). $$
Let $\mathring{D}_u$ denote the open disk of $\mathbb{R}^2$ centered at 0 of radius $u$. Our former hypotheses guarantee transversality of $p(\Gamma, E_C \cup E_x)$ as soon as the $\|w_i\|$ are smaller than $u$ for $i \in E_x$. So the condition $(*) (E_x)$ is realized as soon as $(w_i)_{i \in j_E(E_N)}$ is in an open dense subset $\mathcal{D}(E_x)$ of $\mathring{D}_u^{j_E(E_N)}$. Thus, we have the desired transversality when $(w_i)_{i \in j_E(E_N)}$ is in the intersection of the open dense subsets $\mathcal{D}(E_x)$ over the subsets $E_x$ of $E_N$. $\square$

## 11.4 More on forms dual to transverse propagating chains

Though Lemma 11.7 is not surprising, we prove it and refine it in this section. We use its refinement in Chapter 17. Recall the notation of Definition 11.6, and let $D_\varepsilon$ (resp. $\mathring{D}_\varepsilon$) denote the closed (resp. open) disk of $\mathbb{C}$ centered at 0 with radius $\varepsilon$.

**Lemma 11.12.** Recall that our configuration spaces are equipped with Riemannian metrics. Let $(\tilde{R}, \tau)$ be an asymptotic rational homology $\mathbb{R}^3$. Let $L: \mathcal{L} \to \tilde{R}$ be a link in $\tilde{R}$. Let $n \in \mathbb{N}$, and let $(P(i))_{i \in 3n}$ be a family of propagating chains of $(C_2(R), \tau)$ in general $3n$ position with respect to $L$. For any $\varepsilon > 0$, there exists $\eta > 0$ such that for any $i \in 3n$, for any $\Gamma \in \mathcal{D}_{3n}^e(\mathcal{L})$, and for any $e \in E(\Gamma)$ with associated restriction map $p_e: C(R, L; \Gamma) \to C_2(R)$, we have

$$p_e^{-1}(N_\eta(P(j_E(e)))) \subset N_\varepsilon(p_e^{-1}(P(j_E(e))))$$

**PROOF:** Of course, it is enough to prove the lemma for a fixed $(\Gamma, e)$. Set $i = j_E(e)$. The compact $p_e(C(R, L; \Gamma) \setminus N_\varepsilon(p_e^{-1}(P(i)))))$ does not meet $P(i)$. So there exists $\eta > 0$ such that this compact does not meet $N_\eta(P(i))$ either. This implies $p_e^{-1}(N_\eta(P(i))) \subset N_\varepsilon(p_e^{-1}(P(i)))$. $\square$

**Lemma 11.13.** Let $\Gamma \in \mathcal{D}_{3n}^e(\mathcal{L})$. Assume that the hypotheses of Lemma 11.12 are satisfied. Then the intersection in $C(R, L; \Gamma)$ over the edges $e$ of $E(\Gamma)$ of the codimension 2 rational chains $p_e^{-1}(P(j_E(e)))$ is a finite set $I_S(\Gamma, (P(i))_{i \in 3n})$. Furthermore, for any $\varepsilon > 0$, there exists $\eta > 0$ such that we have

$$\bigcap_{e \in E(\Gamma)} p_e^{-1}(N_\eta(P(j_E(e)))) \subset N_\varepsilon(I_S(\Gamma, (P(i))_{i \in 3n}))$$

for any $\Gamma \in \mathcal{D}_{3n}^e(\mathcal{L})$. So, for any family $(\omega(i))_{i \in 3n}$ of propagating forms of $(C_2(R), \tau)$ $\eta$-dual to the $P(i)$, the form $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e)))$ is supported in $N_\varepsilon(I_S(\Gamma, (P(i))_{i \in 3n}))$. $\square$
Moreover, if $N_{\varepsilon}(I_S(\Gamma, (P(i))))$ is a disjoint union over the points $x$ of $I_S(\Gamma, (P(i)))$ of the $N_{\varepsilon}(x)$, then the integral over $N_{\varepsilon}(x)$ of $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e)))$ is the rational intersection number of the rational chains $p_e^{-1}(P(j_E(e)))$ at $x$. When all the $P(j_E(e))$ are embedded manifolds with coefficient 1 near $p_e(x)$, this intersection number is the sign of $x$ with respect to an orientation $o(\Gamma)$ of $\Gamma$.

**PROOF:** Again, it suffices to prove the lemma for a fixed $\Gamma \in \mathcal{D}_{3n}^e(\mathcal{L})$. We refer to the description of the image under $p(\Gamma) = \prod_{e \in E(\Gamma)} p_e$ of an intersection point $x$ after Definition 11.3.

Fix such an $x$. For each edge $e$, $p_e(x)$ sits inside a nonsingular open 4-dimensional smooth ball $\delta_e$ of a smooth piece $\Delta_{j_E(e),k}$ of $P(j_E(e))$. Consider a tubular neighborhood $N_u(\delta_e)$ whose fibers are disks $D_\theta$ orthogonal to $\delta_e$ of radius $\theta$. The bundle $N_u(\delta_e)$ is isomorphic to $\delta_e \times D_\theta$, with respect to a trivialization of $N_u(\delta_e)$. Another trivialization would compose the diffeomorphism from $N_u(\delta_e)$ to $\delta_e \times D_\theta$ by a map $(x, v) \mapsto (v, \phi(x)(v))$ for some $\phi: \delta_e \to SO(2)$.

The projection $p_e(x)$ may sit simultaneously in different nonsingular 4-dimensional smooth parts $\Delta_{j_E(e),k}$ of $P(j_E(e))$. Let $K(e, x)$ be the finite set of components $\Delta_{j_E(e),k}$ of $P(j_E(e))$ such that $p_e(x) \in \Delta_{j_E(e),k}$. We first focus on one element of $K(e, x)$ for each $e$, and next take the sum over all the choices in $\prod_{e \in E(\Gamma)} K(e, x)$ multiplied by the products of the coefficients of the elements of $K(e, x)$ in the rational chains $P(j_E(e))$. Similarly, our forms $\eta$-dual to the $P(i)$ are thought of and constructed as linear combinations of forms $\eta$-dual to the elements of $K(e, x)$.

Without loss of generality, assume that $\varepsilon$ is small enough so that we have

$$p_e(N_{\varepsilon}(x)) \subset \delta_e \times D_{\theta}$$

for any edge $e$ of $\Gamma$ and for any $\delta_e = \delta_{e,k}$ associated to an element $\Delta_{j_E(e),k}$ of $K(e, x)$, and so that $p_e(N_{\varepsilon}(x))$ does not meet the components $\Delta_{j_E(e),k}$ of $P(j_E(e))$ that are not in $K(e, x)$. Reduce $\varepsilon$ and choose $\eta < \theta$ small enough so that $p_e(N_{\varepsilon}(x))$ does not meet the neighborhoods $N_{\eta}(\Delta_{j_E(e),k})$ of these components, either.

Let $p_{D_{\theta}}: \delta_e \times D_{\theta} \to D_{\theta}$ denote the natural projection. Let $\omega_{\eta}$ be a volume-one form supported on $\tilde{D}_{\eta}$. Forms $\eta$-dual to $P(j_E(e))$ can be constructed by patching forms $(p_{D_{\theta}})^*(\omega_{\eta})$ (multiplied by the coefficients of the $\Delta_{j_E(e),k}$) together, as in Lemma B.4. Conversely, for any form $\omega(j_E(e))$ $\eta$-dual to a piece $\Delta_{j_E(e),k}$ of $P(j_E(e))$ that contains $\delta_e$, there exists a one-form $\alpha_e$ on $\delta_e \times D_{\theta}$, such that $\omega(j_E(e)) = p_{D_{\theta}}^*(\omega_{\eta}) + d\alpha_e$ on $\delta_e \times D_{\eta}$. Then we have

$$\int_{\{x \in \delta_e\} \times \partial D_{\eta}} \alpha_e = \int_{\{x \in \delta_e\} \times D_{\eta}} \omega(j_E(e)) - p_{D_{\theta}}^*(\omega_{\eta}) = 0.$$
So $\alpha_e$ is exact on $\delta_e \times (D_\theta \setminus D_\eta)$, and $\alpha_e$ can and will be assumed to be supported on $\delta_e \times D_\eta$.

In the neighborhood $N_\epsilon(x)$ of $x$, $\prod_{e \in E(\Gamma)} p_{D_\theta} \circ p_e$ is a local diffeomorphism around $x$. Without loss of generality, assume that $\eta$ and $\epsilon$ are small enough so that

$$ \Pi_p = \prod_{e \in E(\Gamma)} p_{D_\theta} \circ p_e : N_\epsilon(x) \to D_\theta^{E(\Gamma)} $$

restricts to a diffeomorphism from $\Pi_p^{-1}(D_{2\eta}^{E(\Gamma)})$ to $D_{2\eta}^{E(\Gamma)}$, for each $x$ (and for each choice in $\prod_{e \in E(\Gamma)} K(e, x)$). If $\prod_{e \in E(\Gamma)} K(e, x)$ has one element, and if the coefficient of the element of $K(e, x)$ in $P(j_E(e))$ is 1 for any edge $e$, then we have

$$ \int_{N_\epsilon(x)} \bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e))) = \int_{N_\epsilon(x)} \bigwedge_{e \in E(\Gamma)} p_e^*(p_{D_\theta}^*(\omega_\eta)). $$

Indeed, changing one $\omega(j_E(e))$ to $(p_{D_\theta})^*(\omega_\eta)$ amounts to add the integral obtained by replacing $\omega(j_E(e))$ by $d\alpha_e$. Since all the forms are closed, this latter integral is the integral over $\Pi_p^{-1}(\partial(D_{2\eta}^{E(\Gamma)}))$ of the form obtained by replacing $d\alpha_e$ by $\alpha_e$, which is zero since the whole form is supported in $\Pi_p^{-1}(D_\eta^{E(\Gamma)})$. Therefore, the integral is the sign of the intersection point $x$ with respect to the given orientation and coorientations.

The open neighborhoods $N_\epsilon(x)$ may be assumed to be disjoint from each other for distinct $x$. Consequently, since $C(R, L; \Gamma)$ is compact, the set of intersection points $x$ is finite. Consider the complement $C^c(\eta_0)$ in $C(R, L; \Gamma)$ of the union over the intersection points $x$ of the $N_\epsilon(x)$. Since $p_{e_1}^{-1}(P(j_E(e_1)))$ does not meet $\bigcap_{e \in E(\Gamma) \setminus \{e_1\}} p_e^{-1}(P(j_E(e)))$ in $C^c(\eta_0)$, there is an $\epsilon_1 > 0$ such that $\overline{N_{\epsilon_1}}(p_{e_1}^{-1}(P(j_E(e_1))))$ does not meet $\bigcap_{e \in E(\Gamma) \setminus \{e_1\}} p_e^{-1}(P(j_E(e)))$ either in $C^c(\eta_0)$. Iterating, we find $\epsilon_2 > 0$ such that

$$ C^c(\eta_0) \cap \bigcap_{e \in E(\Gamma)} N_{\epsilon_2} \left( p_e^{-1} \left( P(j_E(e)) \right) \right) = \emptyset. $$

According to Lemma 11.12, $\eta$ can be reduced so that $p_e^{-1}(N_\eta(P(i))) \subset N_{\epsilon_2}(p_e^{-1}(P(i)))$ for any $i$. Then $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e)))$ is supported where we want it to be. $\square$

Lemma 11.7 follows. $\square$

Theorem 11.8 is now proved. $\square$
## 11.5 A discrete definition of the anomaly $\beta$

In this section, we give a discrete definition of the anomaly $\beta$ and mention a few recent results of Kévin Corbineau on $\beta_3$.

**Lemma 11.14.** Let $n \in \mathbb{N}$. Let $\Gamma \in \mathcal{D}_n^c$. Recall the compactification $\mathcal{S}_{\Gamma(\Gamma)}(\mathbb{R}^3)$ of $\tilde{\mathcal{S}}_{\Gamma(\Gamma)}(\mathbb{R}^3)$ from Theorem 8.11. For any edge $e = j_E^{-1}(i)$ of $\Gamma$, we have a canonical projection

$$p_e: B^3 \times \mathcal{S}_{\Gamma(\Gamma)}(\mathbb{R}^3) \to B^3 \times S^2.$$

Let $i \in \underline{3n}$. When $\Gamma$ is fixed, set $p_i = p_{j_E^{-1}(i)}$. For any $a_i \in S^2$, define the following cooriented chains of $B^3 \times \mathcal{S}_{\Gamma(\Gamma)}(\mathbb{R}^3)$:

$$A(\Gamma, i, a_i) = p_i^{-1}(B^3 \times \{a_i\}),$$

$$B(\Gamma, i, a_i) = p_i^{-1}\left(\left\{\cup_{m \in B^3} (m, \rho(m)(a_i))\right\}\right),$$

and

$$H(\Gamma, i, a_i) = p_i^{-1}(G(a_i)),$$

where $\rho$ is introduced in Definition 4.4, and the chain $G(a_i)$ of $B^3 \times S^2$ is introduced in Lemma 4.12. The codimension of $A(\Gamma, i, a_i)$ and $B(\Gamma, i, a_i)$ is 2, while the codimension of $H(\Gamma, i, a_i)$ is 1. An element $(a_1, \dots, a_{3n})$ of $(S^2)^{3n}$ is $\beta_n$-admissible if for $h \in \underline{3n}$ and for any $\Gamma \in \mathcal{D}_n^c$, the intersection of the $A(\Gamma, i, a_i)$ for $i \in \underline{h-1}$, the $B(\Gamma, i, a_i)$ for $i \in \underline{3n} \setminus \underline{h}$, and $H(\Gamma, h, a_h)$ is transverse. Then the sets of elements of $(S^2)^{3n}$ that are $\beta_n$-admissible is an open dense subset of $(S^2)^{3n}$.

**PROOF:** The principle of the proof is the same as the proof of Proposition 11.1. See also Section 11.3. This lemma is proved in detail in [Cor16]. □

<figure><img src="image_90.png" /></figure>

**Proposition 11.15.** Recall the orientation of $\tilde{\mathcal{S}}_{\Gamma(\Gamma)}(\mathbb{R}^3)$ of Lemma 9.3. For any $\beta_n$-admissible element $(a_1, \dots, a_{3n})$ of $(S^2)^{3n}$, we have

$$\beta_n = \sum_{h=1}^{3n} \sum_{\Gamma \in \mathcal{D}_n^c} \frac{1}{(3n)!2^{3n}} I(\Gamma, h) [\Gamma],$$

with

$$I(\Gamma, h)[\Gamma] = \left\langle \cap_{i=1}^{h-1} A(\Gamma, i, a_i), H(\Gamma, h, a_h), \cap_{i=h+1}^{3n} B(\Gamma, i, a_i) \right\rangle_{B^3 \times \mathcal{S}_{\Gamma(\Gamma)}(\mathbb{R}^3)} [\Gamma].$$
PROOF: For $a \in S^2$ and $t \in [0, 1]$, define the following chain

$$G(a, t) = [0, t] \times B^3 \times \{a\} + (\{t\} \times G(a)) + \{ (u, m, \rho(m)(a)) : u \in [t, 1], m \in B^3 \}$$

of $[0, 1] \times B^3 \times S^2$. Let $(t_i)_{i \in \underline{3n}}$ be a strictly decreasing sequence of $]0, 1[$. Let $\Gamma \in \mathcal{D}_n^c$. For $i \in \underline{3n}$, let $p_i$ also denote the canonical projection associated to $e = j_E^{-1}(i)$ from $[0, 1] \times B^3 \times S^2$ to $[0, 1] \times B^3 \times S^2$.

If $(a_1, \dots, a_{3n})$ is $\beta_n$-admissible, then for any $\Gamma \in \mathcal{D}_n^c$, the intersection of the $p_i^{-1}(G(a_i, t_i))$ is transverse and equal to

$$\sqcup_{h=1}^{3n} \{t_h\} \times \left( \bigcap_{i=1}^{h-1} A(\Gamma, i, a_i) \cap H(\Gamma, h, a_h) \cap \left( \bigcap_{i=h+1}^{3n} B(\Gamma, i, a_i) \right) \right).$$

Indeed, it is clear that the intersection may be expressed as above at the times $t \in \{t_h : h \in \underline{3n}\}$. Since $(a_1, \dots, a_{3n})$ is $\beta_n$-admissible, this intersection is transverse at these times. So it does not intersect the boundaries of the $H(\Gamma, h, a_h)$. Therefore, there is no intersection in $([0, 1] \setminus \{t_h : h \in \underline{3n}\}) \times B^3 \times S^2$.

Then for any $\alpha > 0$, there exist closed 2-forms $\tilde{\omega}(i)$ on $[0, 1] \times \mathbb{R}^3 \times S^2$, as in Proposition 10.7, applied when $\tau_0 = \tau_s$ and $\tau_1 = \tau_0 \circ \psi_{\mathbb{R}}(\rho)$ on $UB^3$, such that $\tilde{\omega}(i)$ is $\alpha$-dual to $G(a_i, t_i)$, for any $i$.

Theorem 4.5 yields $p_1(\tau_0 \circ \psi_{\mathbb{R}}(\rho)) - p_1(\tau_0) = 2 \deg(\rho) = 4$. Therefore, Proposition 10.7 implies

$$\beta_n = \sum_{\Gamma \in \mathcal{D}_n^c} \frac{1}{(3n)!2^{3n}} \int_{[0,1] \times \tilde{S}_{V(\Gamma)}(TB_R)} \bigwedge_{e \in E(\Gamma)} p_e^*(\tilde{\omega}(j_E(e))) [\Gamma],$$

where $\bigwedge_{e \in E(\Gamma)} p_e^*(\tilde{\omega}(j_E(e))) = \bigwedge_{i=1}^{3n} p_i^*(\tilde{\omega}(i))$.

As in Section 11.4, for $\alpha$ small enough, $\int_{[0,1] \times \tilde{S}_{V(\Gamma)}(TB^3)} \bigwedge_{i=1}^{3n} p_i^*(\tilde{\omega}(i))$ is the algebraic intersection of the $p_i^{-1}(G(a_i, t_i))$.

For the signs, note that the coorientation of $\{t_h\} \times G(a_h)$ in $[0, 1] \times B^3 \times S^2$ is represented by the orientation of $[0, 1]$, followed by the coorientation of $G(a_h)$ in $B^3 \times S^2$. $\square$

In his Ph. D. thesis [Cor16, Théorème 2.15], Kévin Corbineau obtained the following simplified expression for $\beta_3$.

**Theorem 11.16.** For $j \in \underline{n}$, set

$$H_h(\Gamma, j, a_j) = p_j^{-1}(G_h(a_j)),$$
where the chain $G_h(a_j)$ of $B^3 \times S^2$ is introduced in Lemma 4.12. Let $\mathcal{D}_3^c(T)$ be the set of numbered graphs in $\mathcal{D}_3$ isomorphic to

<figure><img src="image_91.png" /></figure>

For any element $\beta_3$-admissible $(a_1, \dots, a_9)$ of $(S^2)^9$, we have

$$ \beta_3 = \sum_{j=2}^{8} \sum_{\Gamma \in \mathcal{D}_3^c(T)} \frac{1}{(9)!2^9} I_h(\Gamma, j) [\Gamma], $$

with

$$ I_h(\Gamma, j) [\Gamma] = \langle \cap_{i=1}^{j-1} A(\Gamma, i, a_i), H_h(\Gamma, j, a_j), \cap_{i=j+1}^{9} B(\Gamma, i, a_i) \rangle_{B^3 \times \mathcal{S}_{V(\Gamma)}(\mathbb{R}^3)} [\Gamma]. $$

The Ph. D. thesis of Kévin Corbineau also contains an algorithm to compute $\beta_3$.
<figure><img src="image_92.png" /></figure>
# Part III
## Functoriality
<figure><img src="image_93.png" /></figure>
Recall that $D_1$ denotes the closed disk of $\mathbb{C}$ centered at 0 with radius 1. In this book, a *rational homology cylinder* (or *Q-cylinder*) is a compact oriented 3-manifold, whose boundary neighborhood is identified with a boundary neighborhood $N(\partial(D_1 \times [0, 1]))$ of $D_1 \times [0, 1]$, and which has the same rational homology as a point.

Roughly speaking, *q-tangles* are parallelized cobordisms between limit planar configurations of points up to dilation and translation in rational homology cylinders. We describe the category of *q*-tangles and its structures precisely in Section 13.1. Framed links in rational homology spheres are particular *q*-tangles. They are cobordisms between empty configurations.

In this third part of the book, we define a functorial extension to $q$-tangles of the invariant $\mathcal{Z}^f$ of framed links in $\mathbb{Q}$-spheres defined in Section 7.6, and we prove that it has a lot of useful properties. These properties are listed in Theorem 13.12. They ensure that $\mathcal{Z}^f$ is a functor, which behaves naturally with respect to other structures of the category of $q$-tangles, such as cabling or duplication. They allow one to reduce the computation of $\mathcal{Z}^f$ for links to its computation for elementary pieces of the links.

Section 12.1 introduces particular $q$-tangles, for which the involved planar configurations are injective. Section 12.3 introduces other particular $q$-tangles, for which the involved planar configurations are corners of the Stasheff polyhedra of Example 8.3. In Chapter 12, we define $\mathcal{Z}^f$ for these particular $q$-tangles without proofs. We also state a functoriality result, a monoidality result, and a duplication property, under simple hypotheses, to introduce the involved structures and motivate their introduction. These results are just particular cases of Theorem 13.12.

In Chapter 13, we state our general Theorem 13.12, and we describe our strategy towards a consistent definition of $\mathcal{Z}^f$ for general $q$-tangles in Section 13.2. Our proofs involve convergence results, which rely on intricate compactifications of configuration spaces described in Chapter 14. In Chapter 15, we study $\mathcal{Z}^f$ as a holonomy for the $q$-tangles that are paths in spaces of planar configurations. In Chapter 16, we introduce discretizable versions of $\mathcal{Z}^f$. We use these discretizable versions in the proofs of some important properties of $\mathcal{Z}^f$ given in Chapter 17. The consistency of our strategy for the definition of $\mathcal{Z}^f$ is shown in Chapters 14, 15, and 16. The proof of Theorem 13.12 will be finished in Chapter 17.

This functoriality part contains a generalization of results of Sylvain Poirier [Poi00], who constructed the functor $\mathcal{Z}^f$ and proved Theorem 13.12 for combinatorial $q$-tangles of $\mathbb{R}^3$. We recall his results in Section 12.4.
<figure><img src="image_94.png" /></figure>
# Chapter 12

## A first introduction to the
functor $\mathcal{Z}^f$

In Section 12.1, we extend the definition of the invariant $\mathcal{Z}$ of Theorem 7.20 to long tangle representatives as in Figure 1.12. Then we define the framed version $\mathcal{Z}^f$ of $\mathcal{Z}$ and state that it is multiplicative under the allowed vertical compositions in Section 12.2.

In Section 12.3, we state that $\mathcal{Z}^f$ reaches a limit with nice cabling properties when some vertical infinite strands of the long tangle representatives approach each other. We thus define the restriction of $\mathcal{Z}^f$ to combinatorial $q$-tangles, which are parallelized cobordisms between limit configurations on the real line in rational homology cylinders. This definition is due to Sylvain Poirier [Poi00] when the involved rational homology cylinder is the standard one $D_1 \times [0, 1]$. In Section 12.4, we list sufficiently many properties of the Poirier restriction of $\mathcal{Z}^f$ to characterize the restriction of $\mathcal{Z}^f$ to combinatorial $q$-tangles, in terms of the anomaly $\alpha$.

### 12.1 Extension of $\mathcal{Z}$ to long tangles

View $\mathbb{R}^3$ as $\mathbb{C} \times \mathbb{R}$, where $\mathbb{C}$ is horizontal and $\mathbb{R}$ is vertical, oriented upwards. For a rational homology cylinder $C$, $\tilde{R}(C)$ denotes the asymptotically standard $\mathbb{Q}$-homology $\mathbb{R}^3$ obtained by replacing the standard cylinder $C_0 = D_1 \times [0, 1]$ in $\mathbb{R}^3$ by $C$.

**Definition 12.1.** A long tangle representative (or LTR for short) in $\tilde{R}(C)$ is an embedding $L: \mathcal{L} \hookrightarrow \tilde{R}(C)$ of a one-manifold $\mathcal{L}$, as in Figure 1.12, such that
• $L(\mathcal{L})$ intersects the closure $\check{\mathcal{C}}_0^c$ of the complement of $\mathcal{C}_0$ in $\mathbb{R}^3$ as

$$L(\mathcal{L}) \cap \check{\mathcal{C}}_0^c = (c^-(B^-) \times ]-\infty, 0]) \cup (c^+(B^+) \times [1, \infty]$$

for two finite sets $B^-$ and $B^+$ and two injective maps $c^-$: $B^-$ $\hookrightarrow$ Int($D_1$) and $c^+$: $B^+$ $\hookrightarrow$ Int($D_1$), which are called the bottom configuration and the top configuration of $L$, respectively, and

• $L(\mathcal{L}) \cap \mathcal{C}$ is a compact one-manifold whose unoriented boundary is $(c^-(B^-) \times \{0\}) \cup (c^+(B^+) \times \{1\})$.

For a $3n$-numbered degree $n$ Jacobi diagram with support $\mathcal{L}$ without looped edges, let $\check{C}(\check{R}(\mathcal{C}), L; \Gamma)$ be its configuration space defined as in Section 7.1.<fnref n="1" /> The univalent vertices on a strand, which is the image under $L$ of an open connected component of $\mathcal{L}$ (diffeomorphic to $\mathbb{R}$), move along this whole long component, as in Figure 12.1.

<figure><img src="image_95.png" /><figcaption>Figure 12.1: A (black) Jacobi diagram $\Gamma$ on the domain of an LTR $L$ and a configuration $c$ of $\check{C}(\check{R}(\mathcal{C}), L; \Gamma)$</figcaption></figure>

For any $i \in \underline{3n}$, let $\omega(i)$ be a propagating form of $(C_2(R(\mathcal{C})), \tau)$. Let $o(\Gamma)$ be a vertex-orientation of $\Gamma$. As in Section 7.2, define

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

where $(\check{C}(\check{R}(\mathcal{C}), L; \Gamma), o(\Gamma))$ denotes the manifold $\check{C}(\check{R}(\mathcal{C}), L; \Gamma)$, equipped with the orientation induced by the vertex-orientation $o(\Gamma)$ and by the edge-orientation of $\Gamma$, as in Corollary 7.2, and

$$\underline{I(\mathcal{C}, L, \Gamma, (\omega(i))_{i \in \underline{3n}})}[\Gamma] = \underline{I(\mathcal{C}, L, \Gamma, o(\Gamma), (\omega(i))_{i \in \underline{3n}})}[\Gamma, o(\Gamma)].$$

<footnote n="1">¹The only differences are that $\mathcal{L}$ is not necessarily a disjoint union of circles and that $\check{C}(\check{R}(\mathcal{C}), L; \Gamma)$ was denoted by $\check{C}(R(\mathcal{C}), L; \Gamma)$.</footnote>
**Theorem 12.2.** *The above integral is convergent.*

We prove this theorem in Section 14.2. See Lemma 14.24. Again, its proof involves appropriate compactifications $C(\tilde{R}(\mathcal{C}), L; \Gamma)$ of the configuration spaces $\tilde{C}(\tilde{R}(\mathcal{C}), L; \Gamma)$. The compactifications are more complicated in this case. We study them in Chapter 14.

As an example, let us compute $I(\mathcal{C}, L, \Gamma, o(\Gamma), (\omega(i))_{i \in \underline{3n}})$ when

* $\mathcal{C} = \mathcal{C}_0 = D_1 \times [0, 1]$,

* $L$ is an LTR $\mathcal{L}$ whose bottom and top configurations coincide and map $B^- = B^+$ to $\{-\frac{1}{2}, \frac{1}{2}\}$,

* $L(\mathcal{L}) \cap \mathcal{C}_0$ projects to $\mathbb{R}^2$ as $\mathcal{L}$,

* $(\Gamma, o(\Gamma))$ is the vertex-oriented diagram $\mathcal{L}$ whose chord is oriented and numbered, and

* the propagating forms $\omega(i)$ pull back through $p_{S^2}: C_2(S^3) \to S^2$.

**Lemma 12.3.** *We have*

$$I \left( \mathcal{C}_0, \mathcal{L}, \mathcal{L}, (p_{S^2}^*(\omega_{i,S}))_{i \in \underline{3}} \right) = I \left( \mathcal{C}_0, \mathcal{L}, \mathcal{L}, (p_{S^2}^*(\omega_{i,S}))_{i \in \underline{3}} \right) = 1$$

for any arbitrary numbering of the edge of the involved Jacobi diagram and any choice of volume-one forms $\omega_{i,S}$ of $S^2$.

**PROOF:** Let us compute

$$I \left( \mathcal{L}, \mathcal{L}, (p_{S^2}^*(\omega_{i,S}))_{i \in \underline{3}} \right).$$

The configuration space

$$\tilde{C} = \left( \tilde{C}(\mathbb{R}^3, \mathcal{L}; \Gamma), o(\Gamma) \right)$$

is naturally diffeomorphic to $]-\infty, \infty[ \times ]-\infty, \infty[$, where the first factor parametrizes the height of the vertex on the left strand oriented from bottom to top and the second one parametrizes the height of the vertex on the right strand.
The map $p_{S^2}$ maps $]-\infty, 0]^2$ and $[1, \infty]^2$ to the vertical circle through the horizontal real direction. Therefore, the integral of $p_{S^2}^*(\omega_{i,S})$ vanishes there, and the integral of $p_{S^2}^*(\omega_{i,S})$ over $\tilde{C}$ is the integral of $p_{S^2}^*(\omega_{i,S})$ over $]-\infty, \infty]^2 \setminus (]-\infty, 0[ \cup ]1, \infty]^2$ or over $[-\infty, \infty]^2 \setminus ([-\infty, 0[ \cup ]1, \infty]^2)$, to which $p_{S^2}^*(\omega_{i,S})$ extends naturally. The boundary of this domain, which is drawn in Figure 12.2, is mapped to the vertical half circle between the two vertical directions north $\vec{N}$ and south $(-\vec{N})$ through the horizontal east direction $\vec{E}$ towards the right.

<figure><img src="image_96.png" /><figcaption>Figure 12.2: Images of boundary points of $[-\infty, \infty]^2 \setminus ([-\infty, 0[ \cup ]1, \infty]^2)$ under $p_{S^2}$. The right-hand side shows the computation for the very thick part $[-\infty, 0] \times \{0\}$.</figcaption></figure>

Thus, $p_{S^2}(\partial C(\mathbb{R}^3, \mathcal{H}; \Gamma))$ is algebraically trivial (as in the beginning of Section 7.5), and the differential degree of $p_{S^2}$ is constant on the set of regular values of $p_{S^2}$, according to Lemma 2.3. It can be computed as in Subsection 1.2.3 at the vector that points towards the reader. It is equal to one. Thus we have

$$I\left(\mathcal{H}, \mathcal{H}, (p_{S^2}^*(\omega_{i,S}))_{i \in \mathbb{S}}\right) = 1$$

for any arbitrary numbering of the edge of $\Gamma = \mathcal{H}$ and for any choice of volume-one forms $\omega_{i,S}$ of $S^2$. Similarly, for the opposite orientation of the edge of $\Gamma$, we have

$$I\left(\mathcal{H}, \mathcal{H}, (p_{S^2}^*(\omega_{i,S}))_{i \in \mathbb{S}}\right) = 1.$$

□

**Definition 12.4.** A *parallelization* of $C$ is a parallelization of $\tilde{R}(C)$ that agrees with the standard parallelization of $\mathbb{R}^3$ outside $C$. A *parallelized rational homology cylinder* $(C, \tau)$ is a rational homology cylinder equipped with such a parallelization.
The following lemma shows other important examples of computations.

**Lemma 12.5.** Let $K: \mathbb{R} \hookrightarrow \tilde{R}(C)$ be a component of $L$. Let $\tau$ be a parallelization of $C$ (which is standard near $\partial C$ by Definition 12.4). For any $i \in \mathbb{3}$, let $\omega(i)$ and $\omega'(i)$ be propagating forms of $(C_2(R(C)), \tau)$, which restrict to $\partial C_2(R(C))$ as $p_{\tau}^*(\omega(i)_{S^2})$ and $p_{\tau}^*(\omega'(i)_{S^2})$, respectively. Let $\eta(i)_{S^2}$ be a one-form on $S^2$ such that $\omega'(i)_{S^2} = \omega(i)_{S^2} + d\eta(i)_{S^2}$. Then when $K$ goes from bottom to top or from top to bottom, we have

$$ I \left( k \hat{\zeta}^K, (\omega'(i))_{i \in \mathbb{3}} \right) - I \left( k \hat{\zeta}^K, (\omega(i))_{i \in \mathbb{3}} \right) = \int_{U+K} p_{\tau}^*(\eta(k)_{S^2}) \\ = \int_{p_{\tau}(U+K)} \eta(k)_{S^2}. $$

When $K$ goes from bottom to bottom (resp. from top to top), let $S(K)$ be the half-circle from $-\vec{N}$ to $\vec{N}$ (resp. from $\vec{N}$ to $-\vec{N}$) through the horizontal direction from the initial vertical half-line of $K$ (the first encountered one) to the final one, then we have

$$ I \left( k \hat{\zeta}^K, (\omega'(i))_{i \in \mathbb{3}} \right) - I \left( k \hat{\zeta}^K, (\omega(i))_{i \in \mathbb{3}} \right) = \int_{p_{\tau}(U+K) \cup S(K)} \eta(k)_{S^2}. $$

In particular, $I(k\hat{\zeta}^K, (\omega(i))_{i \in \mathbb{3}})$ depends only on $\omega(k)_{S^2}$. It is also denoted by $I(\hat{\zeta}^K, \omega(k)_{S^2})$.

**PROOF:** In any case, the configuration space $\tilde{C}(\tilde{R}(C), L; \hat{\zeta}^K)$ is identified naturally with the interior of a triangle, as in the left part of Figure 12.3. When $K$ goes from bottom to top, $p_{\tau}$ extends smoothly to the triangle, as a map that sends the horizontal side and the vertical side to $\vec{N}$. Conclude as in Lemmas 7.13 and 7.15. The case in which $K$ goes from top to bottom is similar.

<figure><img src="image_97.png" /><figcaption>Figure 12.3: Compactifications of configuration spaces for the proof of Lemma 12.5</figcaption></figure>

Let us study the case in which $K$ goes from top to top. Let $d_1 = -\{z_1\} \times [1, \infty]$ and $d_2 = \{z_2\} \times [1, \infty]$ denote the vertical half-lines of $K$ above $C$, where $d_1$ is before $d_2$. View $K$ as a path composition $d_1(K \cap (D_1 \times [0, 1]))d_2$ and parametrize

$$ K = d_1 \stackrel{d_2}{\underset{-}{\leftarrow}} \text{ by } m: \begin{array}{rcl} ]0, 1[ & \rightarrow & K \\ t \in ]0, 1/3[ & \mapsto & (z_1, 1/(3t)) \\ t \in [2/3, 1[ & \mapsto & (z_2, 1/(3(1-t))). \end{array} $$
Let $T_0 = \{(t_1, t_2) \in ]0, 1]^2 : t_1 < t_2\}$. We study the integral of $\omega'(k) - \omega(k) = d\eta(k)$ over $\tilde{C}(\tilde{R}(C), L; \zeta^K) = m^2(T_0)$. View $T_0$ as the set

$$T = \left\{ (t_1, t_2, \alpha) \in ]0, 1]^2 \times ]-\frac{\pi}{2}, \frac{\pi}{2}] \mid \text{tan}(\alpha) = \frac{1/(3(1-t_2)) - 1/(3t_1)}{|z_2 - z_1|} \right\}$$

denoted by $T$. Note that when $(t_1, t_2) \in ]0, 1/3[\times]2/3, 1[$, and when $(t_1, t_2, \alpha) \in T$, we have

$$p_{\tau}(m(t_1), m(t_2)) = \cos \alpha \frac{z_2 - z_1}{|z_2 - z_1|} + \sin \alpha \vec{N} \in S(K).$$

Let $\bar{T}$ be the closure of $T$ in $[0, 1]^2 \times [-\frac{\pi}{2}, \frac{\pi}{2}]$. This closure, drawn in the right part of Figure 12.3, is a smooth blow-up of $\{(t_1, t_2) \in [0, 1]^2 : t_1 \le t_2\}$ at $(0, 1)$ (with corners), where $(0, 1)$ lifts as $(0, 1) \times [-\frac{\pi}{2}, \frac{\pi}{2}]$ in $\bar{T}$. The map $m^2$ extends as a smooth map valued in $C_2(R(C))$ on $\bar{T}$. Its composition with $p_\tau$ sends the vertical side of $\bar{T}$ to $-\vec{N}$, its horizontal side to $\vec{N}$, and the blown-up upper-left corner to $S(K)$. The integral of $d\eta(k)$ over $\tilde{C}(\tilde{R}(C), L; \zeta^K)$ is the integral of $d\eta(k)$ over $m^2(\bar{T})$. So it is the integral of $\eta(k)$ over $m^2(\partial\bar{T})$, where $\eta(k)$ can be assumed to be equal to $p_\tau^*(\eta(k)_{S^2})$, as in Lemma 3.17. Furthermore, $p_\tau \circ m^2$ restricts to $\partial\bar{T}$ as a degree one map onto $p_\tau(U^+K) \cup S(K)$. So the stated conclusion follows. The case in which $K$ goes from bottom to bottom can be treated similarly. $\square$

**Definition 12.6.** Recall the notation of Lemma 12.5. For a long component $K: \mathbb{R} \hookrightarrow \tilde{R}(C)$ of a tangle in a parallelized $\mathbb{Q}$-cylinder $(C, \tau)$, define

$$I_{\theta}(K, \tau) = 2I(\zeta^{K}, \omega_{S^2}).$$

Recall the definition of $I_{\theta}(K, \tau)$ for a closed component $K$ of $\tilde{R}(C)$ from Lemma 7.15.

The factor 2 in the definition of $I_{\theta}$ for long components in Definition 12.6 may seem unnatural. It allows to get homogeneous formulas in Theorem 12.7 below. Theorem 12.7 generalizes Theorem 7.20 to long tangle representatives. It will be proved in Section 14.3.

**Theorem 12.7.** Let $C$ be a rational homology cylinder equipped with a parallelization $\tau$ (standard near $\partial C$). Let $L: \mathcal{L} \hookrightarrow \tilde{R}(C)$ be a long tangle representative in $\tilde{R}(C)$. Let $n \in \mathbb{N}$. For any $i \in \underline{3n}$, let $\omega(i)$ be a homogeneous propagating form of $(C_2(R(C)), \tau)$. Set

$$Z_n(C, L, (\omega(i))) = \sum_{\Gamma \in \mathcal{D}_n^e(\mathcal{L})} \zeta_{\Gamma} I(C, L, \Gamma, (\omega(i))_{i \in \underline{3n}}) [\Gamma] \in \mathcal{A}_n(\mathcal{L}),$$
where $\zeta_{\Gamma} = \frac{(3n-|E(\Gamma)|)!}{(3n)!2^{|E(\Gamma)|}}$, with Definition 7.6 for $\mathcal{D}_n^e(\mathcal{L})$. Then $Z_n(\mathcal{C}, L, (\omega(i)))$ is independent of the chosen $\omega(i)$. It depends only on $(\mathcal{C}, L(\mathcal{L}) \cap \mathcal{C})$ up to diffeomorphisms that fix $\partial\mathcal{C}$ (and $L(\mathcal{L}) \cap \partial\mathcal{C})$, pointwise, on $p_1(\tau)$, and on the $I_\theta(K_j, \tau)$ for the components $K_j, j \in \underline{k}$, of $L$. We denote it by $Z_n(\mathcal{C}, L, \tau)$. Set

$$Z(\mathcal{C}, L, \tau) = (Z_n(\mathcal{C}, L, \tau))_{n \in \mathbb{N}} \in \mathcal{A}(\mathcal{L}),$$

and recall the anomalies $\alpha \in \check{\mathcal{A}}(S^1; \mathbb{R})$ and $\beta \in \mathcal{A}(\emptyset; \mathbb{R})$ from Sections 10.3 and 10.2. Then the expression

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

depends only on the boundary-preserving diffeomorphism class of $(\mathcal{C}, L)$.<fnref n="2" /> We denote it by $\mathcal{Z}(\mathcal{C}, L)$.

## 12.2 Definition of $\mathcal{Z}^f$ for framed tangles

Recall Definition 5.35 of knot parallels.

**Definition 12.8.** A parallel $K_\parallel$ of an embedding of a long component $K: \mathbb{R} \hookrightarrow \check{\mathcal{R}}(\mathcal{C})$ parametrized so that $K(\mathbb{R}) \cap \mathcal{C} = K([0, 1])$ is the image of an embedding $K_\parallel: \mathbb{R} \hookrightarrow \check{\mathcal{R}}(\mathcal{C})$ such that there exists an embedding

$$k: [-1, 1] \times \mathbb{R} \to R(\mathcal{C}) \setminus (L(\mathcal{L}) \setminus K(\mathbb{R}))$$

such that $K = k|_{\{0\} \times \mathbb{R}}$, $K_\parallel = k|_{\{1\} \times \mathbb{R}}$ and $k(u, t) = K(t) + u\epsilon(t)(1, 0, 0)$ for any $(u, t) \in [-1, 1] \times (\mathbb{R} \setminus ]0, 1[)$, for a small smooth function $\epsilon: \mathbb{R} \setminus ]0, 1[ \to \mathbb{R} \setminus \{0\}$ such that $\epsilon(0)\epsilon(1)$ is positive for components going from bottom to top or from top to bottom, and negative for components going from bottom to bottom or from top to top. (We push in one of the two horizontal real directions in a way consistent with the orientation.) Parallels are considered up to isotopies that stay within these parallels and up to the exchange of $k|_{\{1\} \times \mathbb{R}}$ and $k|_{\{-1\} \times \mathbb{R}}$. A component $K$ of an LTR is *framed* if $K$ is equipped with such a class of parallels, called a *framing* of $K$. An LTR is *framed* if all its components are. The *self-linking number* of a circle component $K$ in a framed LTR is the linking number $lk(K, K_\parallel)$ of $K$ and its parallel $K_\parallel$.

<footnote n="2">²Again, the subscript $j$ of $\#_j$ indicates that $\exp(-I_\theta(K_j, \tau)\alpha)$ is inserted on the component of $K_j$ of the domain $\mathcal{L}$ of $L$.</footnote>
For a long component $K$ equipped with a parallel, we define its self-linking number $lk(K, K_\parallel)$ in Definitions 12.9 and 12.10 below, and in Definition 13.4, which covers the remaining cases (when $K$ goes from bottom to bottom or from top to top and when $K(1) - K(0)$ is not in the direction of the real line).

**Definition 12.9.** When $K$ goes from bottom to top, and when $\varepsilon(0)$ is positive, let $[K_\parallel(1), (1, 1)]$ be the straight segment from $K_\parallel(1)$ to $(1, 1) \in D_1 \times \{1\}$. Similarly define $[K(1), (-1, 1)]$, $[(1, 0), K_\parallel(0)]$ and $[-1, 0), K(0)]$, and note that they are pairwise disjoint. Define the topological circle embeddings

$$ \hat{K} = K([0, 1]) \cup [K(1), (-1, 1)] \cup (-\{-1\} \times [0, 1]) \cup [-1, 0), K(0)] $$

$$ \hat{K}_{\parallel} = K_{\parallel}([0, 1]) \cup [K_{\parallel}(1), (1, 1)] \cup (-\{1\} \times [0, 1]) \cup [(1, 0), K_{\parallel}(0)], $$

as in the figure below,<fnref n="3" /> and set

<figure><img src="image_98.png" /></figure>

When $\varepsilon(0)$ is negative, the above definition determines $lk(K_\parallel, K)$. Then set

$$ lk(K, K_\parallel) = lk(K_\parallel, K). $$

So $lk(K, K_\parallel)$ is defined when $K$ goes from bottom to top. When $K$ goes from top to bottom, $(-K)$ goes from bottom to top, and $-K_\parallel$ is a parallel of $(-K)$. Then set $lk(K, K_\parallel) = lk(-K, -K_\parallel)$.

**Definition 12.10.** When $K$ goes from bottom to bottom or from top to top, and when $K(1) - K(0)$ is equal to $(v, 0, 0)$ for $v \in \mathbb{R} \setminus \{0\}$, we define the self-linking number $lk(K, K_\parallel)$ as follows. Let us first assume that $v > 0$ and that $\varepsilon(0) > 0$, as in Figure 12.4. In this case, define topological circle embeddings $\hat{K}_\parallel = K_\parallel([0, 1]) \cup [K_\parallel(1), K_\parallel(0)]$, where $[K_\parallel(1), K_\parallel(0)]$ is the straight segment from $K_\parallel(1)$ to $K_\parallel(0)$ in $D_1 \times \{0\}$ or in $D_1 \times \{1\}$, and $\hat{K} = K([0, 1]) \cup \gamma([0, 1])$ for an arbitrary path $\gamma$ from $\gamma(0) = K(1)$ to $\gamma(1) = K(0)$ such that $\gamma([0, 1]) \subset \check{R}(C) \setminus C$ as in Figure 12.4, and set

$$ lk(K, K_\parallel) = lk(\hat{K}, \hat{K}_\parallel). $$

<footnote n="3">³As we often abusively do, we identify embeddings with their images.</footnote>
<figure><img src="image_99.png" /><figcaption>Figure 12.4: Pictures of $\gamma$ and $\hat{K}_{\parallel}$</figcaption></figure>

In the other cases, in which $K$ goes from bottom to bottom or from top to top, and $K(1) - K(0)$ is equal to $(v, 0, 0)$ for $v \in \mathbb{R} \setminus \{0\}$, $lk(K, K_{\parallel})$ is defined so that we have again $lk(K, K_{\parallel}) = lk(K_{\parallel}, K) = lk(-K, -K_{\parallel})$.

**Example 12.11.** Let $K$ be a framed component of an LTR in $\mathbb{R}^3$, with a regular projection on $(\mathbb{R} \subset \mathbb{C}) \times \mathbb{R}$, and such that $K(1) - K(0)$ is equal to $(v, 0, 0)$ for $v \in \mathbb{R} \setminus \{0\}$ if $K$ goes from bottom to bottom or from top to top. The self-linking number of $K$ is its writhe, which is the algebraic number of its crossings, i.e., the number of positive crossings minus the number of negative crossings in its above regular projection. (Check it as an exercise.)

**Definition 12.12.** When a long tangle representative $L = (K_j)_{j \in \underline{k}}$ is framed by some $L_{\parallel} = (K_{j\parallel})_{j \in \underline{k}}$, set

$$ \mathcal{Z}^f(C, (L, L_{\parallel})) = \prod_{j=1}^{k} \left( \exp(lk(K_j, K_{j\parallel})\alpha) \#_j \right) \mathcal{Z}(C, L) $$

with the notation of Theorem 12.7.

We will give general relations between $I_{\theta}$ and self-linking numbers in Section 16.3. See Proposition 16.13.

A tangle representative is a pair $(C, L(\mathcal{L}) \cap C)$ for a rational homology cylinder $C$ and a long tangle representative $L: \mathcal{L} \hookrightarrow \tilde{R}(C)$ as in Definition 12.1. Tangle representatives and LTR are in natural one-to-one correspondence, and we also view $\mathcal{Z}$ as a function of tangle representatives.

A tangle representative $(C_1, L_1)$ is right-composable by a tangle representative $(C_2, L_2)$ when the top configuration of $(C_1, L_1)$ coincides with the bottom configuration of $(C_2, L_2)$. In this case, the product

$$ (C_1C_2, L_1L_2) = (C_1, L_1)(C_2, L_2) $$

is obtained by stacking $(C_2, L_2)$ above $(C_1, L_1)$, after affine reparametrizations of $D_1 \times [0, 1]$, which becomes $D_1 \times [0, 1/2]$ for $(C_1, L_1)$ and $D_1 \times [1/2, 1]$ for $(C_2, L_2)$.

The product of two framed tangle representatives is naturally framed. We will prove the following functoriality theorem for $\mathcal{Z}^f$ in Section 17.2.
**Theorem 12.13.** $\mathcal{Z}^f$ is functorial: For two framed tangle representatives $L_1 = (C_1, L_1)$ and $L_2 = (C_2, L_2)$ such that the top configuration of $L_1$ coincides with the bottom configuration of $L_2$, we have

$$ \mathcal{Z}^f(L_1 L_2) = \mathcal{Z}^f\left(\begin{array}{c|c} L_2 & \\ \hline L_1 & \end{array}\right) = \begin{array}{c} \boxed{\mathcal{Z}^f(L_2)} \\ \hline \boxed{\mathcal{Z}^f(L_1)} \end{array} = \mathcal{Z}^f(L_1)\mathcal{Z}^f(L_2). $$

The product $\mathcal{Z}^f(L_1)\mathcal{Z}^f(L_2)$ is the natural product of Section 6.4. When applied to the case in which $\mathcal{L}_1$ and $\mathcal{L}_2$ are empty, the above theorem implies that the invariant $\mathcal{Z}$ of $\mathcal{Q}$-spheres is multiplicative under connected sum.

**Remark 12.14.** The multiplicativity of Theorem 12.13 does not hold for the unframed version $\mathcal{Z}$ of $\mathcal{Z}^f$. Indeed, as an unframed tangle, the vertical product

$$ \tilde{\mathcal{Z}} \cap \mathcal{Z} $$

equals $\cap$. However, the reader can prove

$$ \mathcal{Z}(\tilde{\mathcal{Z}})\mathcal{Z}(\cap) \neq \mathcal{Z}(\cap) $$

as an exercise, using Lemma 12.3 and the behavior of $\mathcal{Z}^f$ under component orientation reversals described in Theorem 13.12. (Here, the image of any involved boundary planar two-point configuration is $\{-1/2, 1/2\} \subset \mathbb{C}$.)

Lemma 13.7 will imply that $\mathcal{Z}$ is invariant under any isotopy of a tangle representative in a rational homology cylinder during which the bottom and top configurations are constant up to translation and dilation.

**Definition 12.15.** In this book, a *tangle* is an equivalence class of tangle representatives under the equivalence relation that identifies two representatives if and only if they can be obtained from one another by a diffeomorphism $h$ from the pair $(C, L)$ to another such $(C, L')$,

* which fixes a neighborhood of $\partial(D_1 \times [0, 1])$ setwise,

* which fixes a neighborhood of $(\partial D_1) \times [0, 1]$ pointwise,

* such that $h(c^-(B^-) \times \{0\}) \subset D_1 \times \{0\}$ coincides with $c^-(B^-) \times \{0\}$ up to translation and dilation (i.e., as a planar configuration of $\tilde{S}_{c^-(B^-)}(\mathbb{C} \times \{0\})$), and the classes of $h(c^+(B^+) \times \{1\}) \subset D_1 \times \{1\}$ and $c^+(B^+) \times \{1\}$ in $\tilde{S}_{c^+(B^+)}(\mathbb{C} \times \{1\})$ coincide, too,

and which is isotopic to the identity map through such diffeomorphisms.
## 12.3 Defining $\mathcal{Z}^f$ for combinatorial $q$-tangles

The tangles of Definition 12.15 can be framed by parallels of their components as before to become *framed tangles*. These framed tangles are framed cobordisms (up to isotopy) in $\mathbb{Q}$-cylinders from an injective configuration $c^-(B^-) \in \tilde{S}_{c^-(B^-)}(\mathbb{C})$ of points in $\mathbb{C}$, up to dilation and translation, to another planar configuration $c^+(B^+) \in \tilde{S}_{c^+(B^+)}(\mathbb{C})$. Then $\mathcal{Z}^f$ is an invariant of these framed tangles. In Section 16.5, we extend $\mathcal{Z}^f$ to framed cobordisms between limit configurations in the compactifications $S_{c(B)}(\mathbb{C})$ of the spaces $\tilde{S}_{c(B)}(\mathbb{C})$ introduced in Subsection 8.1.2 and studied in Section 8.3. The category of these limit cobordisms called $q$-tangles is equipped with interesting cabling operations described in Section 13.1. As stated in Theorem 13.12, $\mathcal{Z}^f$ behaves nicely under these cabling operations. In this section, we define $\mathcal{Z}^f$ for particular $q$-tangles, called *combinatorial $q$-tangles*, defined below, and we give two examples of cabling properties for these combinatorial $q$-tangles, to motivate and introduce our more general presentation of $q$-tangles in Section 13.1. Dror Bar-Natan, Thang Lê, and Jun Murakami used these combinatorial $q$-tangles to define a functorial extension of the Kontsevich integral of framed oriented links in [BN97] and [LM96].

**Definition 12.16.** A *combinatorial $q$-tangle* is a triple $(L, w^{-}, w^{+})$, where

* $L$ is a framed tangle representative whose bottom and top configurations are on the real line, up to isotopies of $\mathcal{C}$ that globally preserve the intersection of the bottom disk $D_1 \times \{0\}$ with $\mathbb{R} \times \{0\}$ and the intersection of the top disk $D_1 \times \{1\}$ with $\mathbb{R} \times \{1\}$,

* $w^{-}$ and $w^{+}$ are nonassociative words in the letter $\bullet$ (as in Example 8.3) such that the letters of $w^{-}$ (resp. $w^{+}$) are in canonical one-to-one correspondence with the elements of the domain $B^{-}$ (resp. $B^{+}$) of the bottom (resp. top) configuration of $L$.

The nonassociative words $w^{-}$ and $w^{+}$ are respectively called the *bottom and top configurations* of the combinatorial $q$-tangle $(L, w^{-}, w^{+})$. The combinatorial $q$-tangle $(L, w^{-}, w^{+})$ is also called the combinatorial $q$-tangle $L$ from $w^{-}$ to $w^{+}$.

**Examples 12.17.** Combinatorial $q$-tangles in the standard cylinder are unambiguously represented by one of their regular projections to $(\mathbb{R} \subset \mathbb{C}) \times [0, 1]$ such that the parallels of their components are parallel in the figures, together with their bottom and top configurations. Examples of these combinatorial $q$-tangles include

<figure><img src="image_100.png" /></figure>
Recall from Example 8.3 that the involved nonassociative words are corners of $S_{<,k}(\mathbb{R}) \subset S_k(\mathbb{C})$. A combinatorial $q$-tangle $(L, w^{-}, w^{+})$ from a bottom word $w^{-}$ to a top word $w^{+}$ is thought of as the limit, when $t$ tends to 0, of framed tangles $L(t)$ whose bottom and top configurations are the configurations $w^{-}(t)$ and $w^{+}(t)$ defined in Example 8.3, in the isotopy class of $L$ with respect to the isotopies of Definition 12.16.

In Theorem 13.8 and Remark 13.11, following Sylvain Poirier [Poi00], we prove that $\lim_{t \to 0} \mathcal{Z}^f(L(t))$ exists and that the formula

$$ \mathcal{Z}^f(L) = \lim_{t \to 0} \mathcal{Z}^f(L(t)) $$

defines an isotopy invariant $\mathcal{Z}^f$ of these (framed) combinatorial $q$-tangles. The isotopy invariant $\mathcal{Z}^f$ will thus behave naturally with respect to the deletion of components. If $L'$ is a subtangle of a tangle $L$ with domain $\mathcal{L}'$, then $\mathcal{Z}^f(L')$ will be obtained from $\mathcal{Z}^f(L)$ by forgetting $\mathcal{L} \setminus \mathcal{L}'$ and all the diagrams with univalent vertices on $\mathcal{L} \setminus \mathcal{L}'$.

As a first example, let us compute $\mathcal{Z}^f$ for the combinatorial $q$-tangle $\uparrow\uparrow$.

**Lemma 12.18.** *We have*

$$ \mathcal{Z}^f(\uparrow\uparrow) = 1 = [\uparrow\uparrow]. $$

**PROOF:** By definition, the left-hand side is the limit, when $t$ tends to 0, of the evaluation of $Z$ of the LTR whose image is $\{0, t\} \times \mathbb{R}$. There is an action of $\mathbb{R}$ by vertical translation on the involved configuration spaces. The integrated forms factor through the quotients by this action of $\mathbb{R}$ of the configuration spaces, whose dimensions are smaller (by one) than the degrees of the integrated forms. So the integrals vanish for all nonempty diagrams. $\square$

As a second example, we compute $\mathcal{Z}_{\le 1}^f$, which is the truncation of $\mathcal{Z}^f$ in degrees lower than 2, for the combinatorial $q$-tangle $\mathcal{Z}$.

**Lemma 12.19.** *We have*

$$ \mathcal{Z}_{\le 1}^f \left( \mathcal{Z} \right) = \left[ \begin{bmatrix} \uparrow & \uparrow \\ \vdots & \vdots \\ \uparrow & \uparrow \end{bmatrix} \right] + \left[ \begin{bmatrix} \bullet & \bullet \\ \vdots & \vdots \\ \bullet & \bullet \end{bmatrix} \right]. $$

**PROOF:** Since $\mathcal{Z}$ is invariant under the isotopies that preserve the bottom and the top configurations, the contributions of the nonempty diagrams whose univalent vertices are on one strand of the tangle vanish. So we are
left with the contribution of the numbered graphs with one vertex on each strand, treated in Lemma 12.3. □

The obtained invariant $\mathcal{Z}^f$ is still multiplicative under vertical composition as in Theorem 12.13, and we can now define other interesting operations.

For two combinatorial $q$-tangles $L_1 = (C_1, L_1)$ from $w_1^-$ to $w_1^+$, and $L_2 = (C_2, L_2)$ from $w_2^-$ to $w_2^+$, define the product $L_1 \otimes L_2$, from the bottom configuration $w_1^- w_2^-$ to the top configuration $w_1^+ w_2^+$, by shrinking $C_1$ and $C_2$ to make them respectively replace the products by $[0, 1]$ of the horizontal disks with radius $\frac{1}{4}$ and respective centers $-\frac{1}{2}$ and $\frac{1}{2}$.

**Examples 12.20.** We have

$$ \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}]
  \node[draw, circle, fill=white, inner sep=1.2pt] (n) at (0,0) {};
  \node[draw, circle, fill=white, inner sep=1.2pt] (s) at (0,1) {};
  \draw[->] (n) -- (s);
  \node[above=0.2cm of n] at (0.2,0.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[left=0.5cm of n] at (-0.5,0.5) {$\bullet$};
  \node[left=0.5cm of s] at (-0.5,1.5) {$\bullet$};
  \node[above=0.2cm of n] at (0.2,0.5) {$\bullet$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\bullet$};
  \draw[->] (n) -- (s);
\end{tikzpicture} \otimes \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}]
  \node[draw, circle, fill=white, inner sep=1.2pt] (n) at (0,0) {};
  \node[draw, circle, fill=white, inner sep=1.2pt] (s) at (0,1) {};
  \draw[->] (n) -- (s);
  \node[above=0.2cm of n] at (0.2,0.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[left=0.5cm of n] at (-0.5,0.5) {$\bullet$};
  \node[left=0.5cm of s] at (-0.5,1.5) {$\bullet$};
  \node[above=0.2cm of n] at (0.2,0.5) {$\bullet$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\bullet$};
  \draw[->] (n) -- (s);
\end{tikzpicture} = \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}]
  \node[draw, circle, fill=white, inner sep=1.2pt] (n) at (0,0) {};
  \node[draw, circle, fill=white, inner sep=1.2pt] (s) at (0,1) {};
  \draw[->] (n) -- (s);
  \node[above=0.2cm of n] at (0.2,0.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[left=0.5cm of n] at (-0.5,0.5) {$\bullet$};
  \node[left=0.5cm of s] at (-0.5,1.5) {$\bullet$};
  \node[above=0.2cm of n] at (0.2,0.5) {$\bullet$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\bullet$};
  \draw[->] (n) -- (s);
\end{tikzpicture}$, and $\begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}]
  \node[draw, circle, fill=white, inner sep=1.2pt] (n) at (0,0) {};
  \node[draw, circle, fill=white, inner sep=1.2pt] (s) at (0,1) {};
  \draw[->] (n) -- (s);
  \node[above=0.2cm of n] at (0.2,0.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[left=0.5cm of n] at (-0.5,0.5) {$\bullet$};
  \node[left=0.5cm of s] at (-0.5,1.5) {$\bullet$};
  \node[above=0.2cm of n] at (0.2,0.5) {$\bullet$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\bullet$};
  \draw[->] (n) -- (s);
\end{tikzpicture} \otimes \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}]
  \node[draw, circle, fill=white, inner sep=1.2pt] (n) at (0,0) {};
  \node[draw, circle, fill=white, inner sep=1.2pt] (s) at (0,1) {};
  \draw[->] (n) -- (s);
  \node[above=0.2cm of n] at (0.2,0.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[left=0.5cm of n] at (-0.5,0.5) {$\bullet$};
  \node[left=0.5cm of s] at (-0.5,1.5) {$\bullet$};
  \node[above=0.2cm of n] at (0.2,0.5) {$\bullet$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\bullet$};
  \draw[->] (n) -- (s);
\end{tikzpicture} = \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}]
  \node[draw, circle, fill=white, inner sep=1.2pt] (n) at (0,0) {};
  \node[draw, circle, fill=white, inner sep=1.2pt] (s) at (0,1) {};
  \draw[->] (n) -- (s);
  \node[above=0.2cm of n] at (0.2,0.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[left=0.5cm of n] at (-0.5,0.5) {$\bullet$};
  \node[left=0.5cm of s] at (-0.5,1.5) {$\bullet$};
  \node[above=0.2cm of n] at (0.2,0.5) {$\bullet$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\bullet$};
  \draw[->] (n) -- (s);
\end{tikzpicture}$.

We can easily deduce the following theorem from the cabling property and the functoriality property of Theorem 13.12.

**Theorem 12.21.** $\mathcal{Z}^f$ is monoidal: For two combinatorial $q$-tangles $L_1$ and $L_2$, we have

$$ \mathcal{Z}^f(L_1 \otimes L_2) = \mathcal{Z}^f\left(\begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}]
  \node[draw, circle, fill=white, inner sep=1.2pt] (n) at (0,0) {};
  \node[draw, circle, fill=white, inner sep=1.2pt] (s) at (0,1) {};
  \draw[->] (n) -- (s);
  \node[above=0.2cm of n] at (0.2,0.5) {$\begin{pmatrix} L_1 & L_2 \end{pmatrix}$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\begin{pmatrix} L_1 & L_2 \end{pmatrix}$};
  \node[left=0.5cm of n] at (-0.5,0.5) {$\bullet$};
  \node[left=0.5cm of s] at (-0.5,1.5) {$\bullet$};
  \node[above=0.2cm of n] at (0.2,0.5) {$\bullet$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\bullet$};
  \draw[->] (n) -- (s);
\end{tikzpicture}\right) = \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}]
  \node[draw, circle, fill=white, inner sep=1.2pt] (n) at (0,0) {};
  \node[draw, circle, fill=white, inner sep=1.2pt] (s) at (0,1) {};
  \draw[->] (n) -- (s);
  \node[above=0.2cm of n] at (0.2,0.5) {$\begin{pmatrix} \mathcal{Z}^f(L_1) & \mathcal{Z}^f(L_2) \end{pmatrix}$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\begin{pmatrix} \mathcal{Z}^f(L_1) & \mathcal{Z}^f(L_2) \end{pmatrix}$};
  \node[left=0.5cm of n] at (-0.5,0.5) {$\bullet$};
  \node[left=0.5cm of s] at (-0.5,1.5) {$\bullet$};
  \node[above=0.2cm of n] at (0.2,0.5) {$\bullet$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\bullet$};
  \draw[->] (n) -- (s);
\end{tikzpicture} = \mathcal{Z}^f(L_1) \otimes \mathcal{Z}^f(L_2), $$

where $\mathcal{Z}^f(L_1) \otimes \mathcal{Z}^f(L_2)$ denotes the image of $\mathcal{Z}^f(L_1) \otimes \mathcal{Z}^f(L_2)$ under the natural product from $\mathcal{A}(\mathcal{L}_1) \otimes_{\mathbb{R}} \mathcal{A}(\mathcal{L}_2)$ to $\mathcal{A}(\mathcal{L}_1 \sqcup \mathcal{L}_2)$ induced by the disjoint union of diagrams.

We can also double a component $K$ according to its parallelization in a combinatorial $q$-tangle $L$. This operation replaces a component by two parallel components, and if this component has boundary points, it replaces the corresponding letters in the nonassociative words with $(\bullet)$. The obtained combinatorial $q$-tangle is denoted by $L(2 \times K)$. For example, we have

$$ \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}]
  \node[draw, circle, fill=white, inner sep=1.2pt] (n) at (0,0) {};
  \node[draw, circle, fill=white, inner sep=1.2pt] (s) at (0,1) {};
  \draw[->] (n) -- (s);
  \node[above=0.2cm of n] at (0.2,0.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[left=0.5cm of n] at (-0.5,0.5) {$\bullet$};
  \node[left=0.5cm of s] at (-0.5,1.5) {$\bullet$};
  \node[above=0.2cm of n] at (0.2,0.5) {$\bullet$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\bullet$};
  \draw[->] (n) -- (s);
\end{tikzpicture} (2 \times \text{?}) = \begin{tikzpicture}[baseline={([yshift=-0.5ex]current bounding box.center)}]
  \node[draw, circle, fill=white, inner sep=1.2pt] (n) at (0,0) {};
  \node[draw, circle, fill=white, inner sep=1.2pt] (s) at (0,1) {};
  \draw[->] (n) -- (s);
  \node[above=0.2cm of n] at (0.2,0.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\begin{pmatrix} \bullet & \bullet \end{pmatrix}$};
  \node[left=0.5cm of n] at (-0.5,0.5) {$\bullet$};
  \node[left=0.5cm of s] at (-0.5,1.5) {$\bullet$};
  \node[above=0.2cm of n] at (0.2,0.5) {$\bullet$};
  \node[above=0.2cm of s] at (0.2,1.5) {$\bullet$};
  \draw[->] (n) -- (s);
\end{tikzpicture} $$

The following duplication property for $\mathcal{Z}^f$ is a part of Theorem 13.12, which is proved in Section 17.4.

**Theorem 12.22.** Let $K$ be a component of a combinatorial $q$-tangle $L$.<fnref n="4" />

*Recall Notation 6.31.* Then we have

$$ \mathcal{Z}^f(L(2 \times K)) = \pi(2 \times K)^* \mathcal{Z}^f(L). $$

These results are some particular cases of the properties of $\mathcal{Z}^f$, which are listed in Theorem 13.12 and proved in Chapter 17.

<footnote n="4">⁴The theorem holds for components going from bottom to bottom or from top to top. It would not hold in this generality if $\mathcal{Z}^f$ were replaced by certain functorial extensions of the Kontsevich integral.</footnote>
## 12.4 Good monoidal functors for combinatorial $q$-tangles

Recall Notation 6.19. In this section, all combinatorial $q$-tangles are combinatorial $q$-tangles in $\mathbb{R}^3$ or in the standard cylinder, and coefficients for spaces of Jacobi diagrams are in $\mathbb{K} = \mathbb{C}$. In [Poi00], Sylvain Poirier extended the natural projection $\tilde{Z}^f(\mathbb{R}^3, \cdot)$ of $\mathcal{Z}^f(\mathbb{R}^3, \cdot)$ in $\tilde{\mathcal{A}}(\cdot; \mathbb{R}) \subset \tilde{\mathcal{A}}(\cdot; \mathbb{C})$, from framed links of $\mathbb{R}^3$ to these combinatorial $q$-tangles, in an elegant way, and he proved that his extension $Z^l$ is a good monoidal functor with respect to the definition below. Good monoidal functors on the category of combinatorial $q$-tangles (in $\mathbb{R}^3$) are characterized in [Les02]. This section reviews these results of [Poi00] and [Les02]. The quoted results of Sylvain Poirier will be reproved (with much more details) and generalized in this book. In contrast, the proofs of the results of [Les02] will not be reproduced in this book since they do not involve analysis on configuration spaces.

**Definition 12.23.** A good monoidal functor from the category of combinatorial $q$-tangles (in $\mathbb{R}^3$) to the category of spaces of Jacobi diagrams is a map $Y$ sending a combinatorial $q$-tangle $L$ with domain $\mathcal{L}$ to an element of the space $\tilde{\mathcal{A}}(\mathcal{L}; \mathbb{C})$ of Notation 6.19 with the following properties.

* For any combinatorial $q$-tangle $L$, the degree zero part $Y_0(L)$ of $Y(L)$ is 1, which is the class of the empty diagram.

* $Y$ is functorial: For two combinatorial $q$-tangles $L_1$ and $L_2$ such that the top configuration top($L_1$) of $L_1$ coincides with the bottom configuration bot($L_2$) of $L_2$, we have

$$Y(L_1 L_2) = Y\left(\left[\begin{array}{c|c} L_2 & \\ \hline L_1 & \end{array}\right]\right) = \left[\begin{array}{c} Y(L_2) \\ \hline Y(L_1) \end{array}\right] = Y(L_1)Y(L_2).$$

* $Y$ is monoidal: For two combinatorial $q$-tangles $L_1$ and $L_2$, we have

$$Y(L_1 \otimes L_2) = Y\left(\left[\begin{array}{c|c} L_1 & \\ \hline L_2 & \end{array}\right]\right) = \left[\begin{array}{c} Y(L_1) \\ \hline Y(L_2) \end{array}\right] = Y(L_1) \otimes Y(L_2).$$

* $Y$ is compatible with the deletion of a component: If $L'$ is a subtangle of $L$ with domain $\mathcal{L}'$, then $Y(L')$ is obtained from $Y(L)$ by forgetting $\mathcal{L} \setminus \mathcal{L}'$ and all the diagrams with univalent vertices on $\mathcal{L} \setminus \mathcal{L}'$.
* Y is compatible with the duplication of a regular component, which is a component that can be represented without horizontal tangent vectors: For such a component K of a combinatorial q-tangle L, we have

$$Y(L(2 \times K)) = \pi(2 \times K)^* Y(L)$$

with respect to Notation 6.31.

* Y is invariant under the 180-degree rotation around a vertical axis through the real line.

* Let $s_{1/2}$ be the reflection with respect to the horizontal plane at height $1/2$. Let $\sigma_{1/2}$ be the linear endomorphism of the topological vector space $\tilde{\mathcal{A}}(S^1)$ such that $\sigma_{1/2}([\Gamma]) = (-1)^d [\Gamma]$, for any element $[\Gamma]$ of $\tilde{\mathcal{A}}_d(S^1)$. For any framed knot $K = K(S^1)$, $Y(K)$ belongs to $\tilde{\mathcal{A}}(S^1; \mathbb{R})$ and we have

$$Y \circ s_{1/2}(K) = \sigma_{1/2} \circ Y(K).$$

* Y behaves as in Proposition 10.23 with respect to reversals of component orientations. It can be defined as an invariant of unoriented tangles valued in a space of diagrams whose support is the unoriented domain of the tangle, as in Definitions 6.13 and 6.16.

* The degree one part $a_1^Y$ of the element $a^Y \in \tilde{\mathcal{A}}([0, 1])$ such that $a_0^Y = 0$ and

$$Y(\bigcirc) = \exp(a^Y) Y(\bigcap)$$

is

$$a_1^Y = \frac{1}{2} \left[ \hat{\mathcal{C}} \right].$$

As shown in [Les02, Proposition 4.2, 10], a good monoidal functor Y is determined by its values

$$Y(\begin{tikzpicture}[baseline=(current bounding box.center)] 
    
    \node at (0,0) {$Y$};
    \node at (0,1) {$Y$};
    \node at (1,0) {$Y$};
    \node at (1,1) {$Y$};
    \draw[->] (0,0) -- (1,1);
    \draw[->] (0,1) -- (1,0);
\end{tikzpicture}), Y(\begin{tikzpicture}[baseline=(current bounding box.center)] 
    
    \node at (0,0) {$Y$};
    \node at (0,1) {$Y$};
    \node at (1,0) {$Y$};
    \node at (1,1) {$Y$};
    \draw[->] (0,0) -- (1,1);
    \draw[->] (0,1) -- (1,0);
\end{tikzpicture}), \text{ and } Y(\begin{tikzpicture}[baseline=(current bounding box.center)] 
    
    \node at (0,0) {$Y$};
    \node at (0,1) {$Y$};
    \node at (1,0) {$Y$};
    \node at (1,1) {$Y$};
    \draw[->] (0,0) -- (1,1);
    \draw[->] (0,1) -- (1,0);
\end{tikzpicture}).$$

Moreover, Y can be computed combinatorially from these three ingredients.

See also [LM96, BN97]. The value $Y(\begin{tikzpicture}[baseline=(current bounding box.center)] 
    
    \node at (0,0) {$Y$};
    \node at (0,1) {$Y$};
    \node at (1,0) {$Y$};
    \node at (1,1) {$Y$};
    \draw[->] (0,0) -- (1,1);
    \draw[->] (0,1) -- (1,0);
\end{tikzpicture})$ is called an *associator*.

In [LM96], Thang Lê and Jun Murakami constructed the first example of such a good monoidal functor from the Kontsevich integral of links in $\mathbb{R}^3$ described in [CD01].<fnref n="5" /> See also [Les99]. We denote the Lê–Murakami functor

<footnote n="5">⁵They also explain [LM96, Section 5] how to construct such a functor combinatorially, from an element of $\mathcal{A}(|||)$ satisfying some equations [LM96, Section 3, A1–A4], which allow it to be the associator of a good monoidal functor. See also [BN97].</footnote>
**Note 12.30.** Theorem 12.28 would not hold if framed links were replaced by arbitrary tangles. For pure combinatorial *q*-tangles from a nonassociative word to itself, whose components go from top to bottom or from bottom to top and connect a letter of the bottom word to the corresponding letter of the top word, it is true, up to conjugation by a *twisting function* of nonassociative words, introduced by Thang Lê and Jun Murakami in [LM96, Section 7]. A generalization of Theorem 12.28 to all combinatorial *q*-tangles whose components go from top to bottom or from bottom to top is given in [Les02, Theorem 8.5]. It involves the Lê–Murakami twisting functions.
# Chapter 13

## More on the functor $\mathcal{Z}^f$

In this chapter, we state our general Theorem 13.12, which ensures that $\mathcal{Z}^f$ is a functor behaving naturally with respect to various structures of the category of $q$-tangles, such as cabling or duplication. We first describe the category of $q$-tangles in Section 13.1 before stating Theorem 13.12 in Section 13.3.

We describe the main steps of the generalization of the construction of $\mathcal{Z}^f$ to $q$-tangles in Section 13.2. We will give the details of these steps in Chapters 14, 15, and 16. We will finish the proof of Theorem 13.12 in Chapter 17.

### 13.1 Tangles and $q$-tangles

Recall that a tangle representative is a pair $(C, L(\mathcal{L}) \cap C)$ for a rational homology cylinder $C$ and a long tangle representative $L: \mathcal{L} \hookrightarrow \tilde{R}(C)$ as in Definition 12.1.

**Definition 13.1.** In this book, a *braid representative* is a tangle representative $T(\tilde{\gamma})$ of the standard cylinder $D_1 \times [0, 1]$ whose components called *strands* may be expressed as

$$ \{ (\tilde{\gamma}_b(t), t) : t \in [0, 1] \}, $$

for an element $b$ of a finite set $B$, which labels the strands. In the above expression $\tilde{\gamma}_b: [0, 1] \to \tilde{D}_1$ is a path, and, for any $t \in [0, 1]$ and any pair $(b, b')$ of distinct elements of $B$, we have $\tilde{\gamma}_b(t) \neq \tilde{\gamma}_{b'}(t)$ as in Figure 13.1. Such a braid representative can be viewed naturally as a path $\tilde{\gamma}: [0, 1] \to \tilde{C}_B[\tilde{D}_1]$, where $\tilde{C}_B[\tilde{D}_1]$ is the space of injections of $B$ into $\tilde{D}_1$, defined in the beginning of Section 8.6.
<figure><img src="image_101.png" /><figcaption>Figure 13.1: A braid representative with three strands</figcaption></figure>

In this book, a *braid* (resp. a *q-braid*) is a homotopy class of paths $\gamma: [0, 1] \to \dot{S}_B(\mathbb{C})$ (resp. of paths $\gamma: [0, 1] \to S_B(\mathbb{C})$) for some finite set $B$, where $S_B(\mathbb{C})$ is the compactification of $\dot{S}_B(\mathbb{C})$ described in Theorem 8.11. A braid $\gamma$ induces the tangle $T(\gamma)$, which is also called a braid, as above. The path $\bar{\gamma}$ is the path such that $\bar{\gamma}(t) = \gamma(1-t)$. A braid is naturally framed by the parallels obtained by pushing it in the direction of the real line of $\mathbb{C}$.

Tangles, as in Definition 12.15, can be multiplied if they have representatives that can be, i.e., if the top configuration of the first tangle agrees with the bottom configuration of the second one, up to dilation and translation. The product is associative. Framed tangles multiply vertically to give rise to framed tangles.

A *q-tangle* is a framed tangle whose bottom and top configurations are allowed to be limit configurations in some $S_{B-}(\mathbb{C})$ and in some $S_{B+}(\mathbb{C})$. More precisely, a *q-tangle* is represented by a product

$$T(\gamma^{-})(\mathcal{C}, L)T(\gamma^{+}),$$

as in Figure 13.2, where $\gamma^{-}$ and $\gamma^{+}$ are *q-braids*, $(\mathcal{C}, L)$ is a framed tangle whose bottom configuration is $\gamma^{-}(1)$ and whose top configuration is $\gamma^{+}(0)$, and the strands of $T(\gamma^{-})$ and $T(\gamma^{+})$ get their orientations from the orientation of $L$. For consistency, we allow braids with 0 or 1 strand, and we agree that $S_{\emptyset}(\mathbb{C})$ and $S_{\{b\}}(\mathbb{C})$ each have one element, which is the unique configuration of one point in $\mathbb{C}$ up to translation in the latter case. Note that the restriction of a *q-tangle* to one of its components is a framed tangle since configurations of at most two points are always injective. The components of a *q-tangle* representative are framed since braids are.

<figure><img src="image_102.png" /><figcaption>Figure 13.2: A *q-tangle* representative</figcaption></figure>
Now, $q$-tangles are classes of these representatives under the equivalence relation that identifies $T(\gamma^{-})(C, L)T(\gamma^{+})$ with $T(\gamma^{-\prime})(C', L')T(\gamma^{+\prime})$ if and only if $\gamma^{-}(0) = \gamma^{-\prime}(0)$, $\gamma^{+}(1) = \gamma^{+\prime}(1)$, and the framed tangles $(C, L)$ and $T(\alpha)(C', L')T(\beta)$ represent the same framed tangle for any braids $\alpha$ and $\beta$ such that

• the composition $T(\alpha)(C', L')T(\beta)$ is well-defined,

• the path $\alpha$ of $\tilde{S}_{B^{-}}(\mathbb{C})$ is homotopic to $\overline{\gamma^{-}\gamma^{-\prime}}$ in $S_{B^{-}}(\mathbb{C})$, and

• the path $\beta$ of $\tilde{S}_{B^{+}}(\mathbb{C})$ is homotopic to $\gamma^{+\prime}\overline{\gamma^{+}}$ in $S_{B^{+}}(\mathbb{C})$ (by homotopies that fix the boundary points).

The domain of a $q$-tangle (represented by) $T(\gamma^{-})(C, L)T(\gamma^{+})$ is (identified with) the domain $\mathcal{L}$ of $L: \mathcal{L} \hookrightarrow C$, its bottom configuration is $\gamma^{-}(0)$, and its top configuration is $\gamma^{+}(1)$.

**Examples 13.2.** A combinatorial configuration is a nonassociative word in the letter $\bullet$. The combinatorial $q$-tangles $(L, w^{-}, w^{+})$ of Definition 12.16 are examples of $q$-tangles. They are the $q$-tangles of the standard cylinder whose bottom and top configurations are combinatorial. Recall the notation from Example 8.3. A combinatorial $q$-tangle $(L, w^{-}, w^{+})$ may be represented as $T(\gamma^{-})(C, L(1/4))T(\gamma^{+})$ for a $q$-braid $\gamma^{-}$ from $\gamma^{-}(0) = w^{-}$ to $\gamma^{-}(1) = w^{-}(1/4)$ such that $\gamma^{-}([0, 1])$ is $w^{-}([0, 1/4])$, a $q$-braid $\gamma^{+}$ from $\gamma^{+}(0) = w^{+}(1/4)$ to $\gamma^{+}(1) = w^{+}$ such that the sets $\gamma^{+}([0, 1])$ and $w^{+}([0, 1/4])$ are equal, and a representative $L(1/4)$ of $L$ from $w^{-}(1/4)$ to $w^{+}(1/4)$ in the isotopy class of $L$ of Definition 12.16.

We may omit the external pair of parentheses in a combinatorial configuration since it is always present. We may similarly omit the only possible two-point combinatorial configuration $(\bullet)$ from the notation in combinatorial $q$-tangles. So we may represent the examples of Example 12.17 by

$$ \vcentcolon, \stackrel{\bullet}{\leftarrow}^{(\bullet)} \stackrel{\bullet}{\rightarrow}, \circled{\sim}, \text{ and } \circled{\sim}. $$

When the boundary of a $q$-tangle is empty, the $q$-tangle is a framed link in $\tilde{R}(C)$. Conversely, any asymptotically standard $\mathbb{Q}$-homology $\mathbb{R}^3$ equipped with a framed link may be obtained in this way, up to diffeomorphism.

In addition to the (vertical) product, which extends to $q$-tangles naturally, $q$-tangles support a *cabling* operation of a component $K$ of a $q$-tangle $T_m$ (the main tangle) by a $q$-tangle $T_i$ (the inserted tangle), which produces a $q$-tangle $T_m(T_i/K)$. This operation roughly consists in replacing the strand $K$ in $T_m$ by a tangle $T_i$ with respect to the framing of $K$, as in Figure 13.3.
For example, if $T_i$ is the trivial braid $|1|2$ with two strands and if $T_m$ is a combinatorial $q$-tangle, then $T_m(T_i/K)$ is the tangle $T_m(2 \times K)$ described before Theorem 12.22. As another example, the product $L_1 \otimes L_2$ described before Example 12.20 can be written as $(L_1 \otimes |2)(L_2/|2)$, where $L_1 \otimes |2| = (|1|2)(L_1/|1|$.

Let us describe the cabling operation in general in (lengthy) detail.<fnref n="1" />

**Notation 13.3.** A *semi-pure q-tangle* is a $q$-tangle with identical bottom and top configuration (up to dilation and translation). A *pure q-tangle* is a semi-pure $q$-tangle whose interval components connect a bottom configuration point to the top configuration’s corresponding point. The cabling operation that produces $T_m(T_i/K)$ is defined for any pair $(T_m, T_i)$ of $q$-tangles equipped with an interval component $K$ of $T_m$ going from bottom to top. It is also defined for any pair $(T_m, T_i)$ of $q$-tangles equipped with a framed circle component $K$ of $T_m$ provided that $T_i$ is semi-pure.

<figure><img src="image_103.png" /><figcaption>Figure 13.3: Examples of cablings</figcaption></figure>

We begin with the details when $K$ is a circle, because they are lighter in this case. When $K$ is a closed component and $T_i$ is a framed tangle ($C_i, L_i$) with identical injective bottom and top configuration, pick a tubular neighborhood $D^2 \times K$ of $K$, trivialized with respect to the parallelization of $K$, that does not meet the other components of $T_m$. Write this neighborhood as $D^2 \times [0, 1]/(0 \sim 1)$, and replace it by $(C_i, L_i)$ using the identification of $N(\partial(D_1 \times [0, 1]))$ with a neighborhood of $\partial(D_1 \times [0, 1])$ in order to obtain $T_m(T_i/K)$. Note that when $\gamma$ is a braid such that $\gamma(1)$ is the bottom

<footnote n="1">¹Notation 13.3 ends 4 lines before Definition 13.4.</footnote>
configuration of $T_i$, we have

$$T_m \left( \frac{T_i}{K} \right) = T_m \left( \frac{T(\gamma)T_i T(\bar{\gamma})}{K} \right).$$

Any semi-pure $q$-tangle $T_q$ can be written as $T(\gamma)(C_i, L_i)T(\bar{\gamma})$, for some $q$-braid $\gamma$ and some framed tangle $(C_i, L_i)$. For such a tangle, set $T_m(T_q/K) = T_m((C_i, L_i)/K)$. It is easy to check that this definition is consistent.

Let us now define *cabling* or *duplication* for configurations. Let $B$ and $B_i$ be nonempty finite sets, let $b \in B$, and let $B(B_i/b) = (B \setminus \{b\}) \cup B_i$. Let $c_m$ be an element of $S_B(\mathbb{C})$, and let $c_i$ be an element of $S_{B_i}(\mathbb{C})$. The configuration $c_m(c_i/b)$ of $S_{B(B_i/b)}(\mathbb{C})$ is the configuration obtained by letting $c_i$ replace $b$. Up to translation, there is only one configuration of a set of one element. So $c_m(c_i/b)$ is this unique configuration if $|B(B_i/b)| = 1$. If $|B_i| = 1$, then $c_m(c_i/b) = c_m$. If $|B| = 1$, then $c_m(c_i/b) = c_i$ (with natural identifications). Assume that $|B_i| \ge 2$ and $|B| \ge 2$. When $c_m$ and $c_i$ are both combinatorial configurations, it makes natural sense to let $c_i$ replace $b$ to produce $c_m(c_i/b)$. In general, recall Definition 8.24 and define the $\Delta$-parenthesization $\tau(c_m)(\tau(c_i)/b)$ of $B(B_i/b)$ from the respective $\Delta$-parenthesizations $\tau(c_m)$ and $\tau(c_i)$ of $c_m$ and $c_i$ by the following one-to-one correspondence

$$ \phi: \begin{array}{ccc} \tau(c_m) \sqcup \tau(c_i) & \rightarrow & \tau(c_m)(\tau(c_i)/b) \\ A & \mapsto & \begin{cases} A & \text{if } A \in \tau(c_i) \text{ or } (A \in \tau(c_m) \text{ and } b \notin A) \\ A(B_i/b) & \text{if } A \in \tau(c_m) \text{ and } b \in A. \end{cases} \end{array} $$

With the notation of Theorem 8.28, the configuration $c_m(c_i/b)$ is the configuration of $S_{B(B_i/b), \tau(c_m)(\tau(c_i)/b)}(\mathbb{C})$ that restricts to $B_i$ as $c_i$, and to $B(\{b'\}/b)$ as $c_m$ for any element $b'$ of $B_i$.

Assume that $K$ is an interval component going from bottom to top of a $q$-tangle

$$T_m = T(\gamma_m^-)(C_m, L_m)T(\gamma_m^+),$$

and define $T_m(T_i/K)$ for a $q$-tangle $T_i = T(\gamma_i^-)(C_i, L_i)T(\gamma_i^+)$. Let $B_i^-$, (resp. $B_i^+$, $B^-$, $B^+$) denote the set of strand indices of $\gamma_i^-$ (resp. $\gamma_i^+$, $\gamma_m^-$, $\gamma_m^+$). Let $b_K^-$ (resp. $b_K^+$) denote the strand index of $K$ in $B^-$ (resp. $B^+$). Let $c_m^-$ (resp. $c_i^-$) denote the bottom configuration of $T_m$ (resp. $T_i$). Let $c_m^+$ (resp. $c_i^+$) denote the top configuration of $T_m$ (resp. $T_i$). Assume that $\gamma_m^-([0, 1]) \subset \tilde{S}_{B^-}(\mathbb{C})$, $\gamma_i^-([0, 1]) \subset \tilde{S}_{B_i^-}(\mathbb{C})$, $\gamma_m^+([0, 1]) \subset \tilde{S}_{B^+}(\mathbb{C})$, and $\gamma_i^+([0, 1]) \subset \tilde{S}_{B_i^+}(\mathbb{C})$. Let $I_K$ denote the intersection of $K$ with $C_m$. Identify $I_K$ with $[0, 1]$. Let $D^{(i)}$ be a copy of the disk $D^2$. Let $D^{(i)} \times [0, 1]$ be a tubular neighborhood of $I_K$ in $C_m$ that does not meet the other components of $L_m$ and that meets $\partial C_m$ along $D^{(i)} \times \partial [0, 1]$ inside $D_1 \times \partial [0, 1]$, such that $(\{\pm 1\} \times [0, 1]) \subset (\partial D^{(i)} \times [0, 1])$
is the given parallel of $I_K$. Replace $D^{(i)} \times [0, 1]$ by $(C_i, L_i)$ in order to get a tangle $(C_m, L_m)((C_i, L_i)/I_K)$.

Let $\gamma_m^-(\gamma_i^-/K)$ be the path composition $\gamma_m^-(c_i^-/K) (\gamma_m^-(1)(\gamma_i^-/K))$ of the paths $\gamma_m^-(c_i^-/K)$ and $\gamma_m^-(1)(\gamma_i^-/K)$ in $S_{B^-(B_i^-/b_K^-)}(\mathbb{C})$, where $\gamma_m^-(c_i^-/K)(t) = \gamma_m^-(t)(c_i^-/b_K^-)$ for any $t \in [0, 1]$, and the restriction to $[0, 1]$ of $\gamma_m^-(1)(\gamma_i^-/K)$ is represented by a map from $[0, 1]$ to $\tilde{C}_{B^-(B_i^-/b_K^-)}[\tilde{D}_1]$,

* which maps 1 to the bottom configuration of $(C_m, L_m)((C_i, L_i)/I_K)$,

* whose restriction to $B^- \setminus \{b_K^-\}$ is constant,<fnref n="2" /> and,

* whose restriction to $B_i^-$ is a lift of $\gamma_i^-|_{[0,1]}$ in $\tilde{C}_{B_i^-}[p_{\mathbb{C}}(D^{(i)} \times \{0\})]$, such that $\gamma_m^-(1)(\gamma_i^-/K)$ is composable by $\gamma_m^-(c_i^-/K)$ on its left.

The tangle $T_m(T_i/K)$ is defined to be

$$T(\gamma_m^-(\gamma_i^-/K))((C_m, L_m)((C_i, L_i)/I_K))T(\gamma_m^+(\gamma_i^+/K)),$$

where the definition of $T(\gamma_m^+(\gamma_i^+/K))$, similar to that of $T(\gamma_m^-(\gamma_i^-/K))$, follows (and can be skipped...).

The path $\gamma_m^+(\gamma_i^+/K)$ is the path composition $(\gamma_m^+(0)(\gamma_i^+/K)) \gamma_m^+(c_i^+/K)$ in $S_{B^+(B_i^+/b_K^+)}(\mathbb{C})$, where $\gamma_m^+(c_i^+/K)(t) = \gamma_m^+(t)(c_i^+/b_K^+)$ for any $t \in [0, 1]$, and the restriction to $[0, 1[$ of $\gamma_m^+(0)(\gamma_i^+/K)$ is represented by a map from $[0, 1[$ to $\tilde{C}_{B^+(B_i^+/b_K^+)}[\tilde{D}_1]$, mapping 0 to the top configuration of $(C_m, L_m)((C_i, L_i)/I_K)$, whose restriction to $B^+ \setminus \{b_K^+\}$ is constant, and whose restriction to $B_i^+$ is a lift of $\gamma_i^+|_{[0,1]}$ in $\tilde{C}_{B_i^+}[p_{\mathbb{C}}(D^{(i)} \times \{1\})]$, such that $\gamma_m^+(0)(\gamma_i^+/K)$ is composable by $\gamma_m^+(c_i^+/K)$ on its right.

A particular case of cablings is the case in which the inserted $q$-tangle $T_i$ is just the $q$-tangle $y \times [0, 1]$ associated to the constant path of $S_{B_i}(\mathbb{C})$ that maps $[0, 1]$ to a configuration $y$ of $S_{B_i}(\mathbb{C})$. (Formally, this $q$-tangle is represented by $T(\gamma)(\gamma(1) \times [0, 1])T(\bar{\gamma})$ for some path $\gamma$ of $S_{B_i}(\mathbb{C})$ such that $\gamma(0) = y$ and $\gamma(1) \in \tilde{S}_{B_i}(\mathbb{C})$.) Set

$$T_m(y \times K) = T_m((y \times [0, 1])/K).$$

If $K$ is a closed component, then $T_m(y \times K)$ depends only on $|B_i|$. It is denoted by $T_m(|B_i| \times K)$. If $K$ is an interval component and $y$ is the unique configuration of $S_{\{1,2\}}(\mathbb{R})$, then $T_m(y \times K)$ is again denoted by $T_m(2 \times K)$. These special cablings introduced before Theorem 12.22 are called duplications or doublings. Our functor $\mathcal{Z}^f$ will behave well under all cablings.

<footnote n="2">²It is thus located in $\tilde{C}_{B^- \setminus \{b_K^-\}}[\tilde{D}_1 \setminus p_{\mathbb{C}}(D^{(i)} \times \{0\})]$.</footnote>
We end this section by completing Definition 12.10 of the self-linking number for a framed $q$-tangle component, which goes from bottom to bottom or from top to top.

**Definition 13.4.** The self-linking number $lk(K, K_\parallel)$ of a framed component $K$ of a $q$-tangle going from bottom to bottom (resp. from top to top) is defined as follows. The self-linking number depends only on the component. So there is no loss of generality in representing $K$ by a tangle representative with injective bottom (resp. top) configuration whose ends are at a distance bigger than $2\epsilon$ for a small positive $\epsilon$. Also assume that $K_\parallel(0) = K(0) + (\epsilon, 0, 0)$ and $K_\parallel(1) = K(1) - (\epsilon, 0, 0)$. (There is no loss of generality in this assumption either. It suffices to choose a parallel that satisfies this assumption to define the self-linking number. Recall Definition 12.8.) Let $\hat{K} = K([0, 1]) \cup \gamma([0, 1])$ for an arbitrary path $\gamma$ from $\gamma(0) = K(1)$ to $\gamma(1) = K(0)$ such that $\gamma([0, 1]) \subset \tilde{R}(C) \setminus C$.

Let $[K(1), K(0)]$ denote the straight segment from $K(1)$ to $K(0)$ in $D_1$. Let $\alpha_1: [0, 1] \to D_1$ be an arc from $K_\parallel(1)$ to a point $a_1$ inside $[K(1), K(0)]$ such that $\alpha_1(t) = K(1) - \epsilon \exp(2i\pi\theta_1 t)$ for some real number $\theta_1$. Similarly, let $\alpha_0: [0, 1] \to D_1$ be an arc from $K_\parallel(0)$ to a point $a_0$ inside $[K(1), K(0)]$ such that $\alpha_0(t) = K(0) + \epsilon \exp(2i\pi\theta_0 t)$ for some $\theta_0 \in \mathbb{R}$.

<figure><img src="image_104.png" /><figcaption>Figure 13.4: A general picture of $\alpha_0$ and $\alpha_1$, and a picture of $\hat{K}_\parallel$ when $\theta_0 = -\frac{1}{2}$, $\theta_1 = \frac{1}{2}$, and $K$ goes from bottom to bottom</figcaption></figure>

If $K$ goes from bottom to bottom, define

$$ \hat{K}_{\parallel, \theta_0, \theta_1} = K_{\parallel}([0, 1]) \cup ((\alpha_1 \cup [a_1, a_0] \cup \bar{\alpha}_0) \times \{0\}), $$

and set

$$ lk(K, K_\parallel) = lk(\hat{K}, \hat{K}_{\parallel, \theta_0, \theta_1}) + \theta_1 + \theta_0. $$

If $K$ goes from top to top, define

$$ \hat{K}_{\parallel, \theta_0, \theta_1} = K_{\parallel}([0, 1]) \cup ((\alpha_1 \cup [a_1, a_0] \cup \bar{\alpha}_0) \times \{1\}) $$

and set

$$ lk(K, K_\parallel) = lk(\hat{K}, \hat{K}_{\parallel, \theta_0, \theta_1}) - (\theta_1 + \theta_0). $$
Note that these definitions do not depend on the chosen $\theta_1$ and $\theta_0$, which are well-determined modulo $\mathbb{Z}$. The angles $2\pi\theta_1$ and $2\pi\theta_0$ are both congruent to the angle from the oriented real line to $\overrightarrow{K(0)K(1)}$ modulo $2\pi$. So $(lk(K, K_\parallel) - 2\theta_1) \in \mathbb{Q}$ when $K$ goes from bottom to bottom, and $(lk(K, K_\parallel) + 2\theta_1) \in \mathbb{Q}$ when $K$ goes from top to top.

When $\overrightarrow{K(0)K(1)}$ directs and orients the real line, we can choose $\theta_0 = \theta_1 = 0$, and this definition coincides with Definition 12.10. When $\overrightarrow{K(1)K(0)}$ directs and orients the real line, we can choose $\theta_1 = \frac{1}{2} = -\theta_0$ so that $\hat{K}_\parallel$ is simply as in Figure 13.4. So the present definition is again consistent with Definition 12.10.

**Lemma 13.5.** The self-linking number does not depend on the orientations of the components.

**PROOF:** This is easy to see for closed components, and this is part of the definition for components going from bottom to top or from top to bottom. When $K$ goes from top to top, let $K'$ stand for $(-K)$, and let $K'_\parallel$ be the parallel obtained from $K_\parallel$ by a rotation of angle $\pi$ around $K$ (and by reversing the orientation). Choose the corresponding angles $\theta'_0$ and $\theta'_1$ to be $\theta'_0 = \theta_1 - \frac{1}{2}$ and $\theta'_1 = \theta_0 + \frac{1}{2}$. So $\hat{K}'_{\parallel, \theta'_0, \theta'_1}$ is isotopic to $(-\hat{K}_{\parallel, \theta_0, \theta_1})$ in the complement of $\hat{K}$. See Figure 13.5. $\square$

<figure><img src="image_105.png" /><figcaption>Figure 13.5: A general picture of $\theta'_0$ and $\theta'_1$, and a picture of $\hat{K}'_\parallel$ when $\theta'_0 = -\frac{1}{2}$, $\theta'_1 = \frac{1}{2}$, and $K$ goes from top to top</figcaption></figure>

Proposition 16.13 will show that the real-valued self-linking numbers $lk(K, K_\parallel)$ coincide with $I_\theta(K, \tau)$ for the interval components $K$ of the straight tangles of Definition 16.11. Its proof relies only on the beginning of Section 16.3.

## 13.2 Definition of $\mathcal{Z}^f$ for all $q$-tangles

Recall Definition 12.12 of the extension of $\mathcal{Z}^f$ for $q$-tangles whose bottom and top configurations are injective. In this section, we extend the definition of $\mathcal{Z}^f$ to all $q$-tangles.
In Chapter 15, we will prove the following particular case of the functo-
riality property stated in Theorem 12.13. It is a direct corollary of Proposi-
tion 15.19.

**Proposition 13.6.** Let $(C_1, L_1)$ and $(C_2, L_2)$ be two framed tangle representatives such that the bottom of $L_2$ coincides with the top of $L_1$. If one of them is a braid, then we have

$$
\mathcal{Z}^f(C_1C_2, (L_1L_2)_{\parallel}) = \mathcal{Z}^f(C_1, L_1, L_{1\parallel})\mathcal{Z}^f(C_2, L_2, L_{2\parallel}).
$$

The following lemma allows us to consider the tangles’ bottom and top
configurations up to translation and dilation. It will also be a direct corollary
of Proposition 15.19.

**Lemma 13.7.** Let $\gamma: [0, 1] \to \check{C}_B[\check{D}_1]$. Let $p_{CS} \circ \gamma: [0, 1] \to \check{S}_B(\mathbb{C})$ be the composition of $\gamma$ by the natural projection $p_{CS}: \check{C}_B[\check{D}_1] \to \check{S}_B(\mathbb{C})$ (which mods out by translations and dilations). Then $\mathcal{Z}(\gamma)$ and $\mathcal{Z}^f(\gamma)$ depend only on $p_{CS} \circ \gamma$.

Under the assumptions of the lemma, we set $\mathcal{Z}(p_{CS} \circ \gamma) = \mathcal{Z}(\gamma)$ and $\mathcal{Z}^f(p_{CS} \circ \gamma) = \mathcal{Z}^f(\gamma)$. Recall that $\mathcal{Z}$ and $\mathcal{Z}^f$ coincide for braids. We extend the definition of $\mathcal{Z}$ to piecewise smooth paths of $\check{S}_B(\mathbb{C})$ so that $\mathcal{Z}$ is multiplicative with respect to path composition of smooth paths. The following theorem is essentially due to Sylvain Poirier [Poi00]. It allows us to extend $\mathcal{Z}^f$ to $q$-braids, and to $q$-tangles in Definition 13.10.

**Theorem 13.8.** Let $p_{CS} \circ \gamma: [0, 1] \to S_B(\mathbb{C})$ be a path whose restriction to $]0, 1[$ is the projection of some $\gamma$: $]0, 1[ \to \check{C}_B[\check{D}_1]$ that can be described by a collection of piecewise polynomial continuous maps $(\gamma_b: [0, 1] \to \check{D}_1)_{b \in B}$.<fnref n="3" /> Then $\lim_{\epsilon \to 0} \mathcal{Z}(p_{CS} \circ \gamma|_{[\epsilon, 1-\epsilon]})$ makes sense, and it depends only on the homotopy class of $p_{CS} \circ \gamma$ relatively to its boundary. It is denoted by $\mathcal{Z}(p_{CS} \circ \gamma)$ or $\mathcal{Z}^f(p_{CS} \circ \gamma)$.

Theorem 13.8 and Theorem 16.35, which generalizes its homotopy invariance part, will be proved in Section 16.5.

**Proposition 13.9.** Any $q$-braid $\gamma: [0, 1] \to S_B(\mathbb{C})$ is homotopic relatively to its boundary to a $q$-braid $p_{CS} \circ \tilde{\gamma}$ as in the statement of Theorem 13.8. Setting $\mathcal{Z}(\gamma) = \mathcal{Z}(p_{CS} \circ \tilde{\gamma})$ consistently extends the definition of $\mathcal{Z}$ to all $q$-braids. Furthermore, $\mathcal{Z}$ is multiplicative with respect to the $q$-braid composition: For two composable paths $\gamma_1$ and $\gamma_2$ of $S_B(\mathbb{C})$, we have

$$
\mathcal{Z}(\gamma_1 \gamma_2) = \mathcal{Z}(\gamma_1) \mathcal{Z}(\gamma_2).
$$

<footnote n="3">³Every $\gamma_b$ is polynomial over a finite number of intervals covering $[0, 1]$.</footnote>
PROOF: Let us first exhibit a q-braid $p_{CS} \circ \tilde{\gamma}$ homotopic to a given q-braid $\gamma: [0, 1] \to S_B(C)$, with the desired properties. Define a path $\tilde{\gamma}_1: [0, 1/3] \to C_B[\tilde{D}_1]$, such that $p_{CS} \circ \tilde{\gamma}_1(0) = \gamma(0)$, $\tilde{\gamma}_1([0, 1/3]) \subset \tilde{C}_B[\tilde{D}_1]$, and $\tilde{\gamma}_1$ is a path obtained by replacing all the parameters $\mu_A$ in the charts of Lemma 8.27 by $\epsilon t$ for $t \in [0, 1/3]$ for some small $\epsilon > 0$. So $\tilde{\gamma}_1$ is described by a collection of polynomial maps $(\tilde{\gamma}_{1,b}: [0, 1/3] \to \tilde{D}_1)_{b \in B}$. Similarly define a polynomial path $\tilde{\gamma}_3: [2/3, 1] \to C_B[\tilde{D}_1]$ such that $p_{CS} \circ \tilde{\gamma}_3(1) = \gamma(1)$ and $\tilde{\gamma}_3([2/3, 1]) \subset \tilde{C}_B[\tilde{D}_1]$. Define a path $\tilde{\gamma}'_2: [1/3, 2/3] \to C_B[\tilde{D}_1]$, such that $\tilde{\gamma}'_2(1/3) = \tilde{\gamma}_1(1/3)$, $\tilde{\gamma}'_2(2/3) = \tilde{\gamma}_3(2/3)$, and $p_{CS} \circ \tilde{\gamma}'_2$ is a path composition $(p_{CS} \circ \tilde{\gamma}_1)\gamma(p_{CS} \circ \tilde{\gamma}_3)$. This path $\tilde{\gamma}'_2$ of $C_B[\tilde{D}_1]$ is homotopic to a path of the interior $\tilde{C}_B[\tilde{D}_1]$ of the manifold $C_B[D_1]$ with ridges, and it is homotopic to a polynomial path $\tilde{\gamma}_2$ in $\tilde{C}_B[\tilde{D}_1]$. Now, the path composition $\tilde{\gamma} = \tilde{\gamma}_1\tilde{\gamma}_2\tilde{\gamma}_3$ satisfies the hypotheses of Theorem 13.8, and $p_{CS} \circ \tilde{\gamma}$ is homotopic to $\gamma$, relatively to its boundary. Furthermore, for any other path $\tilde{\gamma}'$ satisfying these properties, $p_{CS} \circ \tilde{\gamma}$ and $p_{CS} \circ \tilde{\gamma}'$ are homotopic relatively to the boundary. So the definition of $\mathcal{Z}(\gamma)$ is consistent.

Let us prove the multiplicativity. Pick a piecewise polynomial path $\tilde{\gamma}_1: [0, 1] \to C_B[\tilde{D}_1]$, such that $\gamma_1$ and $p_{CS} \circ \tilde{\gamma}_1$ are homotopic relatively to the boundary, and $\tilde{\gamma}_1([0, 1]) \subset \tilde{C}_B[\tilde{D}_1]$. Next, pick a piecewise polynomial path $\tilde{\gamma}_2: [0, 1] \to C_B[\tilde{D}_1]$, such that, $\tilde{\gamma}_2(t) = \tilde{\gamma}_1(1-t)$ for any $t \in [0, 1/2]$, $\gamma_2$ and $p_{CS} \circ \tilde{\gamma}_2$ are homotopic relatively to the boundary, and $\tilde{\gamma}_2([0, 1]) \subset \tilde{C}_B[\tilde{D}_1]$. Thus we have

$$ \mathcal{Z}(\gamma_1\gamma_2) = \lim_{\epsilon \to 0} \mathcal{Z}\left(p_{CS} \circ \left(\tilde{\gamma}_1|_{[\epsilon, 1/2]}\tilde{\gamma}_2|_{[1/2, 1-\epsilon]}\right)\right). $$

We also have

$$ \begin{aligned} \mathcal{Z}(\gamma_1)\mathcal{Z}(\gamma_2) &= \lim_{\epsilon \to 0} \mathcal{Z}(p_{CS} \circ \tilde{\gamma}_1|_{[\epsilon, 1-\epsilon]})\mathcal{Z}(p_{CS} \circ \tilde{\gamma}_2|_{[\epsilon, 1-\epsilon]}) \\ &= \lim_{\epsilon \to 0} \mathcal{Z}(p_{CS} \circ \tilde{\gamma}_1|_{[\epsilon, 1-\epsilon]}p_{CS} \circ \tilde{\gamma}_2|_{[\epsilon, 1-\epsilon]}) \\ &= \lim_{\epsilon \to 0} \mathcal{Z}(p_{CS} \circ (\tilde{\gamma}_1|_{[\epsilon, 1/2]}\tilde{\gamma}_2|_{[1/2, 1-\epsilon]})), \end{aligned} $$

where the second equality comes from Proposition 13.6. $\square$

Theorem 12.7, Proposition 13.6, Theorem 13.8, and Proposition 13.9 allow us to extend $\mathcal{Z}^f$ unambiguously to the q-tangles of Definition 13.1 as follows.

**Definition 13.10.** Let $\gamma^-$ and $\gamma^+$ be q-braids. Let $(C, L)$ be a framed tangle whose bottom and top configurations are $\gamma^-(1)$ and $\gamma^+(0)$, respectively. Orient the strands of $T(\gamma^-)$ and $T(\gamma^+)$ so that their orientations are consistent with the orientation of $L$. Define $\mathcal{Z}^f(T(\gamma^-)(C, L)T(\gamma^+))$ to be

$$ \mathcal{Z}^f(T(\gamma^-)(C, L)T(\gamma^+)) = \mathcal{Z}(\gamma^-)\mathcal{Z}^f(C, L)\mathcal{Z}(\gamma^+), $$
with respect to Definition 12.12 of $\mathcal{Z}^f(C, L)^4$

**Remark 13.11.** Let $(C, L_q)$ be a $q$-tangle from a bottom limit configuration $c^- \in S_{B-}(\mathbb{C})$ to a top configuration $c^+ \in S_{B+}(\mathbb{C})$. These limit configurations are initial points $\gamma^{\pm}(0) = c^{\pm}$ of polynomial paths $\gamma^{\pm}$ of $S_{B^{\pm}}(\mathbb{C})$, such that $\gamma^{\pm}([0, 1]) \subset \tilde{S}_{B^{\pm}}(\mathbb{C})$ (as in the proof of Proposition 13.9). This allows us to regard $c^{\pm}$ as a limit

$$c^{\pm} = \lim_{t \to 0} \gamma^{\pm}(t)$$

of injective configurations and view $L_q$ as a limit of framed tangles between injective configurations. We can indeed view the $q$-tangle $L_q$ as the limit of

$$L_{q, \varepsilon} = T(\overline{\gamma^{-}|_{[0, \varepsilon]}}) L_q T(\gamma^{+}|_{[0, \varepsilon]})$$

when $\varepsilon$ tends to 0. Then $\mathcal{Z}^f(C, L_q)$ can be defined alternatively to be

$$\mathcal{Z}^f(C, L_q) = \lim_{\varepsilon \to 0} \mathcal{Z}^f(L_{q, \varepsilon}).$$

Indeed, the above consistent definition implies

$$\mathcal{Z}^f(C, L_q) = \mathcal{Z}(\gamma^{-}|_{[0, \varepsilon]}) \mathcal{Z}^f(L_{q, \varepsilon}) \mathcal{Z}(\overline{\gamma^{+}|_{[0, \varepsilon]}}),$$

while Theorem 13.8 and Proposition 13.9 imply

$$\lim_{\varepsilon \to 0} \mathcal{Z}(\gamma^{-}|_{[0, \varepsilon]}) = \mathbf{1} \text{ and } \lim_{\varepsilon \to 0} \mathcal{Z}(\overline{\gamma^{+}|_{[0, \varepsilon]}}) = \mathbf{1}.$$

We will construct variants of $\mathcal{Z}^f$ in the spirit of Theorem 7.40 in Section 16.2. These variants will allow us to prove Theorem 13.12 in Chapter 17.

Proving Theorem 13.12 will require lengthy studies of compactifications of configuration spaces, which are not manifolds with boundaries. These studies will help to get all the nice and natural properties of $\mathcal{Z}^f$, stated in Theorem 13.12. Robin Koytcheff, Brian Munson, and Ismar Volić proposed another approach to obtain invariants of tangles and avoid our complicated configuration spaces in [KMV13].

In the following three chapters, we will show the construction of $\mathcal{Z}^f$ for $q$-tangles in detail, following the outline of this section.

<footnote n="4">⁴Alternatively, we can forget the orientations of $L$, $\gamma^-$, and $\gamma^+$ since $\mathcal{Z}^f$ depends on the component orientations as in Proposition 10.23 for $L$, $\gamma^-$, and $\gamma^+$.</footnote>
## 13.3 Properties of the functor $\mathcal{Z}^f$

In this section, we state the main natural properties of the functor $\mathcal{Z}^f = (\mathcal{Z}_n^f)_{n \in \mathbb{N}}$, whose construction is outlined in the previous section.

**Theorem 13.12.** The invariant $\mathcal{Z}^f$ of $q$-tangles satisfies the following properties.

* $\mathcal{Z}^f$ coincides with the invariant $\mathcal{Z}^f$ of Definition 7.41 for framed links in $\mathbb{Q}$-spheres.

* $\mathcal{Z}^f$ coincides with the Poirier functor $\mathcal{Z}^l$ of [Poi00] for combinatorial $q$-tangles in $\mathbb{R}^3$.

* **Naturality:** If $L$ is a $q$-tangle with domain $\mathcal{L}$, then $\mathcal{Z}^f(L) = (\mathcal{Z}_k^f(L))_{k \in \mathbb{N}}$ is valued in $\mathcal{A}(\mathcal{L}) = \mathcal{A}(\mathcal{L}; \mathbb{R})$, and $\mathcal{Z}_0^f(L)$ is the class of the empty diagram. If $L'$ is a subtangle of $L$ with domain $\mathcal{L}'$, then $\mathcal{Z}^f(L')$ is obtained from $\mathcal{Z}^f(L)$ by mapping all the diagrams with univalent vertices on $\mathcal{L} \setminus \mathcal{L}'$ to zero and by forgetting $\mathcal{L} \setminus \mathcal{L}'$.

* **Dependence on component orientations:** If $\mathcal{L}$ and the components of $L$ are unoriented, then $\mathcal{Z}^f(L)$ is valued in the space $\mathcal{A}(\mathcal{L})$ of Definition 6.16, as in Proposition 10.23. Otherwise, component orientation reversals affect $\mathcal{Z}^f$ as in Proposition 10.23.

* **Framing dependence:** For a $q$-tangle $L = (\mathcal{C}, \bigsqcup_{j=1}^k K_j, \bigsqcup_{j=1}^k K_{j\parallel})$,

$$ \prod_{j=1}^{k} \left( \exp(-lk(K_j, K_{j\parallel})\alpha) \#_j \right) \mathcal{Z}^f(L) $$

is independent of the framing of $L$. It is denoted by $\mathcal{Z}(\mathcal{C}, \bigsqcup_{j=1}^k K_j)^5$.

* **Functoriality:** For two $q$-tangles $L_1$ and $L_2$ such that the bottom configuration of $L_2$ coincides with the top configuration of $L_1$, we have

$$ \mathcal{Z}^f(L_1 L_2) = \mathcal{Z}^f(L_1) \mathcal{Z}^f(L_2), $$

with products obtained by stacking above in natural ways on both sides.

$$ \mathcal{Z}^f(\sim \#_j L_c) = \exp(\alpha) \#_j \mathcal{Z}^f(L_c). $$

We similarly have $\mathcal{Z}^f(\sim \#_j L_c) = \exp(-\alpha) \#_j \mathcal{Z}^f(L_c)$.

<footnote n="5">⁵In particular, when a kink $\sim$ is added to the jth component of a combinatorial $q$-tangle $L_c$, $\mathcal{Z}^f(L_c)$ is changed to</footnote>
* First duplication property: Let $K$ be a component of a $q$-tangle $L$, then we have

$$ \mathcal{Z}^f(L(2 \times K)) = \pi(2 \times K)^* \mathcal{Z}^f(L) $$

with respect to Notation 6.31.

* Second duplication property: Let $B$ be a finite set, let $y$ be an element of $S_B(\mathbb{C})$. Let $K$ be a component going from bottom to top in a $q$-tangle $L$. Recall $L(y \times K) = L((y \times [0, 1])/K)$ from Notation 13.3. Then we have

$$ \mathcal{Z}^f(L(y \times K)) = \pi(B \times K)^* \mathcal{Z}^f(L). $$

* Dependence on ambient orientation: Let $s_{1/2}$ be the orthogonal reflection with respect to the horizontal plane at height $1/2$. Extend $s_{1/2}$ from $\partial C$ to an orientation-reversing diffeomorphism $s_{1/2}$ of $C$. Define the parallales $s_{1/2}(K)_{\parallel}$ of interval components $s_{1/2}(K)$ of $s_{1/2}(L)$ so that they satisfy $lk_{s_{1/2}(C)}(s_{1/2}(K), s_{1/2}(K)_{\parallel}) = -lk_C(K, K_{\parallel})$. Then we have

$$ \mathcal{Z}_n^f(s_{1/2}(C), s_{1/2} \circ L) = (-1)^n \mathcal{Z}_n^f(C, L) $$

for all $n \in \mathbb{N}$.

* Symmetry: Let $\rho$ be a rotation of $\mathbb{R}^3$ that preserves the standard homology cylinder $D_1 \times [0, 1]$ (setwise). Let $L$ be a $q$-tangle of a rational homology cylinder $C$. Extend $\rho$ from $\partial C$ to an orientation-preserving diffeomorphism $\rho$ of $C$. If the angle of $\rho$ is different from $0$ and $\pi$, assume that the interval components of $L$ go from bottom to top or from top to bottom. Then we have

$$ \mathcal{Z}^f(\rho(C), \rho \circ L) = \mathcal{Z}^f(C, L), $$

where the parallels of interval components $\rho(K)$ of $\rho(L)$ are defined so that they satisfy $lk_{\rho(C)}(\rho(K), \rho(K)_{\parallel}) = lk_C(K, K_{\parallel})$.<fnref n="6" />

* Cabling property: Let $B$ be a finite set with cardinality greater than 1. Let $y \in S_B(\mathbb{C})$, let $y \times [0, 1]$ denote the corresponding $q$-braid, and let $K$ be a strand of $y \times [0, 1]$. Let $L$ be a $q$-tangle with domain $\mathcal{L}$. Then $\mathcal{Z}^f((y \times [0, 1]) (L/K))$ is obtained from $\mathcal{Z}^f(L)$ by the natural injection from $\mathcal{A}(\mathcal{L})$ to $\mathcal{A}(\bigsqcup_{b \in B} \mathbb{R}^{\{b\}} (\frac{\mathcal{L}}{K}))$.

<footnote n="6">⁶This statement applies to a diffeomorphism $\rho$ of $C$ restricting to $\partial C$ as the identity map, in particular.</footnote>
• Full twist in degree one: The expansion $\mathcal{Z}_{\le 1}^f$ up to degree 1 of $\mathcal{Z}^f$ satisfies

$$ \mathcal{Z}_{\le 1}^f \left( \mathcal{U} \right) = 1 + \left[ \begin{array}{c|c} \bullet & \bullet \\ \bullet & \bullet \end{array} \right], $$

where the endpoints of the tangle are assumed to lie on $\mathbb{R} \times \{0, 1\}$.

• Group-like behavior: For any $q$-tangle $L$ and any integer $n$, we have

$$ \Delta_n(\mathcal{Z}_n^f(L)) = \sum_{i=0}^{n} \mathcal{Z}_i^f(L) \otimes \mathcal{Z}_{n-i}^f(L) $$

with respect to the coproduct maps $\Delta_n$ of Section 6.5.

The definition of $\mathcal{Z}^f$ in Section 13.2 obviously extends the definition of $\mathcal{Z}^f$ for tangles with empty boundary. Note that the naturality property in Theorem 13.12 easily follows from the definition (which will be justified later). The behavior of $Z(C, L, \tau)$ with respect to the coproduct can be observed from the definition, as in the proof of Proposition 10.3. According to Lemmas 6.37 and 6.38, Definition 10.5, and Proposition 10.13, the correction factors are group-like. So the compatibility between the various products and the coproduct ensures that $\mathcal{Z}^f$ behaves as stated in Theorem 13.12 with respect to the coproduct, for framed tangles between injective configurations. Then Remark 13.11 ensures that this also holds for general $q$-tangles.

**Corollary 13.13.** If $L$ has at most one component, let $p^c$ be the projection given by Corollary 6.40 from $\mathcal{A}(\mathcal{L})$ to the space $\mathcal{A}^c(\mathcal{L})$ of its primitive elements. Set

$$ \mathfrak{z}^f(C, L) = p^c(\mathcal{Z}^f(C, L)). $$

Then we have

$$ \mathcal{Z}^f(C, L) = \exp(\mathfrak{z}^f(C, L)). $$

**PROOF:** In these cases, Lemma 6.37 guarantees that $\mathcal{A}(\mathcal{L})$ is a graded bialgebra, and Theorem 13.12 implies that $\mathcal{Z}^f(C, L)$ is group-like. Conclude with Theorem 6.41. $\square$

The proof of Theorem 13.12 will be finished at the end of Section 17.4. The multiplicativity of $\mathcal{Z}$ under connected sum of Theorem 10.26 is a direct corollary of the functoriality of $\mathcal{Z}^f$ in the above statement. The functoriality also implies that $\mathcal{Z}$ and $\mathcal{Z}^f$ map tangles consisting of vertical segments in the standard cylinder to 1. Consider such a trivial braid consisting of the two vertical segments $\{-\frac{1}{2}\} \times [0, 1]$ and $\{\frac{1}{2}\} \times [0, 1]$. Cable $\{-\frac{1}{2}\} \times [0, 1]$ by a $q$-tangle $(C_1, L_1)$, and cable $\{\frac{1}{2}\} \times [0, 1]$ by a $q$-tangle $(C_2, L_2)$. Call the
<figure><img src="image_106.png" /><figcaption>Figure 13.6: Cabling a component going from bottom to bottom in two different ways</figcaption></figure>

**Remark 13.17.** The behavior of $\mathcal{Z}^f$ when a component $K$ of a link $L: \mathcal{L} \to \tilde{R}(C_m)$ is cabled by a semi-pure $q$-tangle $(C, T_i)$ can be described as follows.

• cut the domain of $K$ to replace it by a copy of $\mathbb{R}$ using Proposition 6.27,

• duplicate the corresponding strand and $\mathcal{Z}^f(R(C_m), L)$, accordingly, as in the duplication property above,

• multiply the obtained element by $\mathcal{Z}^f(C, T_i)$ i.e., concatenate the diagrams, naturally,

• finally, close the domain of $L(T/K)$.

This follows easily from Theorem 13.12, by viewing $(R, L)$ as a vertical composition of two tangles, where the bottom one is just a cup $\cup$. This cup is a trivial strand going from top to top in a standard cylinder. It is a part of $K$. We illustrate the process in Figure 13.7. In order to apply the iterated first duplication property, we change $T_i$ to a conjugate of $T_i^c$ whose bottom (or top) configuration is combinatorial. The result follows with $\mathcal{Z}^f(C, T_i^c)$ instead of $\mathcal{Z}^f(C, T_i)$. Lemma 6.33 guarantees that changing $\mathcal{Z}^f(C, T_i^c)$ to $\mathcal{Z}^f(C, T_i)$ in the above recipe does not change the result. (This is consistent with the fact that the tangles $L(T_i^c/K)$ and $L(T_i/K)$ coincide.)

The first cutting step is not legitimate if $L$ has interval components. (Recall Exercise 6.28). So the above recipe does not generalize to this case.<fnref n="7" />

<footnote n="7">⁷For a circle component $K$ in a general $q$-tangle $T_m$, and a semi-pure $q$-tangle $T_i$ as</footnote>
<figure><img src="image_107.png" /><figcaption>Figure 13.7: Cabling a link component with a semi-pure tangle, step by step</figcaption></figure>

For a $q$-braid $\gamma: [0, 1] \to S_B(\mathbb{C})$, $\mathcal{Z}^f(\gamma) = \mathcal{Z}(\gamma)$ stands for

$$ \mathcal{Z}^f(T(\gamma)) = \mathcal{Z}^f(D_1 \times [0, 1], T(\gamma)). $$

The following proposition, which leads to interesting cablings, is a corollary of Theorem 13.12.

**Proposition 13.18.** Let $q$ be a positive integer. Let $\gamma_{1,q}$ be a braid represented by the map

$$ \begin{array}{ccc} [0, 1] \times \underline{q} & \to & \mathbb{C} \\ (t, k) & \mapsto & \frac{1}{2} \exp \left( \frac{2i\pi(k+\chi(t))}{q} \right) \end{array} $$

for a surjective map $\chi: [0, 1] \to [0, 1]$ with nonnegative derivative, which is constant in neighborhoods of 0 and 1, as in Figure 13.1 of $\gamma_{1,3}$. Recall Notation 6.31 and the anomaly $\alpha$ of Proposition 10.13. Then we have

$$ \mathcal{Z}^f(\gamma_{1,q}) = \exp \left( \pi (q \times \hat{t})^* \left(\frac{1}{q}\alpha\right) \right) \left( \exp(-\frac{1}{q}\alpha) \otimes \dots \otimes \exp(-\frac{1}{q}\alpha) \right). $$

<footnote>above, $T_m(T_i/K)$ is well-defined and it coincides with $T_m(T_i^c/K)$ for any conjugate $T_i^c$ of $T_i$. We can still write $T_m$ as a product $(c_m^- \times [0, 1] \otimes \cup) T_m'$, where $\cup$ is a part of $K$, pick a conjugate $T_i^c$ of $T_i$ whose bottom (or top) configuration is combinatorial, and compute $\mathcal{Z}^f(T_m(T_i^c/K))$ from $\mathcal{Z}^f(T_m')$ and $\mathcal{Z}^f(T_i^c)$. Unfortunately, $\mathcal{Z}^f(T_m')$ is not determined by $\mathcal{Z}^f(T_m)$ anymore.</footnote>
PROOF: Let $\tilde{\gamma}_{1,1}$ be the trivial one-strand braid $K$ in the standard cylinder equipped with its parallel $K_\parallel$ such that $lk(K, K_\parallel) = 1$. The framing dependence property in Theorem 13.12 implies $\mathcal{Z}^f(\tilde{\gamma}_{1,1}) = \exp(\alpha)$. Let $\tilde{\gamma}_{q,q}$ be the $q$-tangle obtained by cabling $\tilde{\gamma}_{1,1}$ as in the second duplication property, by replacing the one-point configuration by the planar configuration of $C$ consisting of the $q$ points $\frac{1}{2}\exp(\frac{2i\pi k}{q})$, for $k \in \underline{q}$. This duplication operation equips each strand $K_k$ of $\tilde{\gamma}_{q,q}$ with a parallel $K_{k\parallel,1}$ such that $lk(K_k, K_{k\parallel,1}) = 1$. The $q$-tangle $\tilde{\gamma}_{q,q}$ coincides with $\gamma_{1,q}^q$ except for the framing since the standard framing of $\gamma_{1,q}^q$ equips $K_k$ with a parallel $K_{k\parallel}$ such that $lk(K_k, K_{k\parallel}) = 0$. According to the second duplication property, we have

$$ \mathcal{Z}^f(\tilde{\gamma}_{q,q}) = \exp \left( \pi (q \times \hat{\uparrow})^* (\alpha) \right), $$

whereas the framing dependence property implies

$$ \mathcal{Z}^f(\gamma_{1,q}^q) = \mathcal{Z}^f(\tilde{\gamma}_{q,q}) (\exp(-\alpha) \otimes \cdots \otimes \exp(-\alpha)). $$

By the invariance of $\mathcal{Z}^f$ under rotation, $\mathcal{Z}^f(\gamma_{1,q})$ is invariant under cyclic permutation of the strands. So the functoriality implies $\mathcal{Z}^f(\gamma_{1,q}^q) = \mathcal{Z}^f(\gamma_{1,q})^q$. The result follows by unicity of a $q$th root of $\mathcal{Z}^f(\gamma_{1,q}^q)$ with 1 as degree 0 part. $\square$

<figure><img src="image_108.png" /></figure>

Dror Bar-Natan, Thang Lê, and Dylan Thurston computed the Kontsevich integral of the trivial knot $O$ in [BNLT03]. Thus, Corollary 12.29 allows one to express $\mathcal{Z}^f$ for the unknot and for the torus knots as a function of the anomaly $\alpha$. Note that the symmetry properties imply that $\mathcal{Z}^f(\cap)$ vanishes in odd degree and that we have $\mathcal{Z}^f(\cap) = \mathcal{Z}^f(\cup) = \sqrt{\mathcal{Z}^f(O)}$, where we implicitly use the natural isomorphism of Proposition 6.22.

**Lemma 13.19.** We have

$$ \mathcal{Z}^f \left( \underbrace{\xrightarrow{\hspace{2em}}}_{\hspace{2em}} \right) = \exp \left( \Psi(2\alpha) \left( \underbrace{\xrightarrow{\hspace{1em}}}_{\xleftarrow{\hspace{1em}}} \xrightarrow{\hspace{1em}} \xrightarrow{\hspace{1em}}} \right) \right). $$

PROOF: This lemma can be deduced from Corollary 12.29. Below, as an exercise, we alternatively deduce it from Proposition 13.18 and Theorem 13.12, assuming that $\alpha$ is a two-leg element of $\tilde{\mathcal{A}}(\hat{\uparrow})$, but without assuming Corollary 12.29. Since $\alpha$ is a two-leg element, we picture it as

$$ \alpha = \boxed{\alpha} \xrightarrow{\hspace{2em}} . $$

So, using Lemma 12.26, we get

$$ \pi (2 \times \hat{\uparrow})^* (\alpha) = 2 \xrightarrow{\hspace{1em}} \boxed{\alpha} \xrightarrow{\hspace{1em}} + \xrightarrow{\hspace{1em}} \boxed{\alpha} \xrightarrow{\hspace{1em}} + \xrightarrow{\hspace{1em}} \boxed{\alpha} . $$
Since $\alpha$ can be slid along its interval, we obtain

$$ \exp\left(\pi(2 \times \uparrow)^*(\alpha)\right) = \exp\left(\Psi(2\alpha)\left(\begin{array}{cc} \uparrow & \uparrow \\ \downarrow & \downarrow \end{array}\right)\right) (\exp(\alpha) \otimes \exp(\alpha)). $$

□
<figure><img src="image_109.png" /></figure>
# Chapter 14

## Invariance of $\mathcal{Z}^f$ for long tangles

In this chapter, we study appropriate compactifications of the configuration space $\tilde{C}(\tilde{R}(\mathcal{C}), L; \Gamma)$ associated to a long tangle representative $L: \mathcal{L} \hookrightarrow \tilde{R}(\mathcal{C})$ and to a Jacobi diagram $\Gamma$ with support $\mathcal{L}$. These compactifications $C_L = C(R(\mathcal{C}), L; \Gamma)$ and $C_L^f = C^f(R(\mathcal{C}), L; \Gamma)$ are introduced in Definition 14.22. They allow us to prove Theorem 12.2, which ensures that the integrals involved in the extension of $\mathcal{Z}^f$ to long tangles converge, in Section 14.2. They also allow us to prove Theorem 12.7, which ensures the topological invariance of this extension of $\mathcal{Z}^f$, in Section 14.3. Our compactifications are locally diffeomorphic to products of smooth manifolds by singular subspaces of $\mathbb{R}^n$ associated to trees. We study these singular subspaces in Section 14.1 and show how Stokes’ theorem applies to them in this preliminary section, which is independent of the rest of the book. We describe the local structure of our compactifications and their codimension-one faces in Theorem 14.23.

### 14.1 Singular models associated to trees

**Definitions 14.1.** In this book, an *oriented tree* is a tree $\mathcal{T}$ as in Figure 14.1 whose edges are oriented so that $\mathcal{T}$ satisfies the following properties.

* There is exactly one vertex without outgoing edges. This vertex $T(\mathcal{T})$ is called the *top* of $\mathcal{T}$. It is also simply denoted by $T$ when $\mathcal{T}$ is fixed.

* The edges of $\mathcal{T}$ are oriented towards $T(\mathcal{T})$. In other words, for any vertex $V$ of $\mathcal{T}$ different from $T(\mathcal{T})$, the orientation of the edges in the injective path $[V, T(\mathcal{T})]$ from $V$ to $T(\mathcal{T})$ is induced by the orientation of $[V, T(\mathcal{T})]$.
Let $\mathcal{T}$ be such an oriented tree. A univalent vertex of $\mathcal{T}$ with one outgoing edge is called a *leaf* of $\mathcal{T}$. The set of leaves of $\mathcal{T}$ is denoted by $L(\mathcal{T})$. A *node* of $\mathcal{T}$ is a vertex with at least two ingoing edges. A *branch* of $\mathcal{T}$ is an oriented injective path of oriented edges going from a leaf $\ell$ to a node $N$ or to the top $T$. Such a branch is denoted by $[\ell, N]$ or by $[\ell, T]$. It is viewed as the subset of the set $E(\mathcal{T})$ of edges of $\mathcal{T}$ between $\ell$ and $N$, or between $\ell$ and $T$. We denote the edge adjacent to a leaf $\ell$ by $\mathbf{e}(\ell)$. For any two vertices $N_1, N_2$ on the same branch $[\ell, T]$, such that $N_2$ is closer to $T$ than $N_1$, $[N_1, N_2]$ is the set of edges between $N_1$ and $N_2$. These edges may contain $N_1$ or $N_2$ as an endpoint. $]N_1, N_2]$ (resp. $[N_1, N_2]$) denotes the set of edges of $[N_1, N_2]$ that do not contain $N_1$ (resp. $N_2$) as an endpoint. For example, the set $[\ell, N] \setminus \{\mathbf{e}(\ell)\}$ is denoted by $]\ell, N]$. Similarly, for any two edges $e_1, e_2$ on the same branch $[\ell, T]$, such that $e_1$ is closer to $\ell$ than $T$, $[e_1, e_2]$ is the set of edges between $e_1$ and $e_2$, including $e_1$ and $e_2$, and $]e_1, e_2]$ (resp. $[e_1, e_2]$) denotes the set $[e_1, e_2] \setminus \{e_1\}$ (resp. $[e_1, e_2] \setminus \{e_2\}$). We also mix edges and vertices in this notation. For example, $]e_1, T]$ is the set of edges between $e_1$ and $T$ different from $e_1$ that may contain $T$. The edges are ordered naturally on such an interval of edges. The first is the first encountered when following the orientation of the interval induced by the orientation of the tree.

For two leaves $\ell_1$ and $\ell_2$ of $\mathcal{T}$, $N(\ell_1, \ell_2)$ denotes the node of $\mathcal{T}$ such that $[\ell_1, T] \cap [\ell_2, T] = [N(\ell_1, \ell_2), T]$. For a subset $\mathcal{E}$ of $E(\mathcal{T})$, $L(\mathcal{E}) = L(\mathcal{E}, \mathcal{T})$ denotes the set of leaves $\ell$ of $\mathcal{T}$ such that $[\ell, T]$ contains at least one edge of $\mathcal{E}$. For a leaf $\ell$ in $L(\mathcal{E})$, the closest edge to $\ell$ in $[\ell, T] \cap \mathcal{E}$ is denoted by $\mathbf{e}(\mathcal{E}, \ell)$.

**Examples 14.2.** In Figure 14.1, the set of leaves is $L(\mathcal{T}) = \{\ell_0, \ell_1, \dots, \ell_5\}$. For $\mathcal{E} = \{e_6, e_{13}\}$, we have $L(\mathcal{E}) = L(\mathcal{T})$, $\mathbf{e}(\mathcal{E}, \ell_0) = \mathbf{e}(\mathcal{E}, \ell_3) = \mathbf{e}(\mathcal{E}, \ell_4) = e_6$, and $\mathbf{e}(\mathcal{E}, \ell_5) = e_{13}$.

<figure><img src="image_110.png" /><figcaption>Figure 14.1: A tree $\mathcal{T}$ with a bold codimension-one system of edges</figcaption></figure>
This section is devoted to the study of the following space $X(\mathcal{T})$ associated to an oriented tree $\mathcal{T}$.

**Definition 14.3.** For $((u_e)_{e \in E(\mathcal{T})}) \in [0, \infty]^{[E(\mathcal{T})]}$ and for a branch $[\ell, N]$ of $\mathcal{T}$, define

$$U([\ell, N]) = \prod_{e \in [\ell, N]} u_e.$$

Define $X(\mathcal{T})$ to be the set of the elements $((u_e)_{e \in E(\mathcal{T})})$ of $[0, \infty]^{[E(\mathcal{T})]}$ such that the equality

$$*(\ell_1, \ell_2) : U([\ell_1, N(\ell_1, \ell_2)]) = U([\ell_2, N(\ell_1, \ell_2)])$$

holds for any two leaves $\ell_1$ and $\ell_2$ of $\mathcal{T}$. Set $\tilde{X}(\mathcal{T}) = X(\mathcal{T}) \cap ]0, \infty[^{[E(\mathcal{T})]}$.

**Example 14.4.** The space $X(\mathcal{T})$ associated to the tree

<figure><img src="image_111.png" /></figure>

is $\{(u_1, u_2, u_3, u_4) \in [0, \infty]^4 : u_1u_2 = u_3u_4\}$. At its point $(0, 0, 0, 0)$, the four half-lines $\mathbb{R}^+(1, 0, 0, 0)$, $\mathbb{R}^+(0, 1, 0, 0)$, $\mathbb{R}^+(0, 0, 1, 0)$, and $\mathbb{R}^+(0, 0, 0, 1)$ embed in $X(\mathcal{T})$. Their four independent unit tangent vectors at $(0, 0, 0, 0)$ generate $\mathbb{R}^4$, but the complement of $X(\mathcal{T})$ is dense in $[0, \infty]^4$. So $X(\mathcal{T})$ is not a submanifold with ridges of $[0, \infty]^4$.

**Remarks 14.5.** Let $[\ell_1, N]$ and $[\ell_2, N]$ be two branches of $\mathcal{T}$ ending at the same vertex $N$. If $((u_e)_{e \in E(\mathcal{T})}) \in X(\mathcal{T})$, then we have $U([\ell_1, N]) = U([\ell_2, N])$.

Let $\ell_0$ be a leaf of $\mathcal{T}$. Then $\tilde{X}(\mathcal{T})$ is the set of the elements $((u_e)_{e \in E(\mathcal{T})})$ of $]0, \infty[^{[E(\mathcal{T})]}$ such that $U([\ell, T]) = U([\ell_0, T])$ for any leaf $\ell$ of $L(\mathcal{T}) \setminus \{\ell_0\}$. Indeed, the equations $*(\ell_1, \ell_2)$ of Definition 14.3 are equivalent to $U([\ell_1, T]) = U([\ell_2, T])$ when no variable is zero.

**Definition 14.6.** A reducing system of edges in an oriented tree $\mathcal{T}$ is a set $\mathcal{E}_r$ of edges such that $L(\mathcal{E}_r) = L(\mathcal{T}) \setminus \{\ell_0\}$ for some leaf $\ell_0$ and $e(\mathcal{E}_r, \cdot)$ is a bijection from $L(\mathcal{E}_r)$ to $\mathcal{E}_r$. For such a reducing system $\mathcal{E}_r$, the leaf $\ell_0$ of $L(\mathcal{T}) \setminus L(\mathcal{E}_r)$ is denoted by $\ell_0(\mathcal{E}_r)$. A maximal free system of edges of $\mathcal{T}$ is the complement $E(\mathcal{T}) \setminus \mathcal{E}_r$ of a reducing system $\mathcal{E}_r$ of edges of $\mathcal{T}$ in $E(\mathcal{T})$.

**Examples 14.7.** For example, for every leaf $\ell_0$ of $\mathcal{T}$, the set of edges adjacent to the leaves of $L(\mathcal{T}) \setminus \{\ell_0\}$ is a reducing system of edges. In Figure 14.1, $\{e_1, e_{10}, e_9, e_{12}, e_{13}\}$ is also a reducing system of edges.
**Lemma 14.8.** Let $\mathcal{E}_r$ be a reducing system of edges of $\mathcal{T}$. Let $e_1$ be an edge of $\mathcal{E}_r$. Let $\ell = \ell(\mathcal{E}_r, e_1)$ be the leaf of $\mathcal{T}$ such that $e_1 = \mathbf{e}(\mathcal{E}_r, \ell)$. There exists a unique leaf $\ell' = \ell'(\mathcal{E}_r, e_1)$ of $\mathcal{T}$ such that $e_1 \in [\ell, N(\ell, \ell')]$ and $e_1$ is the only element of $\mathcal{E}_r$ in $[\ell, N(\ell, \ell')] \cup [\ell', N(\ell, \ell')]$. Then the equation $*(\ell, \ell')$ may be written as

$$u_{e_1} = \frac{U([\ell', N(\ell, \ell')])}{\prod_{e \in [\ell, N(\ell, \ell')] \setminus \{e_1\}} u_e} = u_{e_1} (E(\mathcal{T}) \setminus \mathcal{E}_r; (u_e)_{e \in E(\mathcal{T}) \setminus \mathcal{E}_r})$$

when $x = ((u_e)_{e \in E(\mathcal{T})}) \in \dot{X}(\mathcal{T})$. So it determines $u_{e_1}$ in terms of the variables associated to $E(\mathcal{T}) \setminus \mathcal{E}_r$ in $\dot{X}(\mathcal{T})$. Moreover, we have

$$\frac{du_{e_1}}{u_{e_1}} = \sum_{e \in [\ell', N(\ell, \ell')]} \frac{du_e}{u_e} - \sum_{e \in [\ell, N(\ell, \ell')] \setminus \{e_1\}} \frac{du_e}{u_e}$$

in $\dot{X}(\mathcal{T})$.

**PROOF:** If there is an edge of $\mathcal{E}_r$ after $e_1$ on $[\ell, T]$, let $e_2 = \mathbf{e}(\mathcal{E}_r, \ell')$ be the second edge of $\mathcal{E}_r$ on $[\ell, T]$. This defines $\ell'$. Otherwise, set $\ell' = \ell_0(\mathcal{E}_r)$. See Figure 14.2. This proves the existence of $\ell'$ such that $e_1$ is the only element of $\mathcal{E}_r$ in $[\ell, N(\ell, \ell')] \cup [\ell', N(\ell, \ell')]$. For such an $\ell' \neq \ell_0(\mathcal{E}_r)$, $\mathbf{e}(\mathcal{E}_r, \ell')$ must be the second edge of $\mathcal{E}_r$ on $[\ell, T]$. So the condition of the statement determines $\ell'$. $\square$

<figure><img src="image_112.png" /><figcaption>Figure 14.2: The two cases in Lemma 14.8</figcaption></figure>

**Lemma 14.9.** The set $\dot{X}(\mathcal{T})$ is a smooth manifold of dimension

$$d(\mathcal{T}) = |E(\mathcal{T})| - |L(\mathcal{T})| + 1.$$

For every maximal free system $\mathcal{E}_b$ of edges of $\mathcal{T}$, $\dot{X}(\mathcal{T})$ is freely parametrized by the (variables of) the edges of $\mathcal{E}_b$. For a subset $\mathcal{E}$ of cardinality $d(\mathcal{T})$, the form $\wedge_{e \in \mathcal{E}} du_e$ is a nonvanishing volume form on $\dot{X}(\mathcal{T})$ if and only if $\mathcal{E}$ is a maximal free system of edges of $\mathcal{T}$.
PROOF: Let $\mathcal{E}_b$ be a maximal free system of edges of $\mathcal{T}$, and let $\mathcal{E}_r$ be its complement. An edge $e$ of $\mathcal{E}_r$ may be expressed as $\mathbf{e}(\mathcal{E}_r, \ell(\mathcal{E}_r, e))$ for a unique $\ell(\mathcal{E}_r, e)$ of $L(\mathcal{E}_r)$, and any associated variable $u_e$ can be expressed in terms of the variables associated to $\mathcal{E}_b$ as in Lemma 14.8. Furthermore, any $((u_e)_{e \in E(\mathcal{T})}) \in ]0, \infty[^{E(\mathcal{T})}$ such that

$$U([\ell(\mathcal{E}_r, e), T]) = U([\ell_0(\mathcal{E}_r), T]),$$

for any $e$ in $\mathcal{E}_r$, is in $\dot{X}(\mathcal{T})$. The equation defining $u_{e_1}$ in Lemma 14.8, for $e_1 \in \mathcal{E}_r$, implies $*(\ell(\mathcal{E}_r, e_1), \ell')$, where $\ell' = \ell_0(\mathcal{E}_r)$ if $e_1$ is the only edge of $\mathcal{E}_r$ in $[\ell(\mathcal{E}_r, e_1), T]$. Otherwise, set $\ell_{e_1} = \ell(\mathcal{E}_r, e_1)$ and let $e_1, e_2, \dots, e_k$ denote the edges of $\mathcal{E}_r$ on $[\ell_{e_1}, T]$, where $e_i = \mathbf{e}(\mathcal{E}_r, \ell_{e_i})$ denotes the $i^{th}$ edge that we meet from $\ell_{e_1}$ to $T$. Applying Lemma 14.8 to $e_i$ defines $u_{e_i}$ by an equation, which is equivalent to $*(\ell_{e_i}, \ell_{e_{i+1}})$ if $i < k$, and to $*(\ell_{e_k}, \ell_0(\mathcal{E}_r))$ if $i = k$. These equations together imply $*(\ell_{e_1}, \ell_0(\mathcal{E}_r))$ for any $\ell_{e_1}$ of $L(\mathcal{T})$.

Therefore, any $x = ((u_e)_{e \in E(\mathcal{T})}) \in ]0, \infty[^{E(\mathcal{T})}$ satisfying the equations of Lemma 14.8 defining the $u_e$ for $e \in \mathcal{E}_r$ is in $\dot{X}(\mathcal{T})$. Thus the space $\dot{X}(\mathcal{T})$ is freely parametrized by the (variables of) the edges of $\mathcal{E}_b$. So $\dot{X}(\mathcal{T})$ is a smooth submanifold of $\mathbb{R}^{E(\mathcal{T})}$ of dimension

$$d(\mathcal{T}) = |E(\mathcal{T})| - |L(\mathcal{T})| + 1.$$

Let $\mathcal{E}$ be a subset of cardinality $d(\mathcal{T})$, such that the form $\wedge_{e \in \mathcal{E}} du_e$ is a nonvanishing volume form on $\dot{X}(\mathcal{T})$. Set $\mathcal{E}^c = E(\mathcal{T}) \setminus \mathcal{E}$. Let us show that the map $\mathbf{e}(\mathcal{E}^c, \cdot): L(\mathcal{E}^c) \to \mathcal{E}^c$ is injective. If $\mathbf{e}(\mathcal{E}^c, \ell_1) = \mathbf{e}(\mathcal{E}^c, \ell_2)$, for two leaves $\ell_1$ and $\ell_2$, then $[\ell_1, N(\ell_1, \ell_2)] \cup [\ell_2, N(\ell_1, \ell_2)] \subset \mathcal{E}$ and the relation $U([\ell_1, N(\ell_1, \ell_2)]) = U([\ell_2, N(\ell_1, \ell_2)])$ gives rise to a nontrivial relation between the forms $du_e$ for $e \in [\ell_1, N(\ell_1, \ell_2)] \cup [\ell_2, N(\ell_1, \ell_2)]$.

If $L \setminus L(\mathcal{E}^c)$ has two distinct elements $\ell_1$ and $\ell_2$, then we similarly have a nontrivial linear relation between coordinate forms $du_e$ associated to edges of $[\ell_1, N(\ell_1, \ell_2)] \cup [\ell_2, N(\ell_1, \ell_2)]$. Therefore, the cardinality of $L(\mathcal{E}^c)$ is at least $|L(\mathcal{T})| - 1$, which is the cardinality of $\mathcal{E}^c$. So $\mathbf{e}(\mathcal{E}^c, \cdot)$ is a bijection. $\square$

**Definition 14.10.** A codimension-one system of edges in an oriented tree $\mathcal{T}$ is a set $\mathcal{E}$ of edges such that there is exactly one edge of $\mathcal{E}$ in any path from a leaf to the top of $\mathcal{T}$.

**Examples 14.11.** An example of a codimension-one system of edges of $\mathcal{T}$ is the set of edges that start at leaves. In Figure 14.1, $\mathcal{E} = \{e_6, e_{13}\}, \{e_7\}$, and $\{e_0, e_9, e_{12}, e_5\}$ are other codimension-one systems of edges of $\mathcal{T}$.

**Lemma 14.12.** For any codimension-one system $\mathcal{E}_1$ of $X(\mathcal{T})$ and any edge $e_0 \in \mathcal{E}_1, \mathcal{E}_1 \setminus \{e_0\}$ can be completed to a reducing system that does not contain $e_0$.
PROOF: For each element $e$ of $\mathcal{E}_1$, choose a leaf $\ell_e$ such that $\mathbf{e}(\mathcal{E}_1, \ell_e) = e$. Let $L_1 = \{\ell_e, e \in \mathcal{E}_1\}$ be the set of these leaves, and set $L_2 = L(\mathcal{T}) \setminus L_1$. Let $\mathcal{E}_2 = \mathbf{e}(L_2)$ be the set of edges adjacent to the leaves of $L_2$. Then $\mathcal{E}_1 \cup \mathcal{E}_2 \setminus \{e_0\}$ is a reducing system of $\mathcal{T}$. $\square$

**Lemma 14.13.** The closure of $\mathring{X}(\mathcal{T})$ in $[0, \infty]^{[E(\mathcal{T})]}$ is $X(\mathcal{T})$.

Let $x = ((u_e)_{e \in E(\mathcal{T})})$ be an element of $X(\mathcal{T}) \setminus \mathring{X}(\mathcal{T})$. Let $\mathcal{E}(x)$ denote the set of edges of $\mathcal{T}$ such that $u_e = 0$. Then the image $\mathcal{E}_1(x)$ of $(\mathbf{e}(\mathcal{E}(x), \cdot): L(\mathcal{E}(x)) \to \mathcal{E}(x))$ is a codimension-one system of edges of $\mathcal{T}$.

Let $X_{\mathcal{E}(x)}(\mathcal{T})$ be the set of elements $x'$ of $X(\mathcal{T})$ such that $\mathcal{E}(x') = \mathcal{E}(x)$. Then $X_{\mathcal{E}(x)}(\mathcal{T})$ is a smooth manifold of dimension $d(\mathcal{T})-1-(|\mathcal{E}(x)|-|\mathcal{E}_1(x)|)$.

PROOF: Let $x = ((u_e)_{e \in E(\mathcal{T})}) \in X(\mathcal{T}) \setminus \mathring{X}(\mathcal{T})$. Let us first prove that the image $\mathcal{E}_1(x)$ of $\mathbf{e}(\mathcal{E}(x), \cdot): L(\mathcal{E}(x)) \to \mathcal{E}(x)$ is a codimension-one system of edges of $\mathcal{T}$. Since $\mathcal{E}(x) \neq \emptyset$, there is a leaf $\ell$ of $\mathcal{T}$ such that $U([\ell, T]) = 0$. This implies $U([\ell, T]) = 0$ for all leaves. Thus, $L(\mathcal{E}(x)) = L(\mathcal{T})$. Furthermore no branch $[\ell, T]$ can contain more than one edge of $\mathcal{E}_1(x)$. Otherwise, the first two edges of $\mathcal{E}_1(x)$ on such a branch $[\ell_1, T]$ would be $\mathbf{e}(\mathcal{E}(x), \ell_1)$ and $\mathbf{e}(\mathcal{E}(x), \ell_2)$, respectively, and $\mathbf{e}(\mathcal{E}(x), \ell_1)$ would be the only edge of $\mathcal{E}(x)$ on $[\ell_1, N(\ell_1, \ell_2)] \cup [\ell_2, N(\ell_1, \ell_2)]$. Then we would have $U([\ell_1, N(\ell_1, \ell_2)]) = 0$ and $U([\ell_2, N(\ell_1, \ell_2)]) \neq 0$. So $*(\ell_1, \ell_2)$ would not be satisfied. So $\mathcal{E}_1(x)$ is a codimension-one system of edges of $\mathcal{T}$.

Now, it suffices to prove the following two assertions.

* $x$ is in the closure of $\mathring{X}(\mathcal{T})$ in $[0, \infty]^{[E(\mathcal{T})]}$, and

* $X_{\mathcal{E}(x)}(\mathcal{T})$ is a smooth manifold of dimension $d(\mathcal{T})-1-(|\mathcal{E}(x)|-|\mathcal{E}_1(x)|)$.

Let us first prove them when $\mathbf{e}(\mathcal{E}(x), \cdot): L(\mathcal{T}) \to \mathcal{E}_1(x)$ is injective. Define $x(t) = ((u_e(t))_{e \in E(\mathcal{T})}) \in \mathring{X}(\mathcal{T})$ from $x = ((u_e)_{e \in E(\mathcal{T})})$ for $t \in ]0, \infty[$, as follows. Pick $\ell_0 \in L(\mathcal{T})$, and set $e_0 = \mathbf{e}(\mathcal{E}(x), \ell_0)$. Replace all the variables $u_e$ for $e \in \mathcal{E}(x) \setminus \mathcal{E}_1(x)$ by $t$, replace $u_{e_0}$ by $t^k$ for some positive integer $k$, and leave the variables associated to the edges of $E(\mathcal{T}) \setminus \mathcal{E}(x)$ (which are not zero) unchanged. For an edge $f = \mathbf{e}(\mathcal{E}_1(x), \ell_f)$ of $\mathcal{E}_1(x) \setminus \{e_0\}$, set

$$u_f(t) = \frac{U([\ell_0, T])(t)}{\prod_{e \in [\ell_f, T] \setminus \{f\}} u_e(t)}.$$

Recall that $f$ is the only edge of $\mathcal{E}_1(x)$ on $[\ell_f, T]$ and that $e_0$ is the only edge of $\mathcal{E}_1(x)$ on $[\ell_0, T]$. Then $u_f(t)$ is equal to $\alpha t^{k+r(f)}$ for some $\alpha > 0$ and some $r(f) \in \mathbb{Z}$. Choose $k$ so that $k+r(f) \ge 1$ for any $f \in \mathcal{E}_1(x)$. Then $x(t)$ tends to $x$ when $t$ tends to zero. Furthermore, since all $u_e(t)$ are nonzero, the
defining equations for the $u_f(t)$ are equivalent to the equations $U([\ell, T])(t) = U([\ell_0, T])(t)$. They are satisfied for any $\ell \in L(\mathcal{T})$, so $x(t) \in \dot{X}(\mathcal{T})$. This proves that $x$ is in the closure of $\dot{X}(\mathcal{T})$.

The defining equations of $X(\mathcal{T})$ are satisfied as soon as the $u_e$, for $e \in \mathcal{E}_1(x)$, vanish (still under the assumption that $e(\mathcal{E}(x), .): L(\mathcal{E}(x)) \to \mathcal{E}(x)$ is injective). Therefore, $X_{\mathcal{E}(x)}(\mathcal{T})$ is a manifold freely parametrized by the variables corresponding to the edges of $E(\mathcal{T}) \setminus \mathcal{E}(x)$. Its dimension is $|E(\mathcal{T})| - |\mathcal{E}(x)| - |L(\mathcal{T})| + |\mathcal{E}_1(x)| = d(\mathcal{T}) - 1 - (|\mathcal{E}(x)| - |\mathcal{E}_1(x)|)$. The two assertions are proved when

$$e(\mathcal{E}(x), .): L(\mathcal{E}(x)) \to \mathcal{E}(x)$$

is injective.

In general, choose a leaf $\ell_e$ such that $e(\mathcal{E}(x), \ell_e) = e$ for each element $e$ of $\mathcal{E}_1(x)$. Let $L_1 = \{\ell_e, e \in \mathcal{E}_1(x)\}$ be the set of these leaves. Let $\mathcal{T}_1$ be the subtree of $\mathcal{T}$ such that $E(\mathcal{T}_1) = \cup_{e \in \mathcal{E}_1(x)} [\ell_e, T]$ (so $L(\mathcal{T}_1) = L_1$). Let $x_1$ be the natural projection of $x$ in $[0, \infty^{[E(\mathcal{T}_1)]})$. Note $\mathcal{E}(x) \subseteq E(\mathcal{T}_1)$ and $\mathcal{E}(x) = \mathcal{E}(x_1)$. Also note that the restriction of $e(\mathcal{E}(x), .)$ to $L(\mathcal{T}_1)$ is the map $e_{\mathcal{T}_1}(\mathcal{E}(x_1), .)$ associated to $\mathcal{T}_1$, which is injective.

In particular, the first part of the proof expresses $x_1$ as a limit at 0 of some continuous function $x_1(\cdot): ]0, \infty[ \to \dot{X}(\mathcal{T}_1)$, such that $u_e(t)$ is constant for any edge $e$ of $\mathcal{E}(\mathcal{T}_1) \setminus \mathcal{E}(x)$. Define $x(\cdot): ]0, \infty[ \to ]0, \infty^{[E(\mathcal{T})]}$ so that the variables $u_e(t)$ for $e \notin \mathcal{E}(x)$ are constant (and different from zero) and the variables $u_e(t)$ for $e \in \mathcal{E}(\mathcal{T}_1)$ are the same for $x(t)$ and $x_1(t)$. Let $L_2 = L(\mathcal{T}) \setminus L_1$. For $\ell_2 \in L_2$, we have $e(\mathcal{E}(x), \ell_2) = e(\mathcal{E}(x), \ell_1(\ell_2))$ for a unique $\ell_1(\ell_2)$ of $L_1$. The equation $*(\ell_2, \ell_1(\ell_2))$ between nonvanishing constant products holds for $x(t)$ for any $t \in ]0, \infty[$, and it implies that $U([\ell_2, T])(t) = U([\ell_1(\ell_2), T])(t) = U([\ell_0, T])(t)$. So $x(\cdot)$ is valued in $\dot{X}(\mathcal{T})$. Its limit at 0 is $x$. So $x$ is in the closure of $\dot{X}(\mathcal{T})$.

Let $\mathcal{E}_2 = e(L_2)$ be the set of edges adjacent to the leaves of $L_2$. Set $\mathcal{E}_3 = E(\mathcal{T}) \setminus (E(\mathcal{T}_1) \cup \mathcal{E}_2)$. Then any element $x'$ of $X_{\mathcal{E}(x)}(\mathcal{T})$ is determined by its projection $x'_1 \in X_{\mathcal{E}(x)}(\mathcal{T}_1)$ and by the free nonzero variables associated to the edges of $\mathcal{E}_3$. More precisely, for an edge $e = e(\ell_2 \in L_2)$, the equation $*(\ell_2, \ell_1(\ell_2))$ between nonvanishing products determines $u'_e$ as a function of $x'_1$ and the free nonzero variables associated to the edges of $\mathcal{E}_3$. For elements $((u_e)_{e \in E(\mathcal{T})}) \in \{0\}^{\mathcal{E}(x)} \times ]0, \infty^{[E(\mathcal{T}) \setminus \mathcal{E}(x)}$, if the equations $*(\ell_2, \ell_1(\ell_2))$ are satisfied for all $\ell_2 \in L_2$, then all the equations $*(\ell_2, \ell')$ for $\ell_2 \in L_2$ and $\ell' \in L(\mathcal{T})$ are satisfied, as we prove below. Let $\ell_2 \in L_2$, and let $N(\ell_1(\ell_2))$ denote the closest node to $e(\mathcal{E}(x), \ell_2) = e(\mathcal{E}(x), \ell_1(\ell_2))$ in $[\ell_1(\ell_2), e(\mathcal{E}(x), \ell_2)]$. See Figure 14.3. Then $*(\ell_2, \ell_1(\ell_2))$ is equivalent to $U([\ell_2, N(\ell_1(\ell_2))]) = U([\ell_1(\ell_2), N(\ell_1(\ell_2))])$. In particular, if $e(\mathcal{E}(x), \ell_2) =$
e(\mathcal{E}(x), l') for $l' \in L(\mathcal{T})$ as in Figure 14.3, then $*(l', l_1(l_2))$ is equivalent to $U([l', N(l_1(l_2))]) = U([l_1(l_2), N(l_1(l_2))])$. So $*(l_2, l_1(l_2))$ and $*(l', l_1(l_2))$ imply $*(l_2, l')$. If $e(\mathcal{E}(x), l_2) \neq e(\mathcal{E}(x), l')$, then $e(\mathcal{E}(x), l_2) \in [l_2, N(l_2, l')]$ and $e(\mathcal{E}(x), l') \in [l', N(l_2, l')]$. So $*(l_2, l')$ is equivalent to $0=0$ and is satisfied. Thus $X_{\mathcal{E}(x)}(\mathcal{T})$ is a smooth manifold whose dimension is $|\mathcal{E}_3| + |E(\mathcal{T}_1)| - |\mathcal{E}(x)| - |L(\mathcal{T}_1)| + |\mathcal{E}_1(x)| = |E(\mathcal{T})| - |L_2| - |\mathcal{E}(x)| - |L(\mathcal{T}_1)| + |\mathcal{E}_1(x)|$. $\square$

<figure><img src="image_113.png" /><figcaption>Figure 14.3: Example of leaves $l'_i$ such that $e(\mathcal{E}(x), l'_i) = e(\mathcal{E}(x), l_2)$ for the proof of Lemma 14.13</figcaption></figure>

**Lemma 14.14.** The codimension-one faces of $X(\mathcal{T})$ are in one-to-one correspondence with the codimension-one systems of edges of $\mathcal{T}$. In a neighborhood of such an open face, $X(\mathcal{T})$ has the structure of a smooth manifold with boundary.

**PROOF:** According to Lemma 14.13 above, if $X_{\mathcal{E}}(\mathcal{T})$ is a nonempty manifold of dimension $d(\mathcal{T}) - 1$, then $\mathcal{E}$ is a codimension-one system. Let $\mathcal{E}_c$ be a codimension-one system of edges of $\mathcal{T}$. For any edge of $e_0 \in \mathcal{E}_c$, $\mathcal{E}_c \setminus \{e_0\}$ can be completed to a reducing system $\mathcal{E}_r$ that does not contain $e_0$ as in Lemma 14.12. In particular, $\tilde{X}(\mathcal{T})$ is freely parametrized by the variables associated to the edges of $E(\mathcal{T}) \setminus \mathcal{E}_r$. When all these variables are nonzero except maybe $(u_{e_0} \in [0, \varepsilon])$, we get a local parametrization by $[0, \varepsilon \times ]0, \varepsilon^{[E(\mathcal{T}) \setminus (\mathcal{E}_r \cup \{e_0\})]}$ near the locus $u_{e_0} = 0$. If $e \in \mathcal{E}_c \setminus \{e_0\}$ and $e = e(\mathcal{E}_r, l_e)$, then the variable $u_e$ may be expressed as

$$u_e = \frac{\prod_{f \in [l_0(\mathcal{E}_r), N(l_0(\mathcal{E}_r), l_e)] \setminus \{e_0\}} u_f}{\prod_{f \in [l_e, N(l_0(\mathcal{E}_r), l_e)] \setminus \{e\}} u_f} u_{e_0}$$

with respect to this local parametrization. $\square$

We finish this section by proving a version of Stokes' theorem for spaces modelled by products of $X(\mathcal{T})$ by a manifold. Let us first introduce its statement given in Theorem 14.16.
**Lemma 14.15.** Let $\varepsilon \in ]0, \infty[$. Let $X^\varepsilon(\mathcal{T}) = X(\mathcal{T}) \cap [0, \varepsilon]^{E(\mathcal{T})}$. Assume that $X(\mathcal{T})$ is oriented. Let $n \in \mathbb{N}$. For any smooth form $\Omega$ on $[0, \varepsilon]^{E(\mathcal{T})} \times [0, 1]^n$ of degree $(d(\mathcal{T}) + n)$, the integral $\int_{X^\varepsilon(\mathcal{T}) \times [0,1]^n} \Omega$ of $\Omega$ along the interior of $X^\varepsilon(\mathcal{T}) \times [0, 1]^n$ is absolutely convergent.

For an ordered subset $\mathcal{E}$ of $E(\mathcal{T})$, we set $\Omega_{\mathcal{E}} = \wedge_{e \in \mathcal{E}} du_e$, where the factors are ordered with respect to the order of $\mathcal{E}$.

**PROOF OF LEMMA 14.15:** Any smooth form $\Omega$ on $[0, \varepsilon]^{E(\mathcal{T})} \times [0, 1]^n$ of degree $(d(\mathcal{T}) + n)$ is a sum of forms $g_\varepsilon \Omega_\varepsilon \wedge (\wedge_{i=1}^n dx_i)$, for ordered subsets $\mathcal{E}$ of $E(\mathcal{T})$ of cardinality $d(\mathcal{T})$ and for smooth maps $g_\varepsilon: [0, \varepsilon]^{E(\mathcal{T})} \times [0, 1]^n \to \mathbb{R}$, up to forms that vanish identically on the interior of $X^\varepsilon(\mathcal{T}) \times [0, 1]^n$. These forms are bounded on $[0, \varepsilon]^{E(\mathcal{T})} \times [0, 1]^n$. They are zero on the interior of $X^\varepsilon(\mathcal{T}) \times [0, 1]^n$ unless $\mathcal{E}$ is a maximal free system, according to Lemma 14.9. When $\mathcal{E}$ is a maximal free system, Lemma 14.9 ensures that $\dot{X}^\varepsilon(\mathcal{T})$ is freely parametrized by the variables associated to the edges of $\mathcal{E}$. $\square$

We restrict to the compact subspace $X^\varepsilon(\mathcal{T})$ of $X(\mathcal{T})$. The subspace $X^\varepsilon(\mathcal{T})$ is the closure of the open manifold $\dot{X}(\mathcal{T}) \cap ]0, \varepsilon]^{E(\mathcal{T})}$. For a maximal free system $\mathcal{E}_b$ of edges, $\dot{X}(\mathcal{T})$ is freely parametrized by the variables $u_f$ for $f \in \mathcal{E}_b$ as in Lemma 14.9. For $e_1 \in E(\mathcal{T}) \setminus \mathcal{E}_b$, the variable $u_{e_1}$ is the function $u_{e_1}(\mathcal{E}_b; V \in ]0, \infty[^{\mathcal{E}_b})$ of Lemma 14.8. This function is a smooth quotient of monomials in the $u_f$ for $f \in \mathcal{E}_b$. Then $\dot{X}(\mathcal{T}) \cap ]0, \varepsilon]^{E(\mathcal{T})}$ is parametrized by the set of elements $V \in ]0, \varepsilon]^{\mathcal{E}_b}$ such that $u_e(\mathcal{E}_b; U) \le \varepsilon$ for any $e \in E(\mathcal{T}) \setminus \mathcal{E}_b$. It is a subspace of $]0, \varepsilon]^{\mathcal{E}_b}$ whose boundary can be stratified so that the open strata of the boundary are locally defined by an equation $u_e(\mathcal{E}_b; V \in ]0, \infty[^{\mathcal{E}_b}) = \varepsilon$. Let $\partial_\varepsilon X^\varepsilon(\mathcal{T})$ be the union of the corresponding open codimension-one faces of $X^\varepsilon(\mathcal{T})$. Then the union $\check{\partial} X^\varepsilon(\mathcal{T})$ of the oriented open codimension-one faces of $X^\varepsilon(\mathcal{T})$ is

$$ \check{\partial} X^\varepsilon(\mathcal{T}) = \partial_o X^\varepsilon(\mathcal{T}) \cup \partial_\varepsilon X^\varepsilon(\mathcal{T}), $$

where $\partial_o X^\varepsilon(\mathcal{T})$ is the union of the intersection of the codimension-one faces of $X(\mathcal{T})$ of Lemma 14.14 with $X^\varepsilon(\mathcal{T})$.

**Theorem 14.16.** Let $\varepsilon \in ]0, \infty[$. Recall $X^\varepsilon(\mathcal{T}) = X(\mathcal{T}) \cap [0, \varepsilon]^{E(\mathcal{T})}$. Assume that $X(\mathcal{T})$ is oriented. Let $n \in \mathbb{N}$. Let $\omega$ be a smooth form on $[0, \varepsilon]^{E(\mathcal{T})} \times [0, 1]^n$ of degree $(d(\mathcal{T}) - 1 + n)$. With the notation above, consider the dense part

$$ \check{\partial}(X^{\varepsilon}(\mathcal{T}) \times [0, 1]^n) = (\check{\partial} X^{\varepsilon}(\mathcal{T})) \times [0, 1]^n \cup ((-1)^{d(\mathcal{T})} \dot{X}^{\varepsilon}(\mathcal{T}) \times \partial([0, 1]^n)) \\ \text{of } \partial(X^{\varepsilon}(\mathcal{T}) \times [0, 1]^n). $$
Then the integral $\int_{\partial(X^{\epsilon}(T) \times [0,1]^n)} \omega$ of $\omega$ along $\tilde{\partial}(X^{\epsilon}(T) \times [0,1]^n)$ is absolutely convergent. Furthermore, Stokes' theorem applies to this setting. So we have

$$ \int_{\partial(X^{\epsilon}(T) \times [0,1]^n)} \omega = \int_{X^{\epsilon}(T) \times [0,1]^n} d\omega. $$

Let us prepare the proof of Theorem 14.16 with a few lemmas.

**Lemma 14.17.** Let $\mathcal{E}_1$ be an ordered subset of $E(T)$ of cardinality $d(T)$. Let $g$ be a smooth function on $[0, \varepsilon]^{E(T)} \times [0, 1]^n$. Then Theorem 14.16 holds for $\omega = g\Omega_{\mathcal{E}_1} \wedge (\wedge_{i=2}^n dx_i)$.

**PROOF:** We have

$$ \begin{aligned} \int_{X^{\epsilon}(T) \times [0,1]^n} d\omega &= (-1)^{d(T)} \int_{\dot{X}^{\epsilon}(T) \times [0,1]^n} \frac{\partial g}{\partial x_1} \Omega_{\mathcal{E}_1} \wedge (\wedge_{i=1}^n dx_i) \\ &= (-1)^{d(T)} \int_{\dot{X}^{\epsilon}(T) \times \partial[0,1] \times [0,1]^{n-1}} \omega \\ &= \int_{\partial(X^{\epsilon}(T) \times [0,1]^n)} \omega \end{aligned} $$

since $\dot{X}^{\epsilon}(T) \times \partial[0,1] \times [0,1]^{n-1}$ is the only part of $\partial(X^{\epsilon}(T) \times [0,1]^n)$ where the integral of $\omega$ does not vanish. $\square$

So it suffices to prove Theorem 14.16 for forms $\omega = g\Omega_{\mathcal{E}_2} \wedge (\wedge_{i=1}^n dx_i)$ for ordered subsets $\mathcal{E}_2$ of $E(T)$ of cardinality $d(T) - 1$ such that $\Omega_{\mathcal{E}_2}$ is not zero on $\dot{X}^{\epsilon}(T)$. (If $\Omega_{\mathcal{E}_2}$ is zero on $\dot{X}^{\epsilon}(T)$, then it is also zero on $X^{\epsilon}(T)$, and both sides of the equality to be proved are zero.)

**Lemma 14.18.** Let $\mathcal{E}_2$ be a subspace of $E(T)$ of cardinality $d(T)-1$ such that $\Omega_{\mathcal{E}_2}$ is not zero on $\dot{X}^{\epsilon}(T)$. Let $\mathcal{E}_2^c = E(T) \setminus \mathcal{E}_2$. Then either $L(\mathcal{E}_2^c) = L(T)$ or $L(\mathcal{E}_2^c) = L(T) \setminus \{\ell_0\}$ for a unique $\ell_0 \in L(T)$.

If $L(\mathcal{E}_2^c) = L(T)$, then $e(\mathcal{E}_2^c, \cdot)$ is a bijection from $L(\mathcal{E}_2^c)$ to $\mathcal{E}_2^c$.

If $L(\mathcal{E}_2^c) = L(T) \setminus \{\ell_0\}$, then $e(\mathcal{E}_2^c, \cdot)$ is a bijection from $L(\mathcal{E}_2^c)$ to $\mathcal{E}_2^c \setminus \{f\}$ for a unique element $f$ of $\mathcal{E}_2^c$.

**PROOF:** The set $L(T) \setminus L(\mathcal{E}_2^c)$ cannot contain two distinct leaves $\ell_3$ and $\ell_4$, because Equation $(\ell_3, \ell_4)$ would imply that $\Omega_{\mathcal{E}_2} = 0$. Similarly, $e(\mathcal{E}_2^c, \cdot)$ cannot map two distinct leaves $\ell_3$ and $\ell_4$ of $L(\mathcal{E}_2^c)$ to the same element. So $e(\mathcal{E}_2^c, \cdot)$ is injective. Use that the cardinality of $\mathcal{E}_2^c$ is $|L(T)|$ to conclude. $\square$

For a subset $\mathcal{E}'$ of $\mathcal{E}$, let $p_{\mathcal{E}}: X^{\epsilon}(T) \to [0, \varepsilon]^{\mathcal{E}'}$ denote the composition of the inclusion $X^{\epsilon}(T) \hookrightarrow [0, \varepsilon]^{E(T)}$ with the natural projection.

**Lemma 14.19.** Recall the assumptions of Lemma 14.18. Let $\mathcal{E}_3$ be the set of edges $f$ of $T$ such that $\mathcal{E}_2 \cup \{f\}$ is a maximal free system of edges of $T$. For
an edge f of $\mathcal{E}_3$ and an element $(u_f, U) \in ]0, \varepsilon]^{\{f\} \cup \mathcal{E}_2}$, let $x_f(u_f, U)$ denote the element of $\hat{X}(\mathcal{T})$ such that $p_{\{f\} \cup \mathcal{E}_2}(x_f(u_f, U)) = (u_f, U)$.

Then there exist $f_2 \in \mathcal{E}_3$ and piecewise smooth functions $a$ and $b$ from $]0, \varepsilon]^{\mathcal{E}_2}$ to $]0, \varepsilon]$, such that $X^\varepsilon(\mathcal{T}) \cap p_{\mathcal{E}_2}^{-1}([0, \varepsilon]^{\mathcal{E}_2})$ is the set

$$ \{x_{f_2}(u_{f_2}, U) : U \in ]0, \varepsilon]^{\mathcal{E}_2} \cap p_{\mathcal{E}_2}(X^\varepsilon(\mathcal{T})), u_{f_2} \in [a(U), b(U)]\}. $$

Furthermore, the space $]0, \varepsilon]^{\mathcal{E}_2} \cap p_{\mathcal{E}_2}(X^\varepsilon(\mathcal{T}))$ is a subspace of $]0, \varepsilon]^{\mathcal{E}_2}$ determined by the condition $b(U) - a(U) \ge 0$ and conditions $u_e(U) \le \varepsilon$ for $e \notin \mathcal{E}_3$, for functions $u_e$ of $U$ which are quotients of monomials in the variables $u_g$ for $g \in \mathcal{E}_2$. Moreover, the boundary of $X^\varepsilon(\mathcal{T})$ consists of

* $(d(\mathcal{T}) - 1)$-dimensional strata where $u_z$ vanishes for some $z \in \mathcal{E}_2$,

* (negligible) strata of dimension less than $(d(\mathcal{T}) - 1)$, and

* $(d(\mathcal{T}) - 1)$-dimensional strata of

$$ \partial_b(X^\varepsilon(\mathcal{T})) = \{x_{f_2}(b(U), U) : U \in ]0, \varepsilon]^{\mathcal{E}_2} \cap p_{\mathcal{E}_2}(X^\varepsilon(\mathcal{T}))\}, $$

and

$$ -\partial_a(X^\varepsilon(\mathcal{T})) = -\{x_{f_2}(a(U), U) : U \in ]0, \varepsilon]^{\mathcal{E}_2} \cap p_{\mathcal{E}_2}(X^\varepsilon(\mathcal{T}))\}, $$

which behave as standard codimension-one faces of $X^\varepsilon(\mathcal{T})$,<fnref n="1" />

with respect to some natural stratification.

**PROOF OF THEOREM 14.16 ASSUMING LEMMA 14.19:** Consider a form $\omega = g\Omega_{\mathcal{E}_2} \wedge (\wedge_{i=1}^n dx_i)$, for an ordered subset $\mathcal{E}_2$ of $E(\mathcal{T})$ of cardinality $d(\mathcal{T}) - 1$ and a smooth function $g: [0, \varepsilon]^{E(\mathcal{T})} \times [0, 1]^n \to \mathbb{R}$. Assume that the $du_z$, for $z \in \mathcal{E}_2$, are linearly independent in the $C^\infty(\hat{X}^\varepsilon(\mathcal{T}); \mathbb{R})$-module $\Omega^1(\hat{X}^\varepsilon(\mathcal{T}))$. The integral of

$$ d\omega = \sum_{h \in E(\mathcal{T})} \frac{\partial g}{\partial u_h} du_h \wedge \Omega_{\mathcal{E}_2} \wedge (\wedge_{i=1}^n dx_i) $$

is absolutely convergent over the interior of $X^\varepsilon \times [0, 1]^n$.

With the notation of Lemma 14.19, we have

$$ \int_{\partial(X^\varepsilon(\mathcal{T}) \times [0,1]^n)} \omega = \int_{\partial_b(X^\varepsilon(\mathcal{T})) \times [0,1]^n} \omega - \int_{\partial_a(X^\varepsilon(\mathcal{T})) \times [0,1]^n} \omega $$

<footnote n="1">¹The signs before $\partial_b$ and $\partial_b$ correspond to the orientations provided that $du_{f_2} \wedge \Omega_{\mathcal{E}_2}$ orients $X^\varepsilon(\mathcal{T})$, which we assume without loss.</footnote>
since $\omega$ vanishes along the $(d(\mathcal{T}) - 1)$-dimensional strata where $u_z$ vanishes for some $z \in \mathcal{E}_2$. On the other hand, we have

$$ \int_{X^{\epsilon}(\mathcal{T}) \times [0,1]^n} d\omega = \int_{[0,1]^n} \left( \int_{\hat{X}^{\epsilon}(\mathcal{T})} \frac{\partial g_X}{\partial u_{f_2}} du_{f_2} \wedge \Omega_{\mathcal{E}_2} \right) \wedge \bigwedge_{i=1}^n dx_i, $$

where $g_X$ is the restriction of $g$ to $X^{\epsilon}(\mathcal{T}) \times [0, 1]^n$, where $g_X$ and the $u_h$ are functions of $u_{f_2}$ and of the $u_z$, for $z \in \mathcal{E}_2$. For a fixed implicit $(x_1, \dots, x_n)$, we compute

$$ \int_{\hat{X}^{\epsilon}(\mathcal{T})} \frac{\partial g_X}{\partial u_{f_2}} du_{f_2} \wedge \Omega_{\mathcal{E}_2} = \int_{U \in [0, \epsilon]^{\mathcal{E}_2} \cap p_{\mathcal{E}_2}(X^{\epsilon}(\mathcal{T}))} \left( \int_{u_{f_2} \in [a(U), b(U)]} \frac{\partial g_X}{\partial u_{f_2}} du_{f_2} \right) \Omega_{\mathcal{E}_2} \\ = \int_{U \in [0, \epsilon]^{\mathcal{E}_2} \cap p_{\mathcal{E}_2}(X^{\epsilon}(\mathcal{T}))} (g_X(b(U)) - g_X(a(U))) \Omega_{\mathcal{E}_2}. $$

□

Lemma 14.18 reduces the proof of Lemma 14.19 to the proofs of the following two lemmas.

**Lemma 14.20.** Lemma 14.19 holds when $L(\mathcal{E}_2^c) = L(\mathcal{T}) \setminus \{\ell_0\}$.

**Lemma 14.21.** Lemma 14.19 holds when $L(\mathcal{E}_2^c) = L(\mathcal{T})$.

**PROOF OF LEMMA 14.20:** Recall that $L(\mathcal{E}_2^c) = L(\mathcal{T}) \setminus \{\ell_0\}$ and $e(\mathcal{E}_2^c, \cdot)$ is a bijection from $L(\mathcal{E}_2^c)$ to $\mathcal{E}_2^c \setminus \{f\}$ for a unique element $f$ of $\mathcal{E}_2^c$ from Lemma 14.18. There is a leaf $\ell_1$ such that $f \in [\ell_1, T]$. Let $f_2$ be the last edge (the closest to $f$) of $\mathcal{E}_2^c$ in $[\ell_1, f[$. Then we have $f_2 = e(\mathcal{E}_2^c, \ell_2)$ for some leaf $\ell_2$ of $L(\mathcal{E}_2^c)$. If there is an edge of $\mathcal{E}_2^c$ in $]f, T]$, define the leaf $\ell_3$ such that $e(\mathcal{E}_2^c, \ell_3)$ is the first edge of $\mathcal{E}_2^c$ in $]f, T]$ as in Figure 14.4. Otherwise, set $\ell_3 = \ell_0$. Note that $f$ and $f_2$ are in $\mathcal{E}_3$.

<figure><img src="image_114.png" /><figcaption>Figure 14.4: The edge $f$, when $L(\mathcal{E}_2^c) \neq L(\mathcal{T})$, in the proof of Theorem 14.16</figcaption></figure>

The product $u_{f_2}u_f$ is given by the expression

$$ u_{f_2}u_f = \frac{U([\ell_3, N(\ell_2, \ell_3)])}{\prod_{z \in [\ell_2, N(\ell_2, \ell_3)] \setminus \{f, f_2\}} u_z} = \eta_2(U), $$
in $U \in ]0, \varepsilon]^{{\mathcal{E}}_2} \cap p_{{\mathcal{E}}_2}(X^\varepsilon(\mathcal{T}))$. In particular, we have $u_f = \eta_2(U)/u_{f_2}$, and the conditions $u_f \le \varepsilon$ and $u_{f_2} \le \varepsilon$ imply

$$ \frac{\eta_2(U)}{\varepsilon} \le u_{f_2} \le \varepsilon. $$

Let $e$ be an edge of $\mathcal{E}_2^c$ different from $f$. Then $e$ may be expressed as $e = e(\mathcal{E}_2^c \setminus \{f\}, \ell_e)$ for a unique $\ell_e = \ell(\mathcal{E}_2^c \setminus \{f\}, e)$. As in Lemma 14.8, there is a leaf $\ell' = \ell'(\mathcal{E}_2^c \setminus \{f\}, e)$ of $\mathcal{T}$ such that $e$ is the only element of $\mathcal{E}_2^c \setminus \{f\}$ in $[\ell_e, N(\ell_e, \ell')] \cup [\ell', N(\ell_e, \ell')]$. In particular, if $e \ne f_2$, then $f_2$ does not belong to $[\ell_e, N(\ell_e, \ell')] \cup [\ell', N(\ell_e, \ell')]$. Note that $f$ cannot be on $[\ell', N(\ell_e, \ell')]$. If $f \notin [\ell_e, N(\ell_e, \ell')]$, then $u_e$ is a function of $U \in ]0, \varepsilon]^{{\mathcal{E}}_2}$, $e \notin \mathcal{E}_3$, and $u_e$ is different from zero. If $f \in [\ell_e, N(\ell_e, \ell')]$, then $N(\ell_e, \ell_2)$ is on $]f_2, f[$ (equivalently,

<figure><img src="image_115.png" /><figcaption>Figure 14.5: More figures for the proof of Theorem 14.16</figcaption></figure>

$f_2 \in [\ell_2, N(\ell_e, \ell_2)]$ and $f \notin [\ell_2, N(\ell_e, \ell_2)]$), as in the left part of Figure 14.5, $e \in \mathcal{E}_3$, $f_2$ is the unique element of $\mathcal{E}_2^c$ in $[\ell_2, N(\ell_e, \ell_2)]$, and we have

$$ u_e = \frac{U([\ell_2, N(\ell_e, \ell_2)])}{\prod_{z \in [\ell_e, N(\ell_e, \ell_2)] \setminus \{e\}} u_z}. $$

So, when $U \in ]0, \varepsilon]^{{\mathcal{E}}_2} \cap p_{{\mathcal{E}}_2}(X^\varepsilon(\mathcal{T}))$ is fixed, $u_e$ is a linear function $u_e = \eta_{e,f_2}(U)u_{f_2}$ of $u_{f_2}$. In particular, we have $u_{f_2} \le \varepsilon/\eta_{e,f_2}(U)$. Set $a(U) = \eta_2(U)/\varepsilon$ and $b(U) = \min_{e \in \mathcal{E}_3 \setminus \{f\}}(\varepsilon/\eta_{e,f_2}(U))$, where $\eta_{f_2,f_2} = 1$. Then $U \in ]0, \varepsilon]^{{\mathcal{E}}_2}$ is in $p_{{\mathcal{E}}_2}(X^\varepsilon(\mathcal{T}))$ if and only if $a(U) \le b(U)$ and $u_e(U) \le \varepsilon$ for any $e \in \mathcal{E}_2^c \setminus \mathcal{E}_3$. Furthermore, $X^\varepsilon(\mathcal{T}) \cap p_{{\mathcal{E}}_2}^{-1}([0, \varepsilon]^{{\mathcal{E}}_2})$ is the set

$$ \{x_{f_2}(u_{f_2}, U) : U \in ]0, \varepsilon]^{{\mathcal{E}}_2} \cap p_{{\mathcal{E}}_2}(X^\varepsilon(\mathcal{T})), u_{f_2} \in [a(U), b(U)]\}. $$

The boundary part $\partial_a(X^\varepsilon(\mathcal{T}))$, corresponds to $u_f = \varepsilon$. The boundary part $\partial_b(X^\varepsilon(\mathcal{T}))$, along which $u_e = \varepsilon$ for some $e \in \mathcal{E}_3$, lies in the intersection of
the nonsingular $\tilde{X}$ with loci ($u_e = \varepsilon$) for some $e \in \mathcal{E}_3$. The locus ($u_e = \varepsilon$) is transverse to $\tilde{X}$, for any $e \in \mathcal{E}_3$ since $\mathcal{E}_2 \cup \{e\}$ is a maximal free system for such an $e$. So $\tilde{X} \cap (u_e = \varepsilon)$ is a manifold of dimension ($d(\mathcal{T}) - 1$) for all $e \in \mathcal{E}_3$. Let $e$ and $e'$ be two distinct edges of $\mathcal{E}_3$. Then $u_e = u_{e'} = \varepsilon$ implies a nontrivial equation among variables $u_h$ associated to edges $h$ of $\mathcal{E}_2$ unless $e$ and $e'$ are both adjacent to leaves and meet at a node, which implies that $u_e = u_{e'}$ on $\tilde{X}$. We may assume that this configuration never occurs, without loss of generality, because the space $X(\mathcal{T})$ is canonically diffeomorphic to the space $X(\mathcal{T}')$ obtained from $\mathcal{T}$ by erasing $e'$. Except for this special configuration, the loci $\tilde{X} \cap (u_e = u_{e'} = \varepsilon)$ are of dimension less than ($d(\mathcal{T}) - 1$), and we have a stratification of $\partial_a(X^\varepsilon(\mathcal{T}))$ and $\partial_b(X^\varepsilon(\mathcal{T}))$, where the ($d(\mathcal{T}) - 1$)-dimensional strata are the loci of $\partial_a(X^\varepsilon(\mathcal{T}))$ and $\partial_b(X^\varepsilon(\mathcal{T}))$ where $u_e = \varepsilon$ for (exactly) one $e$ of $\mathcal{E}_3$. These strata are smooth open ($d(\mathcal{T}) - 1$)-manifolds. $\square$

<figure><img src="image_116.png" /></figure>

**PROOF OF LEMMA 14.21:** Let us now assume that $L(\mathcal{E}_2^c) = L(\mathcal{T})$, and recall that $\mathbf{e}(\mathcal{E}_2^c, \cdot)$ is a bijection from $L(\mathcal{T})$ to $\mathcal{E}_2^c$ in this case. The elements of $\mathcal{E}_3$ are the edges $e$ of $\mathcal{E}_2^c$ such that there is no edge of $\mathcal{E}_2^c$ on $]e, T]$. In particular, $\mathcal{E}_3$ is a codimension-one system of edges of $\mathcal{T}$.

Let $e = \mathbf{e}(\mathcal{E}_2^c, \ell_e)$ be an element of $\mathcal{E}_2^c \setminus \mathcal{E}_3$, let $\ell$ be a leaf of $\mathcal{T}$ such that the first edge of $\mathcal{E}_2^c$ on $]e, T]$ is $\mathbf{e}(\mathcal{E}_2^c, \ell)$. Then $e$ is the only element of $\mathcal{E}_2^c$ in $[\ell_e, N(\ell, \ell_e)] \cup [\ell, N(\ell, \ell_e)]$, as in Figure 14.5. So $u_e$ depends only on the fixed variables of $\mathcal{E}_2$, and it is not zero.

If $f \in \mathcal{E}_3$, let $\ell(f)$ denote the unique leaf such that $L(\mathcal{E}_2^c \setminus \{f\}) = L(\mathcal{T}) \setminus \{\ell(f)\}$. The variable $u_f$ is a linear function $u_f = \eta_f(U)u_{f_2}$ of $u_{f_2}$ for one (arbitrary) $f_2 \in \mathcal{E}_3$,

$$ \eta_f(U) = \frac{\prod_{z \in [\ell(f_2), T] \setminus \{f_2\}} u_z}{\prod_{z \in [\ell(f), T] \setminus \{f\}} u_z}. $$

Here, $a(U) = 0$ and $b(U) = \min_{f \in \mathcal{E}_3}(\varepsilon/\eta_f(U))$. So $\partial_a(X^\varepsilon(\mathcal{T}))$ is the codimension-one face associated to $\mathcal{E}_3$, as in Lemma 14.14, while $\partial_b(X^\varepsilon(\mathcal{T}))$ may be stratified as in the previous proof. $\square$

<figure><img src="image_117.png" /></figure>

The proof of Theorem 14.16 is finished. So we have the announced Stokes formula for all forms as in its statement.
## 14.2 Configuration spaces of graphs on long tangles

An $\infty$-component of an LTR $L$ is a connected component of the intersection of the image of $L$ with $R(\mathcal{C}) \setminus \mathcal{C}$. A univalent vertex $v$ of a Jacobi diagram on the domain $\mathcal{L}$ of $L$ approaching $\infty$ moves on such an $\infty$-component. Let $y_{\pm}(v)$ denote the orthogonal projection on $\mathbb{C}$ of the corresponding $\infty$-component of $v$. Then this $\infty$-component may be written as $\{y_{\pm}(v)\} \times ]1, \infty[$ or $\{y_{\pm}(v)\} \times ]-\infty, 0[$. The projection $y_{\pm}(v)$ depends on the considered $\infty$-component of $v$, in general. When such a component is fixed, we simply denote the projection by $y(v)$, and we speak of the $\infty$-component of a univalent vertex mapped to $\infty$.

**Definition 14.22.** Define the two-point compactification of $\mathbb{R} = ]-\infty, +\infty[$ to be $[-\infty, +\infty]$. For a Jacobi diagram $\Gamma$ on the domain $\mathcal{L}$ of $L$, let $U_{ttbb}(\Gamma)$ be the set of vertices $v$ of $U(\Gamma)$ such that the component $L(\mathcal{L}(v))$ of $v$ goes from top to top or from bottom to bottom. Let $\overline{\mathcal{L}}(v)$ denote the two-point compactification of the component $\mathcal{L}(v)$ of $\mathcal{L}$.

The open manifold $\tilde{C}(R(\mathcal{C}), L; \Gamma)$ embeds naturally in the product

$$C_{V(\Gamma)}(R(\mathcal{C})) \times \prod_{v \in U_{ttbb}(\Gamma)} \overline{\mathcal{L}(v)}.$$

Let $C_L^f = C^f(R(\mathcal{C}), L; \Gamma)$ be the closure of $\tilde{C}(R(\mathcal{C}), L; \Gamma)$ in this product. This closure maps naturally onto the closure $C_L = C(R(\mathcal{C}), L; \Gamma)$ of $\tilde{C}(\tilde{R}(\mathcal{C}), L; \Gamma)$ in $C_{V(\Gamma)}(R(\mathcal{C}))$. An element of $C^f(R(\mathcal{C}), L; \Gamma)$ is a configuration $c_{V(\Gamma)}$ of $C(R(\mathcal{C}), L; \Gamma)$ equipped with the additional data of an $\infty$-component for each vertex of $U_{ttbb}(\Gamma) \cap p_b(c_{V(\Gamma)})^{-1}(\infty)^2$.

In this section, we prove the following theorem.

**Theorem 14.23.** Let $L: \mathcal{L} \hookrightarrow \tilde{R}(\mathcal{C})$ be a long tangle representative, and let $\Gamma$ be a numbered degree $n$ Jacobi diagram with support $\mathcal{L}$ without looped edges. For any $c_{V(\Gamma)}^0$ in $C^f(R(\mathcal{C}), L; \Gamma)$, there exist

* a manifold $W$ with boundary and ridges,

* a small $\varepsilon > 0$,

<footnote n="2">²This piece of data is automatically determined by $c_{V(\Gamma)}$ in most cases. However, it may happen that it is not. For example, when $\Gamma$ has a unique univalent vertex on a strand going from top to top, and when this vertex is mapped to $\infty$, the strand of the vertex may not be determined by the configuration in $C_{V(\Gamma)}(R(\mathcal{C}))$.</footnote>
* an oriented tree $\mathcal{T}^0 = \mathcal{T}(c_{V(\Gamma)}^0)$ (described in Notation 14.43), with its associated singular space $X(\mathcal{T}^0)$ defined in Definition 14.3, and

* a smooth map

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

whose restriction to $(X(\mathcal{T}^0) \cap [0, \varepsilon^{[\mathcal{T}^0]}) \times W)$ is injective and induces a map

$$ \varphi^f: (X(\mathcal{T}^0) \cap [0, \varepsilon^{[\mathcal{T}^0]}) \times W \to C^f(R(\mathcal{C}), L; \Gamma), $$

such that $\mathrm{Im}(\varphi^f)$ is an open neighborhood of $c_{V(\Gamma)}^0$ in $C^f(R(\mathcal{C}), L; \Gamma)$ where the $\infty$-component for each univalent vertex of the $p_b(c_{V(\Gamma)}^0)^{-1}(\infty)$ is the same as its $\infty$-component for $c_{V(\Gamma)}^0$.

Let $\vec{N}$ denote the upward unit vertical vector. The codimension-one open faces of $C^f(R(\mathcal{C}), L; \Gamma)$ are

* the faces corresponding to the collapse of a subgraph at one point in $\vec{R}(\mathcal{C})$ as before,

* the faces corresponding to a set of vertices mapped to $\infty$, for which the configuration up to dilation at $\infty$ is injective and does not map a point to 0, as before,

* additional faces called T-faces (for which $C(R(\mathcal{C}), L; \Gamma)$ is not transverse to the ridges of $C_{V(\Gamma)}(R(\mathcal{C}))$), where

- a set of vertices $B \sqcup \sqcup_{j \in I} B_j$ is mapped to $\infty$, for a nonempty set $I$,

- the corresponding configuration up to dilation from $B \sqcup \sqcup_{j \in I} B_j$ to $T_\infty R(\mathcal{C})$ maps each $B_j$ to a nonzero point of the vertical line, and it injects $B$ outside zero and the images of the $B_j$, which are distinct,

- each subset $B_j$ contains univalent vertices of at least 2 distinct $\infty$-components,

- for each $B_j$, the infinitesimal configuration of $B_j$ is an injective configuration of a Jacobi diagram on the lines that extend the half-lines above (resp. below) $\mathcal{C}$, if $B_j$ is mapped to $\lambda \vec{N}$ for some $\lambda > 0$ (resp. for some $\lambda < 0$), up to global vertical translation. (No inversion is involved here.)
Together with Lemma 14.15 and Theorem 14.16, Theorem 14.23 implies the following lemma.

**Lemma 14.24.** *Theorem 12.2 is true.*

Recall $C_L = C(R(C), L; \Gamma)$ and $C_L^f = C^f(R(C), L; \Gamma)$. Let $\eta$ be a form of degree $(\dim(C_L) - 1)$ of $C_{V(\Gamma)}(R(C))$. Then $\int_{C_L} d\eta$ is the sum $\sum_F \int_F \eta$, which runs over the codimension-one faces $F$ of $C_L^f$, oriented as such, and listed in Theorem 14.23.

□

**Example 14.25.** Let $K: ]0, 1[ \hookrightarrow \tilde{R}(C)$ be a (long) component of $L$. Assume that $K: ]0, 1[ \hookrightarrow \tilde{R}(C)$ goes from top to top. Let $d_1 = -\{z_1\} \times [1, \infty]$ and $d_2 = \{z_2\} \times [1, \infty]$ denote the vertical half-lines of $K$ above $C$, where $d_1$ is before $d_2$. Let $G = \{(h, k) \in ]0, 1] \times \mathbb{R} : k + \frac{1}{h} \ge 1\}$. Define the diffeomorphism

$$ g: \begin{array}{rcl} G & \to & d_1 \times d_2 \\ (h, k) & \mapsto & ((z_1, \frac{1}{h}), (z_2, k + \frac{1}{h})). \end{array} $$

This diffeomorphism $g$ extends as a continuous map

$$ g: G \cup (\{0\} \times \mathbb{R}) \to C(R(C), L; \hat{\zeta}^K). $$

It maps $(0, k)$ to the limit $g(0, k)$ at 0 in $C_2(R(C))$ of the $g([0, \varepsilon] \times k)$. The image of $g(0, k)$ under the canonical map from $C_2(R(C))$ to $R(C)^2$ is $(\infty, \infty)$. The configuration in $T_\infty R(C)$ up to dilation corresponding to $g(0, k)$ maps the two points to the same upward vertical vector. We have $p_\tau(g(0, k)) = \frac{(z_2-z_1,k)}{\|(z_2-z_1,k)\|}$. The image of $(-g(\{0\} \times \mathbb{R}))$ under $p_\tau$ is the open half-circle from $\vec{N}$ to $-\vec{N}$ through the direction of $(z_2 - z_1)$. This codimension-one face of $C^f(R(C), L; \hat{\zeta}^K)$ is an example of a $T$-face, for which $B \sqcup \sqcup_{j \in I} B_j = B_1$ is the pair of vertices of the graph. Since this codimension-one face sits in a codimension-two face of $C_2(R(C))$, $C(R(C), L; \hat{\zeta}^K)$ is not transverse to the ridges of $C_2(R(C))$.

Recall that the elements of $C_L^f = C^f(R(C), L; \Gamma)$ are elements of the closure $C_L$ of $\check{C}(\check{R}(C), L; \Gamma)$ in $C_{V(\Gamma)}(R(C))$ equipped with the additional data of the $\infty$-components of the univalent vertices sent to $\infty$.

First note that the configuration space $C_L$ intersects $p_b^{-1}(\check{R}(C)^{V(\Gamma)})$ as a smooth submanifold as in the case of links. The only difference with the case of links occurs when some univalent vertices approach $\infty$. Our configuration space is a local product of the space of the restrictions of the configurations
to the points near $\infty$ and the space of the restrictions of the configurations to the other points, which is a smooth manifold with boundary whose structure has been studied in detail in Chapter 8.

Recall the orientation-reversing embedding $\phi_{\infty}$

$$ \phi_{\infty}: \mathbb{R}^3 \to S^3 \\ \mu(x \in S^2) \mapsto \begin{cases} \infty & \text{if } \mu = 0 \\ \frac{1}{\mu}x & \text{otherwise.} \end{cases} $$

According to Corollary 8.37, with the notation of Chapter 8, and especially those of Section 8.7, an element $c_{V(\Gamma)}$ of $C_{V(\Gamma)}(R(\mathcal{C}))$ consists of

* a subset $V = p_b(c_{V(\Gamma)})^{-1}(\infty)$ of $V(\Gamma)$,

* an element $c_{V(\Gamma)\setminus V} = c_{V(\Gamma)}|_{V(\Gamma)\setminus V}$ of $C_{V(\Gamma)\setminus V}[\check{R}(\mathcal{C})]$,

* an element $c = c_{V(\Gamma)}|_V$ of $C_V(R(\mathcal{C})) \cap p_b^{-1}(\infty^V)$,

* a $\Delta$-parenthesization $\mathcal{P}^+ = \mathcal{P}^+(c)$ of $V^+ = V \sqcup \{v_{\infty}\}$,

* for each $A \in \mathcal{P}^+$, an injective configuration

$$ T_0\phi_{\infty*} \circ w_A \in \check{S}_{K(A)}(T_\infty R(\mathcal{C})), $$

up to dilation and translation (see Proposition 8.36).

**Proposition 8.33** describes the restriction maps, naturally. As reminded above, the configuration space $C_{V(\Gamma)}(R(\mathcal{C}))$ has a natural stratification induced by $V = p_b(c)^{-1}(\infty)$, the parenthesization associated to $c_{V(\Gamma)\setminus V}$ (as before Proposition 8.34) and the above $\Delta$-parenthesization $\mathcal{P}^+$ of $V^+$. Each stratum has a well-defined dimension.

Below, we refine this partition induced on $C^f(R(\mathcal{C}), L; \Gamma)$ by the stratification of $C_{V(\Gamma)}(R(\mathcal{C}))$.

**Notation 14.26.** As in Proposition 8.36, we define the totally ordered subset $\mathcal{P}_s^+$ to be the set of elements of $\mathcal{P}^+$ containing $v_{\infty}$ and $\mathcal{P}_s = \{A \setminus \{v_{\infty}\} : A \in \mathcal{P}_s^+\}$. We have

$$ \mathcal{P}_s = \{V = V(1), V(2), \dots, V(\sigma)\} $$

with $V(i+1) \subset V(i)$. We let $K_d^s(V(i))$ denote the set of kids of $V(i)^+$ that do not contain $v_{\infty}$. We impose $w_{V(i)^+}(v_{\infty}) = 0$ and define $f_i$ to be the restriction of $w_{V(i)^+}$ to $V(i)$. So $f_i$ is an injective map from $K_d^s(V(i))$ to $\mathbb{R}^3 \setminus \{0\}$ up to dilation. It belongs to $\check{S}(\mathbb{R}^3, K_d^s(V(i)))$.

The sets of $\mathcal{P}^+$ that contain a univalent vertex are called *univalent*. As in Proposition 8.36, we set $\mathcal{P}_d = \mathcal{P}^+ \setminus \mathcal{P}_s^+$. A *possibly separating set* associated to the above parenthesization $\mathcal{P}^+$ and to the data of the $\infty$-components of the elements of $V$ is a set $A \in \mathcal{P}_d$ such that
• A has at least two univalent vertices on different $\infty$-components and

• each kid of A has all its univalent vertices on the same $\infty$-component.

Let $\mathcal{P}_X$ denote the set of possibly separating sets associated to the above parenthesization $\mathcal{P}^+$. A set $A$ of $\mathcal{P}_X$ is separating (with respect to $c_{V(\Gamma)}$) if it has at least two univalent kids $A_1$ and $A_2$ such that $w_A(A_1) - w_A(A_2)$ is not vertical. The set of separating sets of $c_{V(\Gamma)}$ is denoted by $\mathcal{P}_x$.

Recall that $\vec{N}$ denotes the upward unit vertical vector. Let $p_{\mathbb{C}}: (\mathbb{R}^3 = \mathbb{C} \times \mathbb{R}) \to \mathbb{C}$ denote the orthogonal projection onto the horizontal plane $\mathbb{C}$, and let $p_{\mathbb{R}}: \mathbb{C} \times \mathbb{R} \to \mathbb{R}$ denote the orthogonal projection onto the vertical line $\mathbb{R}$.

We are going to prove the following two propositions.

**Proposition 14.27.** The space $C_L^f = C^f(R(C), L; \Gamma)$ of Definition 14.22 is the space of configurations $c_{V(\Gamma)}$ of $C_{V(\Gamma)}(R(C))$ as above, equipped with $\infty$-components for the univalent vertices of $V = p_b(c_{V(\Gamma)})^{-1}(\infty)$, such that the following conditions are satisfied.

1. If the configuration $c_{V(\Gamma)}|_{U(\Gamma)\setminus(V\cap U(\Gamma))}$ is injective, then it factors through the restriction to $U(\Gamma) \setminus (V \cap U(\Gamma))$ of a representative of $i_\Gamma$ that maps the univalent vertices of $V \cap U(\Gamma)$ to their $\infty$-components further than the elements of $U(\Gamma) \setminus (V \cap U(\Gamma))$. If the configuration $c_{V(\Gamma)}|_{U(\Gamma)\setminus(V\cap U(\Gamma))}$ is not injective, then it factors through a limit of such restrictions. In any case, the possible infinitesimal configurations of vertices of $U(\Gamma) \setminus (V \cap U(\Gamma))$ are locally ordered on the tangent space to their component, as in the case of links (see Sections 8.4 and 8.8).

2. The $f_i$ map the elements of $K_d^s(V(i))$ that contain a univalent vertex on an $\infty$-component $y \times ]1, \infty[$ (resp. $y \times ]-\infty, 0[$) to the half-line $\mathbb{R}^+\vec{N}$ (resp. $\mathbb{R}^+(-\vec{N})$).

3. If $v_1 \in A_1$ and $v_2 \in A_2$ are two univalent vertices of distinct kids $A_1$ and $A_2$ of an element $A \in \mathcal{P}^+$, fix a normalization of $w_A$, and let $\vec{y} = y(v_2) - y(v_1)$.

• If $\vec{y} = 0$ (that is if $v_1$ and $v_2$ are on the same $\infty$-component), and if $v_1$ is closer to $\infty$ than $v_2$, then $w_A(A_2) - w_A(A_1)$ is a nonzero vertical vector, which may be expressed as $\alpha\vec{N}$, where $\alpha$ is positive when the $\infty$-component is above $C$, and $\alpha$ is negative when the $\infty$-component is under $C$.

• If $A \notin \mathcal{P}_X$, then $w_A(A_2) - w_A(A_1)$ is also a nonzero vertical vector.
* If $A \in \mathcal{P}_X$ and if $\vec{y} \neq 0$, then $w_A(A_2) - w_A(A_1)$ may be expressed as $(\alpha \vec{N} + \beta \vec{y})$ for some nonzero pair $(\alpha, \beta)$ of $\mathbb{R} \times \mathbb{R}^+$. Furthermore, if $v_3 \in A_3$ is a univalent vertex of another kid $A_3$ of $A$, then there exists $\alpha_3 \in \mathbb{R}$ such that $w_A(A_3) - w_A(A_1)$ is equal to $(\alpha_3 \vec{N} + \beta (y(v_3) - y(v_1)))$.

**Proposition 14.28.** The space $C_L^f$ of Proposition 14.27 is partitioned by the data for a configuration $c_{V(\Gamma)}$ of

* the set $V = p_b(c_{V(\Gamma)})^{-1}(\infty)$,

* the parenthesization $\mathcal{P}(c_{V(\Gamma)}|_{V(\Gamma)\setminus V})$ of $V(\Gamma)\setminus V$ associated to $c_{V(\Gamma)}|_{V(\Gamma)\setminus V}$ (as before Proposition 8.34),

and, if $V \neq \emptyset$,

* the $\Delta$-parenthesization $\mathcal{P}^+ = \mathcal{P}^+(c_{V(\Gamma)}|_V)$ of $V^+ = V \cup \{v_\infty\}$

* the data of the $\infty$-components of the univalent vertices that are mapped to $\infty$, and

* the set $\mathcal{P}_x$ of separating sets of $\mathcal{P}^+$.

The part associated to the above data is a smooth submanifold of $C_{V(\Gamma)}(R(\mathcal{C}))$ of dimension

$$|U(\Gamma)| + 3|T(\Gamma)| - |\mathcal{P}(c_{V(\Gamma)}|_{V(\Gamma)\setminus V})| - |\mathcal{P}^+| + |\mathcal{P}_x|.$$

This partition is a stratification of $C_L^f$.

**Remark 14.29.** Proposition 14.28 implies that the only codimension-one new parts –which come necessarily from strata for which $V \neq \emptyset$– come from the parts such that $\mathcal{P}^+ = \{V^+\} \sqcup \mathcal{P}_x$ and $\mathcal{P}_d = \mathcal{P}_x$. They are the $T$-faces of Theorem 14.23, for which $\mathcal{P}_s = \{V\} = \{B \sqcup \sqcup_{j \in I} B_j\}$ and $\mathcal{P}_d = \mathcal{P}_X = \mathcal{P}_x = \{B_j : j \in I\}$.

The rest of this section is devoted to the proofs of Theorem 14.23 and the above two propositions.

Let $c_{V(\Gamma)}^0$ be a configuration of $C_L^f$ in a stratum as in the statement of Proposition 14.28. Let $c^0 = c_{V(\Gamma)}^0|_V$ denote the restriction of $c_{V(\Gamma)}^0$ to $V(=V(c^0))$.

It is easy to see that $c_{V(\Gamma)}^0$ has a neighborhood $N_\Gamma(c_{V(\Gamma)}^0)$ in $C_L^f$ consisting of configurations mapping

<footnote n="3">³When $V = \emptyset$, the structure of $C_L^f$ near $c_{V(\Gamma)}$ is already known from Chapter 8.</footnote>