path. In particular, any two points $a$ and $b$ of $S_B(\mathbb{C})$ are connected by a semi-algebraic path $\gamma$: $[0, 1] \to S_B(\mathbb{C})$ such that $\gamma(0) = a$ and $\gamma(1) = b$. Furthermore, any path from $a$ to $b$ of $S_B(\mathbb{C})$ can be $C^0$-approximated by a homotopic semi-algebraic path. So, any homotopy class of paths from $a$ to $b$ has a semi-algebraic representative.

Now Theorem 13.8 is a direct corollary of the following one, which will be proved after Lemma 16.41. This theorem is a mild generalization of [Poi00, Proposition 9.2], thanks to Lemma 16.33.

**Theorem 16.35.** Let $\gamma: [0, 1] \to [0, 1] \times S_B(\mathbb{C})$ be a semi-algebraic path. Then $\widetilde{\text{hol}}_{\gamma}(\eta_{B,A})$ depends only on $\gamma(0), \gamma(1)$, the $\tilde{\omega}(i, S^2)$ for $i \in A$, and the homotopy class of $\gamma$ relatively to $\partial\gamma$.

According to Lemmas 16.21 and 16.34, Theorem 16.35 holds for smooth paths $\gamma$ of $\{u\} \times \check{S}_B(\mathbb{C})$ and their piecewise smooth compositions. We now prove the following other particular case of Theorem 16.35.

**Lemma 16.36.** Let $\gamma: [0, 1]^2 \to [0, 1] \times S_B(\mathbb{C})$ be a semi-algebraic homotopy such that

* $\gamma$ is injective on $[0, 1] \times ]0, 1[$,

* $\gamma_u(t) = \gamma(u, t)$ belongs to $[0, 1] \times \check{S}_B(\mathbb{C})$ for any $(u, t) \in ]0, 1] \times ]0, 1[$,

* we have $\gamma_u(0) = \gamma_0(0)$ and $\gamma_u(1) = \gamma_0(1)$ for all $u \in [0, 1]$,

* $\gamma_0(t)$ is in a fixed stratum of $[0, 1] \times S_B(\mathbb{C})$ for $t \in ]0, 1[$, where a stratum of $[0, 1] \times S_B(\mathbb{C})$ is the product by $[0, 1]$ of a stratum of $S_B(\mathbb{C})$ associated to a $\Delta$-parenthesization.

Then we have

$$ \widetilde{\text{hol}}_{\gamma_0}(\eta_{B,A}) = \widetilde{\text{hol}}_{\gamma_1}(\eta_{B,A}). $$

Lemma 16.36 is the direct consequence of Lemmas 16.39 and 16.40 below. The proof of Lemma 16.39 uses Lemma 16.38 and the following sublemma.

**Sublemma 16.37.** Under the assumptions of Lemma 16.36, let $\Gamma^{(r)}$ be an element of $\mathcal{D}_{n,A}^{e,r}(\bigsqcup_{b \in B} \mathbb{R}_b)$, recall $C(\Gamma^{(r)}, \gamma_u) = P_{\Gamma^{(r)}}^{-1}(\gamma_u^{(r)}(\Delta^{(r)}))$, and let

$$ C(\Gamma^{(r)}, (\gamma_u)) = \cup_{u \in [0,1]} C(\Gamma^{(r)}, \gamma_u) $$

be the associated semi-algebraic set of dimension $2|E(\Gamma^{(r)})| + 1$. Then the codimension-one boundary of $C(\Gamma^{(r)}, (\gamma_u))$ is

$$ C(\Gamma^{(r)}, \gamma_1) - C(\Gamma^{(r)}, \gamma_0) - \cup_{u \in [0,1]} \partial C(\Gamma^{(r)}, \gamma_u), $$
with

$$ \partial C(\Gamma^{(r)}, \gamma_u) = \partial_C C(\Gamma^{(r)}, \gamma_u) + \partial_{\Delta C}(\Gamma^{(r)}, \gamma_u), \\ \partial_C C(\Gamma^{(r)}, \gamma_u) = \pm \cup_{(t_1, \dots, t_r) \in \Delta^{(r)}} \partial P_{\Gamma^{(r)}}^{-1}(\gamma_u^{(r)}((t_1, \dots, t_r))), $$

and

$$ \partial_{\Delta C}(\Gamma^{(r)}, \gamma_u) = \pm P_{\Gamma^{(r)}}^{-1}(\gamma_u^{(r)}(\partial \Delta^{(r)})). $$

PROOF: When the image of $(\gamma_u)$ is in $[0, 1] \times \tilde{S}_B(\mathbb{C})$, it follows from Lemmas 15.13 and 14.14. Let us prove that it is still true for our homotopies $(\gamma_u)$. The part $C(\Gamma^{(r)}, \gamma_1)$ is in the boundary as before. We can ignore the contributions of the extremities of $\gamma_1$ since they belong to parts of dimension at most $2|E(\Gamma^{(r)})| - 1$, thanks to Lemma 15.12. For the part coming from $\cup_u \partial C(\Gamma^{(r)}, \gamma_u)$ in the $2|E(\Gamma)|$-dimensional boundary, we may restrict to $u \in ]0, 1[$ for dimension reasons, which we do. So this part is in the boundary as before, too

The part over $\gamma_0$ of the codimension-one boundary of $C(\Gamma^{(r)}, (\gamma_u))$ is included in $C(\Gamma^{(r)}, \gamma_0)$. Let us prove that the corresponding algebraic boundary is indeed $-C(\Gamma^{(r)}, \gamma_0)$ when $\gamma_0([0, 1])$ is in some stratum of $[0, 1] \times \partial \tilde{S}_B(\mathbb{C})$, associated to a parenthesization $\mathcal{P}_B$ of $B$.

Let $t_i \in ]0, 1[$. In a neighborhood $[0, \eta[ \times N(t_i)$ of $(0, t_i)$ in $[0, 1]^2$, $\gamma_u(t) = \gamma(u, t)$ is expressed as

$$ \gamma_u(t) = \left( (y_D(u, t))_{D \in \mathcal{P}_B}, (u_D(u, t))_{D \in \mathcal{P}_B \setminus \{B\}} \right), $$

where we have $u_D(0, t) = 0$ for all $t \in N(t_i)$.

Let $c^0$ be a point of the $(2|E(\Gamma_i)| - 1)$-dimensional open part of $\mathcal{V}(\gamma_0(t_i), \Gamma_i)$ with $\gamma'_0(t_i) \neq 0$. According to Lemma 15.12, $\mathcal{P}(c^0)$ equals $\mathcal{P}'_x(c^0)$. We express a neighborhood of $c^0$ in $\cup_{(u,t) \in [0, \eta[ \times N(t_i)} \mathcal{V}(\gamma_u(t), \Gamma_i)$, as a product by $[0, \eta[ \times N(t_i)$, as follows. We use the above coordinates $y_D(u, t)$, $u_D(u, t)$ of the base, and the parameters $c_Z, p_{\mathbb{R}} \circ c_Z(Y)$, $\lambda(Y)$ listed in the fourth, fifth and sixth sets of variables of Lemma 15.6. We have $\lambda(V(\Gamma))(c^0) = \lambda(c^0)$ and $\lambda(c^0) \neq 0$ since we have $\mathcal{P} = \mathcal{P}'_x$. Let $Y$ be an element of $\mathcal{P} \setminus \{V(\Gamma)\}$. We similarly have $\lambda(Y)(c^0) \neq 0$. Let $Y^+$ be the smallest set of $\mathcal{P}$ that strictly contains $Y$. Set $B_1 = \hat{B}(Y)$ and $B'_1 = B_1^+ = \hat{B}(Y^+)$. Equation 15.2 in Lemma 15.6 applied to these sets when $n = 1$ may be written as $\lambda(Y)\mu_Y = \lambda(Y^+)u_{\hat{B}(Y)}$. It implies that $\mu_Y = (\lambda(Y^+)/\lambda(Y))u_{\hat{B}(Y)}$ is determined by the listed parameters, and that all $\mu_Y$ are zero over $\gamma_0(N(t_i))$ in our neighborhood of $c^0$. So we have $\mathcal{P} = \mathcal{P}'_x = \mathcal{P}(c^0)$ over $\gamma_0(N(t_i))$ in our neighborhood of $c^0$. In particular, if $Y \subsetneq Y'$, then $\hat{B}(Y) \subsetneq \hat{B}(Y')$, and all equations 15.2 are satisfied, there. Over $\gamma([0, \eta[ \times N(t_i))$, no $u_D$ vanishes so that no $\mu_Y$ vanishes either in our neighborhood of $c^0$, and the
equations 15.2 are implied by the equations $*(Y)$, which are implied by the equations $\mu_Y = (\lambda(Y^+)/\lambda(Y))u_{\tilde{B}(Y)}$ and $\lambda(V(\Gamma)) = \lambda$. So we get a neighborhood of $c^0$ in $\cup_{(u,t)\in[0,\eta[\times N(t_i))}V(\gamma_u(t), \Gamma_i)$ parametrized by $(u, t)$ and by the parameters $c_Z$, $p_{\mathbb{R}} \circ c_Z(Y)$, $\lambda(Y)$ in the fourth, fifth and sixth lists of variables of Lemma 15.6. For any horizontally normalizing kid $Y$ of $\mathcal{P}$, remove $\lambda(Y) = \lambda^0(Y)$ from the parameters. Use the first and second constraints of Lemma 15.6 to remove other superfluous parameters $p_{\mathbb{R}} \circ c_Z(Y)$, $|p_{\mathbb{C}} \circ c_Z(Y)|$ and get a free system of parameters. Thus, we obtain an open $(2|E(\Gamma_i)|)$-dimensional neighborhood $\mathcal{O}$ of $c^0$ in $\cup_{t\in N(t_i)}V(\gamma_0(t), \Gamma_i)$ and a local open embedding of the product of $[0, \eta[\times \mathcal{O}$ into $\cup_{(u,t)\in[0,\eta[\times N(t_i))}V(\gamma_u(t), \Gamma_i)$. So $(-C(\Gamma^{(r)}, \gamma_0))$ is the algebraic boundary of $C(\Gamma^{(r)}, (\gamma_u))$ over $\gamma_0$. $\square$

**Lemma 16.38.** For a Jacobi diagram in $\mathcal{D}_{n,A}^e(\sqcup_{b\in B}\mathbb{R}_b)$, for an element $(t, y)$ of $[0, 1] \times \tilde{S}_B(\mathbb{C})$, $d\eta_\Gamma(t, y)$ is the integral of $(-\bigwedge_{e\in E(\Gamma)} p_{e,S^2}^*(\tilde{\omega}(j_E(e), S^2)))$ along the interiors of the codimension-one faces of $\{t\} \times \mathcal{V}(y, \Gamma)$.

PROOF: See the proof of Lemma 15.23. $\square$

Recall from the beginning of Section 15.2 that the fiber $\mathcal{V}(y, \Gamma)$ is oriented so that the orientation of $\mathcal{V}(y, \Gamma)$ preceded by the upward translation parameter—which replaces the parametrization of the paths along which we integrate—matches the usual orientation of our configuration spaces, induced as in Lemma 7.1.

For a subset $A_i$ of $A$, we have

$$d\eta_{B,A_i} = \sum_{\Gamma_i \in \mathcal{D}_{n_i,A_i}^e (\sqcup_{b \in B} \mathbb{R}_b)} \zeta_{\Gamma_i} d\eta_{\Gamma_i} [\Gamma_i],$$

and we set

$$(\eta \wedge \eta)_{B,A_i} = \sum_{(\Gamma, \Gamma') \in \mathcal{D}_{n_i,A_i}^{e,2} (\sqcup_{b \in B} \mathbb{R}_b)} \zeta_{\Gamma \sqcup \Gamma'} \eta_\Gamma \wedge \eta_{\Gamma'} [\Gamma] [\Gamma'].$$

**Lemma 16.39.** Let $(\gamma_u(t))_{u\in[0,1]}$ be a semi-algebraic homotopy satisfying the assumptions of Lemma 16.36 or a smooth homotopy valued in $[0, 1] \times \tilde{S}_B(\mathbb{C})$. Assume that $\gamma_u(0)$ and $\gamma_u(1)$ do not depend on $u$.

Then we have

$$\begin{aligned} \widetilde{\mathrm{hol}}_{\gamma_1}(\eta_{B,A}) - \widetilde{\mathrm{hol}}_{\gamma_0}(\eta_{B,A}) &= \sum_{r=0}^{\infty} \sum_{(A_1, \dots, A_r) \in P_r(A)} \frac{\prod_{i=1}^{r} |A_i|!}{|A|!} \int_{[0,1] \times \Delta^{(r)}} \sum_{i=1}^{r} \delta(i, A_1, \dots, A_r), \end{aligned}$$
with

$$ \delta(i, A_1, \dots, A_r) = (-1)^{i-1} \bigwedge_{j=1}^{r} (\gamma \circ p_j)^*(\eta_{B,A_j}) \left( \frac{(\gamma \circ p_i)^*(d\eta_{B,A_i} + (\eta \wedge \eta)_{B,A_i})}{(\gamma \circ p_i)^*(\eta_{B,A_i})} \right), $$

where the fraction means that $(\gamma \circ p_i)^*(\eta_{B,A_i})$ is replaced by $(\gamma \circ p_i)^*(d\eta_{B,A_i} + (\eta \wedge \eta)_{B,A_i})$.

**PROOF:** Set $\partial_C C(\Gamma^{(r)}, (\gamma_u)) = - \cup_{u \in [0,1]} \partial_C C(\Gamma^{(r)}, \gamma_u)$ and

$$ \partial_C C(\Gamma^{(r)}, (\gamma_u)) = - \cup_{u \in [0,1]} \partial_C C(\Gamma^{(r)}, \gamma_u). $$

Since

$$ \partial C(\Gamma^{(r)}, (\gamma_u)) = C(\Gamma^{(r)}, \gamma_1) - C(\Gamma^{(r)}, \gamma_0) + \partial_C C(\Gamma^{(r)}, (\gamma_u)) + \partial_C C(\Gamma^{(r)}, (\gamma_u)) $$

is a null-homologous cycle, we get

$$ \widetilde{\mathrm{hol}}_{\gamma_1}(\eta_{B,A}) - \widetilde{\mathrm{hol}}_{\gamma_0}(\eta_{B,A}) = \tilde{\delta}_{(\gamma_u)}(d\eta_{B,A}) + \tilde{\delta}_{(\gamma_u)}((\eta \wedge \eta)_{B,A}), $$

with

$$ \tilde{\delta}_{(\gamma_u)}(d\eta_{B,A}) \stackrel{\mathrm{def}}{=} - \sum_{r=0}^{\infty} \sum_{\Gamma^{(r)} \in \mathcal{D}_{n,A}^{e,r} (\bigsqcup_{b \in B} \mathbb{R}_b)} \zeta_{\Gamma^{(r)}} \int_{\partial_C C(\Gamma^{(r)}, (\gamma_u))} \Omega_{\Gamma^{(r)}} [\Gamma^{(r)}] $$

and

$$ \tilde{\delta}_{(\gamma_u)}((\eta \wedge \eta)_{B,A}) \stackrel{\mathrm{def}}{=} - \sum_{r=0}^{\infty} \sum_{\Gamma^{(r)} \in \mathcal{D}_{n,A}^{e,r} (\bigsqcup_{b \in B} \mathbb{R}_b)} \zeta_{\Gamma^{(r)}} \int_{\partial \Delta C(\Gamma^{(r)}, (\gamma_u))} \Omega_{\Gamma^{(r)}} [\Gamma^{(r)}]. $$

Let us study these terms. Since

$$ P_{\Gamma^{(r)}}^{-1}(\gamma_u^{(r)}((t_1, \dots, t_r))) = \pm \prod_{i=1}^{r} \left( \left\{ p_{[0,1]}(\gamma_u(t_i)) \right\} \times \mathcal{V}(p_{\tilde{S}_B(\mathbb{C})}(\gamma_u(t_i)), \Gamma_i) \right) $$

is diffeomorphic to $\pm \prod_{i=1}^{r} \mathcal{V}(\gamma_u(t_i), \Gamma_i)$ (forgetting the natural $p_{\tilde{S}_B(\mathbb{C})}$), we have

$$ \partial P_{\Gamma^{(r)}}^{-1}(\gamma_u^{(r)}((t_1, \dots, t_r))) = \pm \sum_{i=1}^{r} \partial \mathcal{V}(\gamma_u(t_i), \Gamma_i) \times \prod_{j \in \underline{n} \setminus \{i\}} \mathcal{V}(\gamma_u(t_j), \Gamma_j). $$

According to Lemma 16.38, we have

$$ \tilde{\delta}_{(\gamma_u)}(d\eta_{B,A}) = \sum_{r=0}^{\infty} \sum_{(A_1, \dots, A_r) \in P_r(A)} \frac{\prod_{i=1}^{r} |A_i|!}{|A|!} \int_{[0,1] \times \Delta^{(r)}} \sum_{i=1}^{r} \alpha(i, A_1, \dots, A_r), $$
with

$$ \alpha(i, A_1, \dots, A_r) = (-1)^{i-1} \bigwedge_{j=1}^{r} (\gamma \circ p_j)^*(\eta_{B,A_j}) \left( \frac{(\gamma \circ p_i)^*(d\eta_{B,A_i})}{(\gamma \circ p_i)^*(\eta_{B,A_i})} \right) $$

and $d\eta_{B,\emptyset} = 0$.

Let us give a few details about the involved signs. Recall that the space $C(\Gamma^{(r)}, \gamma_u)$ is oriented locally as the product of the $C(\Gamma_i^{(1)}, \gamma_u)$ for $i \in \underline{r}$, which are oriented so that the parameter $t$ replaces the translation parameter in $\mathcal{V}(\gamma_u(t), \Gamma_i)$. Thus, the boundary $\partial_C$, along which we integrate, is locally diffeomorphic to

$$ \left( - \cup_{u \in [0,1]} \left( \prod_{j=1}^{i-1} C(\Gamma_j^{(1)}, \gamma_u) \right) \times \partial_C C(\Gamma_i^{(1)}, \gamma_u) \times \left( \prod_{j=i+1}^{r} C(\Gamma_j^{(1)}, \gamma_u) \right) \right), $$

where the dimension of $C(\Gamma_j^{(1)}, \gamma_u)$ is even. When rewriting such an integral as an integral over $[0, 1] \times \Delta^r$ of the two-form $(\gamma \circ p_i)^*(d\eta_{B,A_i})$ and one-forms $(\gamma \circ p_j)^*(\eta_{B,A_j})$, one must take into account the fact that the two-form will be integrated along the product by $[0, 1]$ of the interval parametrized by $t_i$. This gives rise to the factor $(-1)^{i-1}$.

Recall $\partial\Delta^{(r)} = \sum_{i=0}^{r} (-1)^{i+1} F_i(\Delta^{(r)})$, with $F_0(\Delta^{(r)}) = \{(0, t_2, \dots, t_r) \in \Delta^r\}$, $F_r(\Delta^{(r)}) = \{(t_1, t_2, \dots, t_{r-1}, 1) \in \Delta^{(r)}\}$, and

$$ F_i(\Delta^{(r)}) = \{(t_1, \dots, t_i, t_{i+1}, \dots, t_{r-1}) \in \Delta^{(r)}\} $$

for $i \in \underline{r-1}$. Observe that the faces $F_0$ and $F_r$ do not contribute to $\tilde{\delta}_{(\gamma_u)}((\eta \wedge \eta)_{B,A})$. Indeed for $F_0$, the directions of the edges of $\Gamma_1$ are in the image of $\mathcal{V}(\gamma_0(0), \Gamma_1)$, which is $(2|E(\Gamma_1)| - 1)$-dimensional. The contribution of the faces $F_i$ yields

$$ \tilde{\delta}_{(\gamma_u)}((\eta \wedge \eta)_{B,A}) = \sum_{r=1}^{\infty} \sum_{(A_1, \dots, A_{r-1}) \in P_{r-1}(A)} \frac{\prod_{i=1}^{r-1} |A_i|!}{|A|!} \int_{[0,1] \times \Delta^{r-1}} \sum_{i=1}^{r-1} \tilde{\delta}_d(i, A_1, \dots, A_{r-1}), $$

with

$$ \tilde{\delta}_d(i, A_1, \dots, A_{r-1}) = (-1)^{i-1} \bigwedge_{j=1}^{r-1} (\gamma \circ p_j)^*(\eta_{B,A_j}) \left( \frac{(\gamma \circ p_i)^*((\eta \wedge \eta)_{B,A_i})}{(\gamma \circ p_i)^*(\eta_{B,A_i})} \right). \quad \square $$

<figure><img src="image_133.png" /></figure>

Note that $(\eta \wedge \eta)_{B,A_i} = \sum_{(\Gamma, \Gamma') \in \mathcal{D}_{n_i, A_i}^{e,2}(\bigcup_{b \in B} \mathbb{R}_b)} \zeta_{\Gamma \sqcup \Gamma'} \eta_{\Gamma} \wedge \eta_{\Gamma'} [\Gamma] [\Gamma']$ is valued in the space of primitive elements of $\mathcal{A}_{n_i}(\bigcup_{b \in B} \mathbb{R}_b)$. (Use $\eta_{\Gamma} \wedge \eta_{\Gamma'} [\Gamma] [\Gamma'] + \eta_{\Gamma'} \wedge \eta_{\Gamma} [\Gamma'] [\Gamma] = \eta_{\Gamma} \wedge \eta_{\Gamma'} ([\Gamma] [\Gamma'] - [\Gamma'] [\Gamma]).)$
**Lemma 16.40.** The form $d\eta_{B,A} + (\eta \wedge \eta)_{B,A}$ on $[0, 1] \times \tilde{S}_B(\mathbb{C})$ vanishes identically for any subset $A$ of $3N$ with cardinality $3n$.

**PROOF:** We proceed by induction on $n$ as in the proof of Corollary 15.22, using Lemma 16.39 and Lemma 16.21, which guarantees that $\text{hol}(\eta_{B,A})$ vanishes along homotopically trivial loops in $\{u\} \times \tilde{S}_B(\mathbb{C})$ and along boundaries of squares $[w, w + \varepsilon] \times p_{CS} \circ \gamma([t, t + \varepsilon])$. (This flatness condition can also be proved directly as an exercise, as in Remark 15.24.) $\square$

Lemma 16.36 is proved. $\square$

We now generalize Lemma 16.36 as follows.

**Lemma 16.41.** Let $\gamma$ and $\delta$ be two semi-algebraic paths of $[0, 1] \times S_B(\mathbb{C})$ homotopic relatively to $\{0, 1\}$. Assume $\gamma([0, 1]) \subset \tilde{S}_B(\mathbb{C})$ and $\delta([0, 1]) \subset \tilde{S}_B(\mathbb{C})$. Then we have

$$ \widetilde{\text{hol}}(\eta_{B,A})(\gamma) = \widetilde{\text{hol}}(\eta_{B,A})(\delta). $$

**PROOF:** According to Lemma 16.21, it suffices to take care of homotopies near the endpoints. Thanks to Lemma 16.34, it suffices to prove that there exist $t, t' > 0$, and a path $\varepsilon$ from $\gamma(t)$ to $\delta(t')$ in $[0, 1] \times \tilde{S}_B(\mathbb{C})$ such that $\gamma|_{[0,t]} \varepsilon \delta|_{[0,t']}$ is null-homotopic and the holonomy along $\gamma|_{[0,t]} \varepsilon \delta|_{[0,t]}$ is one. When $t$ and $t'$ are small enough, the images of $\gamma|_{[0,t]}$ and $\delta|_{[0,t']}$ lie in a subset equipped with a local semi-algebraic chart as in Theorem 8.28, from which it is easy to construct semi-algebraic interpolations in products of sphere pieces and intervals. Furthermore, there is no loss of generality in assuming that $\gamma|_{[0,t]} \varepsilon$ and $\delta|_{[0,t']}$ meet only at $\gamma(0)$ and $\delta(t')$, and that straight interpolation provides a boundary-fixing semi-algebraic homotopy from $\gamma|_{[0,t]} \varepsilon$ to $\delta|_{[0,t']}$, which satisfies the injectivity hypotheses of Lemma 16.36. (Otherwise, we could use an intermediate $\gamma'|_{[0,t]}$.) Thus, Lemma 16.36 allows us to prove $\widetilde{\text{hol}}_{\gamma|_{[0,t]} \varepsilon}(\eta_{B,A}) = \widetilde{\text{hol}}_{\delta|_{[0,t']}}(\eta_{B,A})$. $\square$

**PROOF OF THEOREM 16.35:** Lemma 16.41 allows us to define a map $\tilde{h}_A$ induced by $\widetilde{\text{hol}}(\eta_{B,A})$ from homotopy classes of paths with fixed boundaries of $[0, 1] \times S_B(\mathbb{C})$ to $A_{|A|/3}(\bigcup_{b \in B} \mathbb{R}_b)$ as follows. For a path $\gamma: [0, 1] \to S_B(\mathbb{C})$, set $\tilde{h}_A(\gamma) = \widetilde{\text{hol}}(\eta_{B,A})(\delta)$ for any semi-algebraic path $\delta$ of $[0, 1] \times S_B(\mathbb{C})$ that is homotopic to $\gamma$ relatively to $\{0, 1\}$ and such that $\delta([0, 1]) \subset \tilde{S}_B(\mathbb{C})$.

Now, it suffices to prove that $\widetilde{\text{hol}}_\gamma(\eta_{B,A})$ coincides with $\tilde{h}_A(\gamma)$ for any semi-algebraic path $\gamma: [0, 1] \to [0, 1] \times S_B(\mathbb{C})$. Recall that a stratum of $[0, 1] \times S_B(\mathbb{C})$ is the product by $[0, 1]$ of a stratum of $S_B(\mathbb{C})$ associated to a $\Delta$-parenthesization. The preimage under $\gamma$ of such a stratum of $[0, 1] \times S_B(\mathbb{C})$ is semi-algebraic. So $\gamma$ is a finite composition of paths whose interiors lie in a
fixed stratum of $\mathcal{S}_B(\mathbb{C})$ (according to the Lojasiewicz theorem 16.32). Thus, thanks to Lemma 16.34, it suffices to prove that $\widetilde{\mathrm{hol}}_\gamma(\eta_{B,A})$ coincides with $\tilde{h}_A(\gamma)$ for any injective semi-algebraic path $\gamma$ whose interior lies in a fixed stratum of $\mathcal{S}_B(\mathbb{C})$ and in a subset equipped with a local semi-algebraic chart as in Theorem 8.28. Such a path can be deformed by sending the vanishing coordinates of $\gamma(t)$ in the $[0, \varepsilon]$ factors in such a chart to $\varepsilon(\frac{1}{2} - |\frac{1}{2} - t|)u$ for $u \in [0, 1]$ giving rise to a semi-algebraic homotopy $(\gamma_u(t))_{u \in [0,1]}$ such that $\tilde{h}_A(\gamma) = \widetilde{\mathrm{hol}}_{\gamma_1}(\eta_{B,A})$. So Lemma 16.36 implies $\widetilde{\mathrm{hol}}_{\gamma_0}(\eta_{B,A}) = \widetilde{\mathrm{hol}}_{\gamma_1}(\eta_{B,A})$. $\square$

Theorem 16.35 allows us to set the following definition.

**Definition 16.42.** For any continuous path $\gamma: [0, 1] \to [0, 1] \times \mathcal{S}_B(\mathbb{C})$, define $\widetilde{\mathrm{hol}}(\eta_{B,A})(\gamma)$ to be

$$ \widetilde{\mathrm{hol}}(\eta_{B,A})(\gamma) = \widetilde{\mathrm{hol}}(\eta_{B,A})(\delta) $$

for any semi-algebraic path $\delta$ of $[0, 1] \times \mathcal{S}_B(\mathbb{C})$ homotopic to $\gamma$ relatively to $\{0, 1\}$.

We can now generalize [Poi00, Proposition 1.18] for braids.

**Proposition 16.43.** Let $B$ and $C$ be two finite sets. Let $b_0 \in B$.

Let $\gamma_B$ be a path of $[0, 1] \times \mathcal{S}_B(\mathbb{C})$ and let $\gamma_C$ be a path of $[0, 1] \times \mathcal{S}_C(\mathbb{C})$. Let $\gamma_B(\gamma_C/b_0) = \gamma_B(\gamma_C/K_{b_0})$ be the q-braid obtained by cabling the strand $K_{b_0}$ of $b_0$ in $T(\gamma_B)$ by $T(\gamma_C)$, as in Notation 13.3. Then we have

$$ \widetilde{\mathrm{hol}}_{\gamma_B(\gamma_C/b_0)}(\eta_B(\frac{C}{b_0},)) = (\pi(C \times b_0)^* (\widetilde{\mathrm{hol}}_{\gamma_B}(\eta_{B,})) \widetilde{\mathrm{hol}}_{\gamma_C}(\eta_{C,}))_{\sqcup}, $$

where $\pi(C \times b_0)^*$ denotes the duplication of the strand $\mathbb{R}_{b_0}$ for diagrams as in Notation 6.31 and we use the product of Definition 16.10.

**PROOF:** Thanks to Definition 16.42, we assume that $\gamma_C$ and $\gamma_B$ are semi-algebraic, without loss of generality. Since a semi-algebraic path is a path composition of finitely many semi-algebraic paths whose interiors lie in a fixed stratum of $[0, 1] \times \mathcal{S}(\mathbb{C})$ (according to the Lojasiewicz theorem 16.32), the functoriality of Lemma 16.34 allows us to assume furthermore that the image of the interior of $\gamma_C$ lies in a fixed stratum of $[0, 1] \times \mathcal{S}_C(\mathbb{C})$ and that the image of the interior of $\gamma_B$ lies in a fixed stratum of $[0, 1] \times \mathcal{S}_B(\mathbb{C})$, for the proof. (Recall the commutation lemma 6.33.)

Let $B(C/b_0)$ be the set obtained from $B$ by replacing $b_0$ by $C$. We refer to Lemma 15.12 for the description of the stratification of $\mathcal{V}(\Gamma)$ and the dimensions of the fibers for connected diagrams on $B(C/b_0) \times \mathbb{R}$. When computing the “holonomy” of $\eta_{B(C/b_0),}$ we integrate over products of one-parameter families $\mathcal{V}(y, \Gamma)$ with $y \in \gamma_B(\gamma_C/b_0)([t_i - \varepsilon, t_i + \varepsilon])$, locally. We
may restrict to the strata of $\mathcal{V}(y, \Gamma)$ of dimension $(2|E(\Gamma)| - 1)$, which are described in Lemma 15.12.

Consider a connected diagram $\Gamma$ on $B(C/b_0) \times \mathbb{R}$ together with a $\Delta$-parenthesization $\mathcal{P}$ of its vertices corresponding to such a stratum of configurations. Since $\mathcal{P}$ is equal to $\mathcal{P}'_x$, all elements of $\mathcal{P}$ are univalent. Assume $\mathcal{P} \neq \{V(\Gamma)\}$. Let $\Gamma_{\mathcal{P}}$ be obtained from $\Gamma$ by identifying all the vertices in a daughter $A$ of $V(\Gamma)$ to a single vertex $v_A$ and by erasing the edges between two elements in $A$, for each $A$ of $D(V(\Gamma))$. Then $\Gamma_{\mathcal{P}}$ is connected and its vertices $v_A$ move along vertical lines. Let $U(\Gamma_{\mathcal{P}})$ and $T(\Gamma_{\mathcal{P}})$ respectively denote the set of univalent vertices of $\Gamma_{\mathcal{P}}$ distinct from the $v_A$ and the set of trivalent vertices of $\Gamma_{\mathcal{P}}$ distinct from the $v_A$. The dimension of the one-parameter family of configurations of the vertices of $\Gamma_{\mathcal{P}}$ up to vertical translation is $|D(V(\Gamma))| + |U(\Gamma_{\mathcal{P}})| + 3|T(\Gamma_{\mathcal{P}})|$. The form $\bigwedge_{e \in E(\Gamma_{\mathcal{P}})} p_{e,S^2}(\tilde{\omega}(j_E(e), S^2))$ factors through this one-parameter family. Thus, a count of half-edges shows that the stratum cannot contribute unless the vertices $v_A$ are univalent in $\Gamma_{\mathcal{P}}$.

Now assume that all the vertices $v_A$ are univalent in $\Gamma_{\mathcal{P}}$. Let $e_A$ be the edge of $\Gamma_{\mathcal{P}}$ with maximal label such that $e_A$ is adjacent to a vertex $v_A$ for some $A \in D(V(\Gamma))$. The subgraph $\Gamma_A$ of $\Gamma$ consisting of the vertices of $A$ and the edges of $\Gamma$ between two such vertices is connected. It has one bivalent vertex $b$ in $\Gamma_A$ (which is the end of $e_A$ in $\Gamma \cap A$). Its configurations are considered up to vertical translations. Their contribution is opposite to that of the configurations of the graph $\Gamma'_A$ obtained from $\Gamma_A$ by exchanging the labels and possibly the orientations of the two edges of $\Gamma_A$ that contain $b$, as in Lemma 9.11.

Thus, the $\Delta$-parenthesization of $\Gamma$ is $\{V(\Gamma)\}$ in the strata that may contribute. If $p_{B(C/b_0)}(U(\Gamma)) \subset C$, then $\Gamma$ is a diagram on $C \times \mathbb{R}$, which contributes as in $\mathrm{hol}_{\gamma_C}(\eta_{C,\cdot})$. Otherwise, the projection to the horizontal plane of the vertices of $U(\Gamma) \cap p_{B(C/b_0)}^{-1}(C)$ is reduced to a point. So all diagrams obtained from these diagrams $\Gamma$ by changing the map from $U(\Gamma) \cap p_{B(C/b_0)}^{-1}(C)$ to $C$ arbitrarily contribute together to $\pi(C \times b_0)^*(\mathrm{hol}_{\gamma_B}(\eta_{B,\cdot}))$ as desired, locally. We get the proposition since the two kinds of diagrams commute thanks to Lemma 6.33. $\square$

**Definition 16.44.** Recall Proposition 16.20 and Definition 16.42. For a $q$-braid (representative) $\gamma: [0, 1] \to S_B(\mathbb{C})$, set

$$ \mathcal{Z}^f(\gamma, \cdot, (\tilde{\omega}(i, 1, S^2))_{i \in \overline{3N}}) = \widetilde{\mathrm{hol}}_{\{1\} \times \gamma}(\eta_{B,\cdot}). $$

For a $q$-tangle

$$ T = T(\gamma^{-})(C, L, L_{\parallel})T(\gamma^{+}) $$
such that $\gamma^-$ and $\gamma^+$ are $q$-braids, and $(C, L)$ is a $J_{bb,tt}$-oriented framed tangle whose bottom configuration is $\gamma^-$ (1) and whose top configuration is $\gamma^+$ (0), as in Definition 13.1, for $N \in \mathbb{N}$, and for a family $(\omega(i, S^2))_{i \in \underline{3N}}$ of volume-one forms of $S^2$, set

$$ \mathcal{Z}^f(T, \cdot, (\omega(i, S^2))_{i \in \underline{3N}}) = \left( \mathcal{Z}^f(\gamma^-, \cdot, (\omega(i, S^2))) \mathcal{Z}^f(C, L, L_\parallel, \cdot, (\omega(i, S^2))) \mathcal{Z}^f(\gamma^+, \cdot, (\omega(i, S^2))) \right)_{\sqcup} $$

with the notation of Definition 16.10 and Definition 16.17.

Lemma 16.34 and the isotopy invariance of Theorems 16.16 and 16.35 ensure that the definition above is consistent.

Theorem 16.9 allows us to express the variation of

$$ \mathcal{Z}^f(T, \cdot, (\omega(i, S^2))_{i \in \underline{3N}}) $$

when $(\omega(i, S^2))_{i \in \underline{3N}}$ varies for framed straight tangles with injective top and bottom configurations.<fnref n="6" /> As a corollary of Theorem 16.35, this expression generalizes to $q$-tangles. We get the following theorem.

**Theorem 16.45.** Let $N \in \mathbb{N}$. For $i \in \underline{3N}$, let $\tilde{\omega}(i, S^2) = (\tilde{\omega}(i, t, S^2))_{t \in [0,1]}$ be a closed 2-form on $[0,1] \times S^2$ with $\int_{\{0\} \times S^2} \tilde{\omega}(i, 0, S^2) = 1$. Let $T$ denote a $J_{bb,tt}$-oriented $q$-tangle. Assume that the bottom and top configurations of $T$ are elements $y^-$ of $S_{B^-}(\mathbb{C})$ and $y^+$ of $S_{B^+}(\mathbb{C})$, respectively. Then we have

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

□

Recall that the forms $\eta$ and their holonomies introduced in Definitions 16.1 and 16.6 depend on the forms $\tilde{\omega}(i, S^2)$. In the end of this chapter, we will apply Theorem 16.45, only when $\tilde{\omega}(i, 0, S^2)$ is the standard form $\omega_{S^2}$. So $\mathcal{Z}^f(T, \cdot, (\tilde{\omega}(i, 0, S^2)))$ is simply $\mathcal{Z}^f(T, \cdot)$.

The following lemma is also easy to prove.

<footnote n="6">⁶See also the proof of Theorem 16.16.</footnote>
**Lemma 16.46.** The behavior of $\mathcal{Z}^f$ under reversing the orientation of a closed component is the same as that described in Proposition 10.23. Its behavior under reversing the orientation of component going from bottom to top or from top to bottom in a $q$-tangle $T$ is similar: Let $K$ be an oriented component going from bottom to top or from top to bottom in a $q$-tangle $T$. For a Jacobi diagram $\Gamma$ on the domain $\mathcal{L}$ of $T$, let $U_K(\Gamma)$ denote the set of univalent vertices of $\Gamma$ mapped to the domain $\mathbb{R}_K$ of $K$. This set is ordered by the orientation of $\mathbb{R}_K$. When the orientation of $K$ is changed, $\mathcal{Z}^f(T)$ is modified by reversing the orientation of $\mathbb{R}_K$ (that is reversing the order of $U_K(\Gamma)$) in classes $[\Gamma]$ of diagrams $\Gamma$ on $\mathcal{L}$, and multiplying these classes by $(-1)^{|U_K(\Gamma)|}$ in $\mathcal{A}(\mathcal{L})$, simultaneously.

□

In other words, we can forget the orientation of closed components and components going from bottom to top or from top to bottom, and regard $\mathcal{Z}^f(\mathcal{L})$ as valued in spaces of diagrams where the domains of these components are not oriented, as in Definitions 6.13 and 6.16. However, we a priori need a $J_{bb,tt}$-orientation as Lemma 16.47 below indicates.

**Lemma 16.47.** Let $A$ be a set of cardinality 3k. Recall Definition 16.6. The form $\eta(A, S_{WE} \cup (-S_{EW}))$ is zero when $k$ is odd. If $k$ is even, then we have

$$ \eta(A, S_{WE} \cup (-S_{EW})) = -s_* (\eta(A, S_{WE} \cup (-S_{EW})))^7 $$

If $\eta(A, S_{WE} \cup (-S_{EW}))$ is zero, then $\mathcal{Z}^f(\dots, (\tilde{\omega}(i, 1, S^2))_{i \in \mathbb{N}})$ is independent of the $J_{bb,tt}$-orientation of the tangles (as in Proposition 10.23).

**PROOF:** The circle $S_{WE} \cup (-S_{EW})$ is the great circle $\partial D$ of $S^2$ that is the boundary of the hemisphere $D$ of $S^2$ centered at $(-i)$. By Definition 16.6, we have $\eta(A, \partial D)(t) = \int_{\{t\} \times \partial D} \omega(A)$ with

$$ \int_{[0,t] \times \partial D} \omega(A) = \int_{(\partial[0,t]) \times D} \omega(A) = \int_{\{t\} \times D} \omega(A) - \int_{\{0\} \times D} \omega(A) $$

and

$$ \int_{\{t\} \times S^2} \omega(A) = \int_{\{t\} \times D} (\omega(A) - \iota^*(\omega(A))) = (1 + (-1)^{k+1}) \int_{\{t\} \times D} \omega(A) $$

according to Lemma 16.5. In particular, when $k$ is odd, the integral

$$ \int_{\{t\} \times D} \omega(A) = \frac{1}{2} \int_{\{t\} \times S^2} \omega(A) $$

<footnote n="7">⁷In particular, $\eta(A, S_{WE} \cup (-S_{EW}))$ is also zero when $k$ is even, if $s_*$ is the identity map. But this is unknown to me.</footnote>
is independent of $t$ and $\eta(A, \partial D)$ is zero. When $k$ is even, Lemma 16.5 implies $\omega(A) = -s_*(\omega(A))$.

Changing the orientation of a component $K$ of $J_{bb}$ amounts to construct the invariant by imposing the condition $p_\tau(U^+K) \subseteq S_{EW}$ rather than $p_\tau(U^+K) \subseteq S_{WE}$. So this replaces the factor $\widetilde{\mathrm{hol}}_{[0,1]}(-\eta(\cdot, S_{WE}))$ associated to $K$ with $\widetilde{\mathrm{hol}}_{[0,1]}(-\eta(\cdot, S_{EW}))$ in the formula of Theorem 16.45. See Theorem 16.16 and Definition 16.17. This amounts to multiply by $\widetilde{\mathrm{hol}}_{[0,1]}(\eta(\cdot, S_{WE} \cup -S_{EW}))$ on the component of $K$. $\square$

**Remarks 16.48.** Similarly, if we had imposed that $p_\tau$ maps the unit tangent vectors to components of $J_{bb} \cup J_{tt}$ to the vertical half great circle $S(\theta)$ from $-\vec{N}$ to $\vec{N}$ that contains the complex direction $\exp(2i\pi\theta)$, for $\theta \in ]0, 1[$, in our definition of straight tangles in Section 16.3, then $S(\theta)$ would replace $S_{WE}$ in the formula of Theorem 16.45, and $\mathcal{Z}^f(T, \cdot, (\tilde{\omega}(i, 1, S^2))_{i \in \overline{3N}})$ would have been multiplied by

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

In particular, when $T$ is a tangle with only one component going from bottom to bottom, $\mathcal{Z}^f(T, \cdot, (\tilde{\omega}(i, 1, S^2))_{i \in \overline{3N}})$ would have been multiplied by $\widetilde{\mathrm{hol}}_{[0,1]}(\eta(\cdot, S_{WE} \cup (-S(\theta)))$.

With the notation of Definition 16.6 and Lemma 16.5,

$$ \widetilde{\mathrm{hol}}_{[0,1]} \eta(\underline{3}, S_{WE} \cup (-S(\theta))) = \int_{[0,1] \times (S_{WE} \cup (-S(\theta)))} \omega(\underline{3}) \\ = \frac{1}{6} \sum_{i=1}^{3} \int_{[0,1] \times (S_{WE} \cup (-S(\theta)))} (\tilde{\omega}(i, S^2) - \iota^* \tilde{\omega}(i, S^2)) [\hat{\zeta}] $$

would have been added to $\mathcal{Z}^f(T, \underline{3}, (\tilde{\omega}(i, 1, S^2))_{i \in \overline{3}})$. For a chain $D$ of $S^2$ bounded by $(S_{WE} \cup (-S(\theta)))$, we have

$$ \int_{[0,1] \times (S_{WE} \cup (-S(\theta)))} \tilde{\omega}(i, S^2) = \int_{(\partial[0,1]) \times D} \tilde{\omega}(i, S^2). $$

In particular, if $\theta$ is in $]1/2, 1[$, then we can choose $D$ and $\tilde{\omega}(i, 1, S^2)$ so that $(\tilde{\omega}(i, 1, S^2) - \iota^* \tilde{\omega}(i, 1, S^2))$ is supported outside $D \cup \iota S^2(D)$ for any $i$. In this
case, we get

$$ \widetilde{\mathrm{hol}}_{[0,1]}\eta(\underline{3}, S_{WE} \cup (-S(\theta))) = - \int_{\{0\} \times D} \omega_{S^2} = \theta - 1. $$

Thus, as claimed in Remark 16.18, Definition 16.17 is not canonical.

The above calculation does not rule out the alternative choice of the vertical half great circle $S(\frac{1}{2}) = S_{EW}$ for our definition. This choice would multiply $\mathcal{Z}^f(T, \cdot, (\tilde{\omega}(i, 1, S^2))_{i \in \underline{3N}})$ by $\widetilde{\mathrm{hol}}_{[0,1]}(\eta(\cdot, S_{WE} \cup (-S_{EW})))$, which is zero in degree 1, according to Lemma 16.47.

It might be tempting to modify the definition of $\mathcal{Z}^f(T, \cdot, (\tilde{\omega}(i, 1, S^2))_{i \in \underline{3N}})$, by multiplying it by

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

Unfortunately, $\widetilde{\mathrm{hol}}_{[0,1]}(\eta(\cdot, S_{WE}))$ depends on the closed 2-forms $\tilde{\omega}(i, S^2)$ of $[0, 1] \times S^2$, and not only on the $\tilde{\omega}(i, 1, S^2)$. Indeed, assume that all $\tilde{\omega}(i, S^2)$ coincide with each other and change all of them by adding $d\eta_S$, for a one-form $\eta_S$ of $[0, 1] \times S^2$ supported on the product of $[1/4, 3/4]$ by a small neighborhood of $\vec{N}$. Then the variation of the degree one part of $2\widetilde{\mathrm{hol}}_{[0,1]}(\eta(\cdot, S_{WE}))$ maps $\underline{3}$ to $\int_{[0,1] \times S_{WE}} (d\eta_S - \iota^*(d\eta_S))$, with

$$ \int_{[0,1] \times S_{WE}} d\eta_S = \int_{\partial([0,1] \times S_{WE})} \eta_S = - \int_{[0,1] \times \{\vec{N}\}} \eta_S $$

and

$$ \int_{[0,1] \times S_{WE}} -\iota^*(d\eta_S) = - \int_{[0,1] \times \iota_{S^2}(S_{WE})} d\eta_S = \int_{\partial([0,1] \times -\iota_{S^2}(S_{WE}))} \eta_S \\ = - \int_{[0,1] \times \{\vec{N}\}} \eta_S. $$

(In Theorem 16.9, the factors $\widetilde{\mathrm{hol}}_{[t,0] \times y^-}(\eta_{B^-, \cdot})$ and $\widetilde{\mathrm{hol}}_{[0,t] \times y^+}(\eta_{B^+, \cdot})$ also depend on $\tilde{\omega}(i, S^2)$. However, both types of dependences cancel each other.)
<figure><img src="image_134.png" /></figure>
Chapter 17

Justifying the properties of Z $^f$

Recall Definition 13.10 of the invariant Z $^f$. So far we have succeeded in constructing this invariant Z $^f$ of q-tangles, invariant under boundary-fixing diffeomorphisms, generalizing both the invariant Z $^f$ for framed links in Q-spheres and the Poirier functor Z l for q-tangles in R $^{3}$. The framing depen-dence of Theorem 13.12 comes from Definition 12.12.

The behavior of Z and Z f under orientation changes of the components
described in the statement of Theorem 13.12 can be justified as in the case
of links in rational homology spheres treated in Section 10.6.

**Lemma 17.1.** The invariant $\mathcal{Z}^f$ behaves as prescribed by Theorem 13.12 under the diffeomorphisms $s_{\frac{1}{2}}$ and $\rho$.

PROOF: Let $L$ be a tangle representative as in Theorem 12.7. We proceed as in the proof of Proposition 10.1, except that we need to take care of the facts that, for $\psi = s_{\frac{1}{2}}$ or $\psi = \rho$, $\psi_*(\tau)$ is not asymptotically standard, and that $s_{\frac{1}{2}}$ reverses the orientation. Therefore, we use $\tau' = \psi_*(\tau) \circ (\mathbf{1}_{\tilde{R}(C)} \times \psi_{\mathbb{R}^3}^{-1}) = T\psi \circ \tau \circ (\psi^{-1} \times \psi_{\mathbb{R}^3}^{-1})$, as an asymptotically standard parallelization of $R = R(C)$. So we have $(\rho_*^{-1})^*p_\tau^*(\omega_{S^2}) = p_{\tau'}^*(\omega_{S^2})$ and $(\sigma_{\frac{1}{2}*}^{-1})^*p_\tau^*(\omega_{S^2}) = -p_{\tau'}^*(\omega_{S^2})$. If $\omega$ is a homogeneous propagating form of $(C_2(R), \tau)$, then $(\rho_*^{-1})^*(\omega)$ is a homogeneous propagating form of $(C_2(\rho(R)), \tau')$ and $(-\sigma_{\frac{1}{2}*}^{-1})^*(\omega)$ is a homogeneous propagating form of $(C_2(s_{\frac{1}{2}}(R)), \tau')$. Let us now focus on the case $\psi = s_{\frac{1}{2}}$, since the case $\psi = \rho$ is similar, but simpler. For any Jacobi diagram $\Gamma$ on the domain of $L$, equipped with an implicit orientation $o(\Gamma)$, compute
$I = I(s_{\frac{1}{2}}(R), s_{\frac{1}{2}}(L), \Gamma, (-\sigma_{\frac{1}{2}*}^{-1})^*(\omega))$. We have

$$
\begin{align*}
I &= \int_{\tilde{C}(s_{\frac{1}{2}}(R), s_{\frac{1}{2}}(L); \Gamma)} \bigwedge_{e \in E(\Gamma)} p_e^* \left( -\left(\sigma_{\frac{1}{2}*}^{-1}\right)^*(\omega) \right) \\
&= (-1)^{|E(\Gamma)|} \int_{\tilde{C}(s_{\frac{1}{2}}(R), s_{\frac{1}{2}}(L); \Gamma)} \left(\sigma_{\frac{1}{2}*}^{-1}\right)^* \left( \bigwedge_{e \in E(\Gamma)} p_e^*(\omega) \right) \\
&= (-1)^{|E(\Gamma)| + |T(\Gamma)|} I(R, L, \Gamma, \omega).
\end{align*}
$$
Thus, we get $Z_n(s_{1/2}(\mathcal{C}), s_{1/2}(L), \tau') = (-1)^n Z_n(\mathcal{C}, L, \tau)$ for all $n \in \mathbb{N}$. In particular, Corollary 4.8 and Propositions 5.15 and 7.17 imply $p_1(\tau') = -p_1(\tau)$ and $I_\theta(K_j, \tau') = -I_\theta(K_j, \tau)$ for any component $K_j$ of $L$. So we have $Z_n(s_{1/2}(\mathcal{C}), s_{1/2}(L)) = (-1)^n Z_n(\mathcal{C}, L)$ for all $n \in \mathbb{N}$ since the anomalies $\alpha$ and $\beta$ vanish in even degrees, thanks to Propositions 10.7 and 10.13. Now, recall Proposition 16.13, and note that if a component $K$ is straight with respect to $\tau$, then $s_{1/2}(K)$ is straight with respect to $\tau'$. In particular, the condition

$$lk_{s_{1/2}(\mathcal{C})} (s_{1/2}(K), s_{1/2}(K)_{\parallel}) = -lk_{\mathcal{C}}(K, K_{\parallel})$$

is realizable and natural, and we get the desired equality for a framed tangle from an injective bottom configuration to an injective top configuration. Thanks to Remark 13.11, it is still true for a $q$-tangle. $\square$

We are thus left with the proofs of the functoriality, the duplication properties, and the cabling property to finish the proof of Theorem 13.12. These proofs will occupy four sections of this chapter, which will end with a section describing other properties of $\mathcal{Z}^f$. The corresponding properties of variants of $\mathcal{Z}$ and $\mathcal{Z}^f$ involving nonhomogeneous propagating forms will be treated simultaneously since they are often easier to prove, and since we are going to use them to prove some of the results for homogeneous propagating forms.

## 17.1 Transversality and rationality

In this section, we generalize the rationality results of Chapter 11 to the tangle case. The generalization will be useful in the proofs of the properties later.

Let $S_H^2$ denote the subset of $S^2$ consisting of the vectors whose vertical coordinate is in $] -\frac{1}{2}, \frac{1}{2}[$.

**Proposition 17.2.** Let $(\mathcal{C}, \tau)$ be a parallelized rational homology cylinder. Let $L: \mathcal{L} \hookrightarrow \mathcal{C}$ be a long tangle of $\tilde{R}(\mathcal{C})$. Let $N$ be an integer greater than 1. Then there exist $(X_1, X_2, \dots, X_{3N}) \in (S_H^2)^{3N}$, $M \in [1, +\infty[$, and propagating chains $P(i)$ of $(C_2(\tilde{R}(\mathcal{C})), \tau)$ for $i \in \underline{3N}$ such that

* $P(i)$ intersects the domain $D(p_\tau)$ of Notation 16.14 as $p_\tau^{-1}(X_i)$,

* the $P(i) \cap C_2(R_{M,[-M,M]}(\mathcal{C}))$ are in general 3N-position with respect to $L$, with the natural generalization of the notion of Definition 11.3 (where $R_{M,[-M,M]}(\mathcal{C})$ replaces $R$, with Notation 16.14),
• the intersections

$$I_S(\Gamma, (P(i))_{i \in \mathcal{3N}}) = \bigcap_{e \in E(\Gamma)} p_e^{-1}(P(j_E(e)))$$

in $C(R(\mathcal{C}), L; \Gamma)$ are transverse and located in $C(R_{M,[-M,M]}(\mathcal{C}), L; \Gamma)$ for any $\Gamma \in \mathcal{D}_{\mathcal{3N}}^e(\mathcal{L}) = \cup_{k \in \mathbb{N}} \mathcal{D}_{k,\mathcal{3N}}^e(\mathcal{L}) = \cup_{k=0}^{3N} \mathcal{D}_{k,\mathcal{3N}}^e(\mathcal{L})$,

• for any $\alpha > 0$, there exists $\beta > 0$ such that

$$\bigcap_{e \in E(\Gamma)} p_e^{-1}\left(N_{\beta}\left(P(j_E(e))\right)\right) \subset N_{\alpha}\left(I_S\left(\Gamma, (P(i))_{i \in \mathcal{3N}}\right)\right)$$

for any $\Gamma \in \mathcal{D}_{\mathcal{3N}}^e(\mathcal{L})$, with the notation of Definition 11.6,

• there exists an open ball $B_X$ around $(X_1, X_2, \dots, X_{3N})$ in $(S^2)^{3N}$ such that $B_X \subset (S_H^2)^{3N}$ and for any $(Y_1, Y_2, \dots, Y_{3N}) \in B_X$, there exist propagating chains $P(i)(Y_i)$ of $(C_2(\tilde{R}(\mathcal{C})), \tau)$ satisfying all the above conditions with respect to $Y_i$ with the same $M$.

The set of $(X_1, X_2, \dots, X_{3N}) \in (S_H^2)^{3N}$ such that there exist

$$M = M(X_1, X_2, \dots, X_{3N}) \in [1, +\infty[$

and propagating chains $P(i)$ of $(C_2(\tilde{R}(\mathcal{C})), \tau)$ satisfying the above conditions is dense in $(S_H^2)^{3N}$.

In order to prove the proposition, we begin by producing some

$$(W_1, W_2, \dots, W_{3N}) \in (S_H^2)^{3N}$$

(in a given neighborhood of some $(W_1^0, W_2^0, \dots, W_{3N}^0)$ in $(S_H^2)^{3N}$) and

$$M(W_1, W_2, \dots, W_{3N}) \in [1, +\infty[$$

For a one-manifold $\mathcal{L}$ and a finite set $A$, the set of connected $A$-numbered degree $k$ Jacobi diagrams with support $\mathcal{L}$ without looped edges is denoted by $\mathcal{D}_{k,A}^e(\mathcal{L})$.

**Lemma 17.3.** Let $N$ be an integer greater than 1. Let $B$ be a finite set. Let $y: B \hookrightarrow D_1$ be a planar configuration. For a $\mathcal{3N}$-numbered Jacobi diagram $\Gamma$ on $\sqcup_{b \in B} \mathbb{R}_b$, define the semi-algebraic map

$$g(\Gamma): C(S^3, y(B) \times \mathbb{R}; \Gamma) \times (S^2)^{\mathcal{3N} \setminus j_E(E(\Gamma))} \to (S^2)^{\mathcal{3N}}$$
to be the product $(\prod_{e \in E(\Gamma)} p_{e, S^2}) \times \mathbf{1}((S^2)^{3N \setminus j_E(E(\Gamma))})$.

The subset $\mathcal{O}(N, y)$ of $(S_H^2)^{3N}$ of points that are in the complement of the images of the maps $g(\Gamma)$ for all $\Gamma \in \cup_{k=1}^{3N} \mathcal{D}_{k, \underline{3N}}^c (\cup_{b \in B} \mathbb{R}_b)$ is dense and open.<fnref n="1" />

**PROOF:** It suffices to prove that the complement of the image of the map $g(\Gamma)$ is open and dense for any of the finitely many graphs $\Gamma \in \cup_{k=1}^{3N} \mathcal{D}_{k, \underline{3N}}^c(\mathbb{R})$. The dimension of $C(S^3, y(B) \times \mathbb{R}; \Gamma)$ is the same as the dimension of $(S^2)^{j_E(E(\Gamma))}$. The quotient of $\tilde{C}(S^3, y(B) \times \mathbb{R}; \Gamma)$ by global vertical translations is a semi-algebraic set with dimension one less. Thus, the image of $g(\Gamma)$ is a compact semi-algebraic subset of $(S_H^2)^{3N}$ of codimension at least one. Its complement is thus an open dense semi-algebraic subset of $(S_H^2)^{3N}$. □

Note the following easy lemma.

**Lemma 17.4.** Let $a$ and $h$ denote two vectors of $\mathbb{R}^n$ such that $a$ and $a+h$ are different from 0. Then we have

$$ \left\| \frac{1}{\|a+h\|} (a+h) - \frac{1}{\|a\|} a \right\| \le 2 \frac{\|h\|}{\|a\|}. $$

**PROOF:** The left-hand side can be written as

$$ \left\| \frac{1}{\|a\|} h + \left( \frac{1}{\|a+h\|} - \frac{1}{\|a\|} \right) (a+h) \right\|. $$

So it is less than or equal to

$$ \left( \frac{\|h\|}{\|a\|} + \frac{\|\|a\| - \|a+h\|\|}{\|a\|} \right). $$

□

**Lemma 17.5.** Under the assumptions of Lemma 17.3, assume that the image of the configuration $y: B \to D_1$ contains 0. Equip $(S_H^2)^{3N}$ with the distance coming from the Euclidean norm of $(\mathbb{R}^3)^{3N}$. Let $\varepsilon \in ]0, \frac{1}{20^{12N}}[$ be such that the ball $B(W_1, W_2, \dots, W_{3N})$ centered at $(W_1, W_2, \dots, W_{3N})$ of radius $24N\varepsilon^{\frac{1}{12N}}$ of $(S_H^2)^{3N}$ sits in the subset $\mathcal{O}(N, y)$ of Lemma 17.3.

Let $L: \mathcal{L} \hookrightarrow \mathcal{C}$ be a long tangle of $\tilde{R}(\mathcal{C})$ whose bottom and top planar configurations are subconfigurations (i.e., restrictions) of $y$.

Let $D(W_i, \varepsilon)$ be the disk of radius $\varepsilon$ centered at $W_i$ in $S^2$. For $i \in \underline{3N}$ and $Y_i \in D(W_i, \varepsilon)$, let $P(Y_i)$ be a propagating chain of $(C_2(R(\mathcal{C})), \tau)$ that

<footnote n="1">¹Though we are going to study $\mathcal{Z}^f$ up to degree $N$, higher degree diagrams will occur in our proofs. See the proof of Lemma 17.5.</footnote>
coincides with $p_{\tau}^{-1}(Y_i)$ on the domain $D(p_{\tau})$ of Notation 16.14. Then for any
$\Gamma \in \bigcup_{k=1}^{3N} \mathcal{D}_{k,3N}^{c}(\mathcal{L})$, we have

$$
\bigcap_{e \in E(\Gamma)} p_e^{-1} \left( \bigcup_{Y_{j_E(e)} \in B(W_{j_E(e)}, \epsilon)} P(Y_{j_E(e)}) \right) \subset C \left( R_{\frac{1}{\sqrt{\epsilon}}, [-\frac{1}{\sqrt{\epsilon}}, \frac{1}{\sqrt{\epsilon}}]} (\mathcal{C}), L; \Gamma \right).
$$

For $i \in \underline{3N}$, let $\omega_{i,S^2}$ be a two-form of $S^2$ supported in $D(W_i, \varepsilon)$. Then for any family $(\omega_i)_{i \in \underline{3N}}$ of closed propagating forms $\omega_i$ of $(C_2(R(\mathcal{C})), \tau)$ restricting to $D(p_\tau)$ as $p_\tau^*(\omega_{i,S^2})$, the support of $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e)))$ is included in $C(R_{\frac{1}{\sqrt{\varepsilon}}, [-\frac{1}{\sqrt{\varepsilon}}, \frac{1}{\sqrt{\varepsilon}}]}(\mathcal{C}), L; \Gamma)$.

PROOF: Fix a connected 3N-numbered Jacobi diagram Γ on the domain L of L. Note that Γ has at most 6N vertices. Define a sequence α₁, ..., α₆N by

$$
\alpha_k = \varepsilon^{\frac{1-k}{12N}}.
$$

Since $\varepsilon$ is smaller than $\frac{1}{20^{12N}}$, we have $\varepsilon^{-\frac{1}{12N}} > 20$ and $\alpha_2 > 20$.

Define an open covering of $\check{C}(\check{R}(\mathcal{C}), L; \Gamma)$ associated to colorings of the
vertices of $\Gamma$ by colors blue and $k$, for $k \in \underline{6N}$, such that

- vertices of color 1 go to (or belong to) $\check{R}_{3,[-2,3]}(\mathcal{C})$,

- blue vertices and vertices of color $k \ge 2$ do not go to $R_{2,[-1,2]}(\mathcal{C})$, with Notation 16.14,

- any vertex of color 2 is connected to a vertex of color 1 by an edge of
  Γ,

- any vertex of color 2 is at a distance smaller than $5\alpha_2$ from $(0,0) \in \mathbb{R}^3 = \mathbb{C} \times \mathbb{R}$ (with respect to the Euclidean norm of $\mathbb{R}^3$),

- for $k$ such that $2 \le k \le 6N - 1$, any vertex of color $(k+1)$ is connected to a vertex of color $k$ by an edge of length smaller than $5\alpha_{k+1}$,<fnref n="2" />

- when there is an edge of $\Gamma$ between a blue vertex and a vertex colored by 1, the distance between the blue vertex and (0,0) is greater than $3\alpha_2$ (with respect to the Euclidean distance of $\mathbb{R}^3$),

- when there is an edge of $\Gamma$ between a blue vertex and a vertex colored by $k$ for $2 \le k \le 6N - 1$, the distance between the two vertices is greater than $3\alpha_{k+1}$. (Since $\Gamma$ has at most $6N$ vertices, if there is a blue vertex, then no vertex can be colored by $6N$.)

<footnote n="2">²This edge length makes sense since vertices of color $k \ge 2$ belong to $R_{2,[-1,2]}^c(\mathcal{C}) \subset \mathbb{R}^3$.</footnote>
The subset $U(\mathbf{c})$ of $\check{C}(\check{R}(\mathcal{C}), L; \Gamma)$, consisting of the configurations that satisfy the above conditions with respect to a coloring $\mathbf{c}$ of the vertices is open. Let us prove that $\check{C}(\check{R}(\mathcal{C}), L; \Gamma)$ is covered by these sets. For a configuration $\mathbf{c}$, color by 1 its vertices that are in $\check{R}_{3,[-2,3]}(\mathcal{C})$. Then color by 2 all the still uncolored vertices $v$ connected to a vertex of color 1 (by an edge of $\Gamma$) such that $d(v, (0, 0)) < 5\alpha_2$. Continue by coloring all the possible uncolored vertices connected to a vertex of color 2 by an edge of length smaller than $5\alpha_3$ by 3, and so on, in order to end up with a coloring, which obviously satisfies the above conditions, by coloring the uncolored vertices blue.

Note that the distance between a vertex colored by $k \ge 2$ and the point $(0, 0) \in \mathbb{R}^3$ is less than

$$ \left( 5 \sum_{i=2}^{k} \alpha_i = 5 \frac{\alpha_{k+1} - \alpha_2}{\varepsilon^{-\frac{1}{12N}} - 1} \right) < 5 \frac{\alpha_{k+1}}{\frac{5}{6}\varepsilon^{-\frac{1}{12N}}} \le 6\alpha_k \le 6\alpha_{6N}, $$

where since $\varepsilon^{-\frac{1}{12N}}$ is greater than 20, we have

$$ (6\alpha_{6N} = 6(\varepsilon^{-\frac{1}{12N}})^{6N-1}) \le \left( (\varepsilon^{-\frac{1}{12N}})^{6N} = \frac{1}{\sqrt{\varepsilon}} \right). $$

Thus, the vertices colored by some $k$ are in $R_{\frac{1}{\sqrt{\varepsilon}}, [-\frac{1}{\sqrt{\varepsilon}}, \frac{1}{\sqrt{\varepsilon}}]}(\mathcal{C})$.

Let us prove that an open set $U(\mathbf{c})$ associated to a coloring $\mathbf{c}$ for which the color blue appears cannot intersect $\bigcap_{e \in E(\Gamma)} p_e^{-1}(\cup_{Y_{jE(e)} \in B(W_{jE(e)}, \varepsilon)} P(Y_{jE(e)}))$. Fix such a coloring $\mathbf{c}$. Remove from $\Gamma$ all the edges that do not contain a blue vertex (without removing their ends). Let $\Gamma_b$ be a connected component with at least one blue vertex of the obtained graph. It has blue vertices, which are trivalent or univalent (in $\Gamma$ and $\Gamma_b$). The blue univalent vertices go to $y \times \mathbb{R}$. Color its other vertices colored by some $k$ red. Red vertices may have 1, 2, or 3 adjacent edges in $\Gamma_b$. Let $\Gamma'_b$ be the uni-trivalent graph obtained by blowing up $\Gamma_b$ at its red vertices by replacing such a vertex by a red univalent vertex for each adjacent edge. Color the edges between blue vertices blue, and the edges between a blue vertex and a red one purple. To a configuration of $U(\mathbf{c})$ in $\bigcap_{e \in E(\Gamma)} p_e^{-1}(\cup_{Y_{jE(e)} \in B(W_{jE(e)}, \varepsilon)} P(Y_{jE(e)}))$ associate the configuration of $\Gamma'_b$ obtained by sending all the red vertices to $\mathbf{o} = (0, 0)$, leaving the positions of the blue vertices unchanged. Thus,

* the direction of a blue edge numbered by $i$ is at a distance less than $\varepsilon$ from $W_i$,

* the direction of a purple edge numbered by $p$ is at a distance less than $(\varepsilon + 4\varepsilon^{\frac{1}{12N}})$ from $W_p$.
Let us justify the second assertion. Let **b** denote the blue vertex of the purple edge numbered by *p*, and let **r** be its red vertex in the configured graph Γ. Assume that **r** is colored by *k* with *k* ≥ 2. Then we have

$$d(\mathbf{b}, \mathbf{r}) > 3\alpha_{k+1}, \quad (\mathbf{o}, \mathbf{b}) \in D(p_{\tau}), \quad (\mathbf{r}, \mathbf{b}) \in D(p_{\tau}), \quad \text{and} \quad d(\mathbf{o}, \mathbf{r}) < 6\alpha_k.$$

Since the configuration consisting of **b** and **r** in Γ is in some $P(Y_p)$, we have

$$\left| \pm \frac{1}{\|\vec{rb}\|} \vec{rb} - W_p \right| \le \varepsilon.$$

Apply Lemma 17.4 with $a = \vec{rb}$ and $h = \vec{or}$ to obtain

$$\left\| \frac{1}{\|\vec{ob}\|} \vec{ob} - \frac{1}{\|\vec{rb}\|} \vec{rb} \right\| \le \frac{2\|\vec{or}\|}{\|\vec{rb}\|} < \frac{4\alpha_k}{\alpha_{k+1}} \le 4\varepsilon^{\frac{1}{12N}}.$$

This proves the second assertion when **r** is colored by *k* ≥ 2.

Assume that **r** is colored by 1. Then we have $d(\mathbf{b}, \mathbf{o}) > 3\alpha_2$. When the configuration consisting of **b** and **r** in Γ is in some $P(Y_p)$, there exists an **s** in $D_3 \times [-2, 3]$ such that the direction of $\pm \vec{sb}$ is at a distance less than $\varepsilon$ from $W_p$. Here, we have $d(\mathbf{o}, \mathbf{s}) < 5\alpha_1$. So the direction of $\pm \vec{ob}$ is still at a distance less than $(\varepsilon + 4\varepsilon^{\frac{1}{12N}})$ from $W_p$. Indeed Lemma 17.4 applied with $a = \vec{ob}$ and $h = \vec{so}$ yields

$$\left\| \frac{1}{\|\vec{ob}\|} \vec{ob} - \frac{1}{\|\vec{sb}\|} \vec{sb} \right\| \le \frac{2\|\vec{os}\|}{\|\vec{ob}\|} \le \frac{10\alpha_1}{3\alpha_2} < 4\varepsilon^{\frac{1}{12N}}.$$

Therefore, the directions of the edges numbered by *i* of the configured graph $\Gamma'_b$ are at a distance less than $(\varepsilon + 4\varepsilon^{\frac{1}{12N}} \le 8\varepsilon^{\frac{1}{12N}})$ from the $W_i$. But the directions of these edges cannot be in the image of $\prod_{e \in E(\Gamma'_b)} p_{e,S^2}$ according to our conditions. Indeed, together with $(W_i)_{i \in \overline{3N \setminus j_E(E(\Gamma'_b))}}$, they form a 3N-tuple that is at a distance less than $3N \times (8\varepsilon^{\frac{1}{12N}})$ from $(W_i)_{i \in \overline{3N}}$.

Thus $\bigcap_{e \in E(\Gamma)} p_e^{-1}(\cup_{j \in E(E(e), \varepsilon)} P(Y_{j \in E(E)}))$ does not intersect the open subsets of the coverings that use the blue color.

It is now easy to conclude.

<figure><img src="image_135.png" /></figure>

PROOF OF PROPOSITION 17.2: Fix an $\varepsilon$ as in Lemma 17.5. For a diagram Γ of $\mathcal{D}_{3N}^e(\mathcal{L})$ and a subset E of E(Γ), the map

$$q(\Gamma, E) = \prod_{e \in E} p_{\tau} \circ p_e : C(R_{\frac{1}{\varepsilon}, [-\frac{1}{\varepsilon}, \frac{1}{\varepsilon}]}(\mathcal{C}), L; \Gamma) \cap \bigcap_{e \in E} p_e^{-1}(D(p_{\tau})) \to (S^2)^E$$
has an open dense set of regular values. The product of this set by $(S^2)^{3N\setminus E}$ is also open and dense. So is the intersection $\mathcal{I}_q$ over all such pairs $(\Gamma, E)$. Thus, there exist $(X_1, X_2, \dots, X_{3N})$ in this intersection and $\alpha \in ]0, \varepsilon]$ such that $\prod_{i=1}^{3N} B(X_i, \alpha) \subset \mathcal{I}_q \cap \prod_{i=1}^{3N} B(W_i, \varepsilon)$ for the $B(W_i, \varepsilon)$ of Lemma 17.5. For $(Y_1, Y_2, \dots, Y_{3N}) \in \prod_{i=1}^{3N} B(X_i, \alpha)$, let $P(Y_i)$ be a propagator of $(C_2(\tilde{R}(\mathcal{C})), \tau)$ restricting to $D(p_\tau)$ as $p_\tau^{-1}(Y_i)$, for each $i \in \underline{3N}$, as in Lemma 16.15. Then the $P(Y_i)$ can be put in general $3N$ position as in Section 11.3, by changing them only on $\tilde{C}_2(R_{2,[-1,2]}(\mathcal{C}))$ since they satisfy the general position conditions on the boundaries. Thus, Proposition 17.2 holds with $M = \frac{1}{\sqrt{\varepsilon}}$. For a given $\alpha > 0$, the existence of a $\beta > 0$ such that

$$ \bigcap_{e \in E(\Gamma)} p_e^{-1} \left( N_{\beta} \left( P(j_{E}(e)) \right) \right) \subset N_{\alpha} \left( I_{S} \left( \Gamma, (P(i))_{i \in \underline{3N}} \right) \right) $$

for any $\Gamma \in \mathcal{D}_{\underline{3N}}^e(\mathcal{L})$ can be proved as in the end of the proof of Lemma 11.13. $\square$

<figure><img src="image_136.png" /></figure>

**Corollary 17.6.** For forms $\omega_i$ $\beta$-dual (as in Definition 11.6) to the $P(i)$ of Proposition 17.2, for any subset $A$ of $\underline{3N}$ with cardinality $3k$,

$$ Z(C, L, \tau, A, (\omega_i)_{i \in \underline{3N}}), $$

which is defined in Theorem 16.9, is rational.

**PROOF:** As in Lemma 11.7, the involved configuration space integrals can be computed as algebraic intersections of rational preimages of the $P(i)$. $\square$

## 17.2 Functionality

In this chapter, we prove the functoriality of $\mathcal{Z}^f$, which implies the multiplicativity of $\mathcal{Z}$ under connected sum. A reader only interested in the latter proof can read the proof by replacing the set $B$ of strands by $\{0\}$ and by viewing $\tilde{R}(C_j)$ as an asymptotically standard $\mathbb{R}^3 = \mathbb{C} \times \mathbb{R}$, identified with $\mathbb{C} \times \mathbb{R}$ outside $D_1 \times [0, 1]$, where $D_1$ is the unit disk of $\mathbb{C}$, for $j \in \underline{2}$.

**Proposition 17.7.** Let $N \in \mathbb{N}$. Let $(C_1, L_1, \tau_1)$ and $(C_2, L_2, \tau_2)$ be two composable tangles. There exist volume-one forms $\omega(i, S^2)$ of $S^2$ for $i \in \underline{3N}$ such that

$$ Z(C_1C_2, L_1L_2, \tau_1\tau_2, A, (\omega(i, S^2))_{i \in A}) = \sum_{(A_1, A_2) \in P_2(A)} \frac{|A_1|! |A_2|!}{|A|!} Z(C_1, L_1, \tau_1, A_1, (\omega(i, S^2))) Z(C_2, L_2, \tau_2, A_2, (\omega(i, S^2))) $$
for any subset A of $\underline{3N}$ with cardinality $3n$, with the notation of Theorem 16.9.

**PROOF:** Let $y: B \hookrightarrow D_1$ be a planar configuration whose image contains 0 and the images of the bottom and top configurations of $L_1$ and $L_2$. Let $B(X_1, X_2, \dots, X_{3N})$ be a ball centered at $(X_1, X_2, \dots, X_{3N})$ of radius $24N\epsilon^{1/12N}$ of $(S_H^2)^{3N}$ sitting in the subset $O(N, y)$ of Lemma 17.3, with $\epsilon \in ]0, \frac{1}{20^{12N}}[$ as in Lemma 17.5. For $i \in \underline{3N}$, let $\omega(i, S^2)$ be a volume-one form on $S^2$ supported on a disk $D(X_i, \epsilon)$ of $S_H^2$ of radius $\epsilon$ centered at $X_i$. Define an extension $\omega_1(i)$ of $p_{\tau_1}^*(\omega(i, S^2))$ to $C_2(R(C_1))$ as in Lemma 16.15. Let $m_\epsilon$ denote the multiplication by $\epsilon$ in $\mathbb{R}^3$. Let $m_{1,\epsilon}$ be a diffeomorphism from $\check{R}(C_1)$ to a manifold denoted by $\check{R}(\epsilon C_1)$ such that $m_{1,\epsilon}$ coincides with $m_\epsilon$ on $\mathbb{R}^3 \setminus \text{Int}(D_1) \times ]0, 1[$. Call $(\epsilon C_1, \epsilon L_1)$ the intersection of $m_{1,\epsilon}((C_1, L_1))$ with the part that replaces $D_1 \times [0, 1]$ in $\check{R}(\epsilon C_1)$. Note that $\check{R}(\epsilon C_1)$ is standard outside $D_\epsilon \times [0, \epsilon]$. Use forms $(m_{1,\epsilon}^{-1})^*(\omega_1(i))$ for $(\epsilon C_1, \epsilon L_1)$. So we have

$$Z(\epsilon C_1, \epsilon L_1, m_{1,\epsilon*}(\tau_1), A_1, (\omega(i, S^2))) = Z(C_1, L_1, \tau_1, A_1, (\omega(i, S^2)))$$

for any subset $A_1$ of $\underline{3N}$ with cardinality multiple of 3, with $m_{1,\epsilon*}(\tau_1) = Tm_{1,\epsilon} \circ \tau_1 \circ (m_{1,\epsilon}^{-1} \times m_\epsilon^{-1})$. Define $\omega_2(i)$ on $C_2(\check{R}(C_2))$, so that $\omega_2(i)$ extends $p_{\tau_2}^*(\omega(i, S^2))$, as in Lemma 16.15. Let $T_{2,\epsilon}$ be the vertical translation of $\mathbb{R}^3$ by $(0, 0, 1 - \epsilon)$. Let $m_{2,\epsilon}$ be a diffeomorphism from $\check{R}(C_2)$ to a manifold denoted by $\check{R}(\epsilon C_2)$ such that $m_{2,\epsilon}$ coincides with the composition $T_{2,\epsilon} \circ m_\epsilon$ on $\mathbb{R}^3 \setminus \text{Int}(D_1) \times ]0, 1[$. Call $(\epsilon C_2, \epsilon L_2)$ the intersection of $m_{2,\epsilon}((C_2, L_2))$ with the part that replaces $D_1 \times [0, 1]$ in $\check{R}(\epsilon C_2)$. Note that $\check{R}(\epsilon C_2)$ is standard outside $D_\epsilon \times [1 - \epsilon, 1]$. Use forms $(m_{2,\epsilon}^{-1})^*(\omega_2(i))$ for $(\epsilon C_2, \epsilon L_2)$. So we have

$$Z(\epsilon C_2, \epsilon L_2, m_{2,\epsilon*}(\tau_2), A_2, (\omega(i, S^2))) = Z(C_2, L_2, \tau_2, A_2, (\omega(i, S^2)))$$

for any subset $A_2$ of $\underline{3N}$ with cardinality multiple of 3, where $m_{2,\epsilon*}(\tau_2) = Tm_{2,\epsilon} \circ \tau_2 \circ (m_{2,\epsilon}^{-1} \times m_\epsilon^{-1})$.

Now, let $(\epsilon C_1 C_2, \epsilon L_1 L_2)$ be obtained from $(\epsilon C_1, \epsilon L_1)$ by inserting

$$(\epsilon C_2, \epsilon L_2) \cap R_{\epsilon, [1-\epsilon, 1]}(\epsilon C_2)$$

instead of $D_\epsilon \times [1 - \epsilon, 1]$. Here and below, we use a natural extension of the $R_{\cdot, \cdot}$ notation introduced in 16.14. Define the propagator $\omega(i)$ of $(R(\epsilon C_1 C_2), \tau_1 \tau_2)$,

* to coincide with $(m_{1,\epsilon}^{-1})^*(\omega_1(i))$ on $C_2(R(\epsilon C_1 C_2) \setminus R_{2\epsilon, [1-2\epsilon, 1+\epsilon]}(\epsilon C_1 C_2))$,

* to coincide with $(m_{2,\epsilon}^{-1})^*(\omega_2(i))$ on $C_2(R(\epsilon C_1 C_2) \setminus R_{2\epsilon, [-\epsilon, 2\epsilon]}(\epsilon C_1 C_2))$, and
• to be zero on $R_{.1,[-.1,.1]}(\varepsilon C_1 C_2) \times R_{.1,[.9,.1]}(\varepsilon C_1 C_2)$ and

$$R_{.1,[.9,.1]}(\varepsilon C_1 C_2) \times R_{.1,[-.1,.1]}(\varepsilon C_1 C_2).$$

This definition is consistent because the form $\omega(i, S^2)$ is supported in $S_H^2$. Then compute $Z(C_1C_2, L_1L_2, \tau_1\tau_2, A, (\omega(i, S^2))_{i \in A})$ as

$$Z(\varepsilon C_1 C_2, \varepsilon L_1 L_2, \tau_1 \tau_2, A, (\omega(i, S^2))_{i \in A})$$

with these propagators $\omega(i)$. We are going to prove the following lemma under the above hypotheses.

**Lemma 17.8.** For any $3N$-numbered Jacobi diagram $\Gamma_1$ on the domain of $L_1$ of degree at most $N$, the form on $C(R(\varepsilon C_1), \varepsilon L_1; \Gamma_1)$

$$\bigwedge_{e \in E(\Gamma_1)} (p_e m_{1,\varepsilon}^{-1})^* (\omega_1(j_E(e)))$$

is supported on $C_{V(\Gamma_1)}(R_{.1,[-.1,.1]}(\varepsilon C_1)) \cap C(R(\varepsilon C_1), \varepsilon L_1; \Gamma_1)$.

For any $3N$-numbered Jacobi diagram $\Gamma_2$ on the domain of $L_2$ of degree at most $N$, the form $\bigwedge_{e \in E(\Gamma_2)} (p_e m_{2,\varepsilon}^{-1})^* (\omega_2(j_E(e)))$ on $C(R(\varepsilon C_2), \varepsilon L_2; \Gamma_2)$ is supported on $C_{V(\Gamma_2)}(R_{.1,[.9,.1]}(\varepsilon C_2)) \cap C(R(\varepsilon C_2), \varepsilon L_2; \Gamma_2)$.

For any $3N$-numbered Jacobi diagram $\Gamma$ on the domain of $L_1L_2$ of degree at most $N$, the form $\bigwedge_{e \in E(\Gamma)} p_e^* (\omega(j_E(e)))$ on $C(R(\varepsilon C_1C_2), \varepsilon L_1L_2; \Gamma)$ is supported on

$$\cup_{\{V_1,V_2\} \in P_2(\Gamma)} C_{V_1} (R_{.1,[-.1,.1]}(\varepsilon C_1 C_2)) \times C_{V_2} (R_{.1,[.9,.1]}(\varepsilon C_1 C_2)),$$

where $P_2(\Gamma)$ denotes the set of partitions $\{V_1, V_2\}$ of $V(\Gamma)$ into two disjoint subsets $V_1$ and $V_2$ such that no edge of $\Gamma$ has one vertex in $V_1$ and the other in $V_2$.

Assuming Lemma 17.8, we can conclude the proof of Proposition 17.7 as follows. Lemma 17.8 implies that both sides of the equality to be proved are sums over pairs $(\Gamma_1, \Gamma_2)$ of A-numbered diagrams such that $\Gamma_1$ is a diagram on the domain of $L_1$, $\Gamma_2$ is a diagram on the domain of $L_2$, and $j_E(E(\Gamma_1)) \cap j_E(E(\Gamma_2)) = \emptyset$, of terms

$$I(C_1, L_1, \Gamma_1, (\omega_1(i))_{i \in A}) I(C_2, L_2, \Gamma_2, (\omega_2(i))_{i \in A}) [\Gamma_1] [\Gamma_2].$$

Using Lemma 16.2 to identify the coefficients finishes the proof of Proposition 17.7 up to the proof of Lemma 17.8, which follows. $\square$
PROOF OF LEMMA 17.8: The first two assertions follow from Lemma 17.5. Let us focus on the third one. Fix a $\underline{3N}$-numbered Jacobi diagram $\Gamma$ of degree at most $N$ on the domain of $L_1L_2$. For $i \in \underline{6N}$, set $\beta_i = \varepsilon\alpha_i$ with the sequence $\alpha_i = \varepsilon^{\frac{1-i}{12N}}$ of the proof of Lemma 17.5. Define an open covering of $\check{C}(\check{R}(\varepsilon C_1 C_2), \varepsilon C_1 C_2; \Gamma)$ associated to colorings of the vertices by colors blue, (1, k), and (2, k), with $k \in \underline{6N}$, such that

* blue vertices and vertices of color $(j, k)$ with $j \in \underline{2}$ and $k \ge 2$ do not go to $R_{2\varepsilon,[-\varepsilon,2\varepsilon]}(\varepsilon C_1 C_2) \cup R_{2\varepsilon,[1-2\varepsilon,1+\varepsilon]}(\varepsilon C_1 C_2)$,

* vertices of color (1, 1) go to $\check{R}_{3\varepsilon,[-2\varepsilon,3\varepsilon]}(\varepsilon C_1 C_2)$,

* vertices of color (2, 1) go to $\check{R}_{3\varepsilon,[1-3\varepsilon,1+2\varepsilon]}(\varepsilon C_1 C_2)$,

* for $j \in \underline{2}$, any vertex of color $(j, 2)$ is connected by an edge of $\Gamma$ to a vertex of color $(j, 1)$ and is at a distance smaller than $5\beta_2$ from $(0, j-1)$ (with respect to the Euclidean norm of $\mathbb{R}^3$),

* for $j \in \underline{2}$ and $k$ such that $2 \le k \le 6N - 1$, any vertex of color $(j, k+1)$ is connected to a vertex of color $(j, k)$ by an edge of length smaller than $5\beta_{k+1}$,

* when there is an edge of $\Gamma$ between a blue vertex and a vertex colored by $(j, 1)$ for $j \in \underline{2}$, the distance between the blue vertex and $(0, j-1)$ is greater than $3\beta_2$, and

* when there is an edge of $\Gamma$ between a blue vertex and a vertex colored by $(j, k)$ with $j \in \underline{2}$ and $2 \le k \le 6N - 1$, the distance between the two vertices is greater than $3\beta_{k+1}$.

The subset $U(\mathbf{c})$ of $\check{C}(\check{R}(\varepsilon C_1 C_2), \varepsilon C_1 C_2; \Gamma)$ consisting of the configurations that satisfy the above conditions with respect to a coloring $\mathbf{c}$ of the vertices is open, and $\check{C}(\check{R}(\varepsilon C_1 C_2), \varepsilon C_1 C_2; \Gamma)$ is covered by these sets as in the proof of Lemma 17.5. The only additional thing to notice is that a vertex could not be simultaneously colored by $(1, k)$ and by $(2, k')$ since a vertex colored by $(j, k)$ is at a distance less than

$$6\beta_k \le \sqrt{\varepsilon} \le \frac{1}{20^6}$$

from $(0, j-1)$. In particular, the vertices colored by $(1, k)$ are in

$$R_{.1,[-.1,.1]}(\varepsilon C_1 C_2),$$

and the vertices colored by $(2, k)$ are in $R_{.1,[.9,.1]}(\varepsilon C_1 C_2)$.
So the form $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e)))$ vanishes on open sets corresponding to colorings for which a vertex $(1, k)$ is connected to a vertex $(2, k')$ (by some edge of $\Gamma$), according to the conditions before Lemma 17.8.

As in the proof of Lemma 17.5, we prove that our form vanishes on open sets $U(\mathbf{c})$ associated to colorings for which the color blue appears. Fix such a coloring $\mathbf{c}$ and remove from $\Gamma$ all the edges that do not contain a blue vertex. Let $\Gamma_b$ be a connected component of this graph with at least one blue vertex. It has blue vertices, which are trivalent or univalent in $\Gamma$ and $\Gamma_b$. The blue univalent vertices lie on $\varepsilon y \times \mathbb{R}$. The other vertices of $\Gamma_b$ are either colored by some $(1, k)$, in which case we color them yellow, or by some $(2, k)$, in which case we color them red. Red and yellow vertices may have 1, 2, or 3 adjacent edges in $\Gamma_b$. Let $\Gamma'_b$ be the uni-trivalent graph obtained by blowing up $\Gamma_b$ at its yellow and red vertices by replacing such a vertex with a univalent vertex of the same color for each adjacent edge. Color the edges between blue vertices blue, the edges between a blue vertex and a yellow one green, and the edges between a blue vertex and a red one purple. To a configuration of $U(\mathbf{c})$ in the support of $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e)))$, associate the configuration of $\Gamma'_b$ obtained by sending all the yellow vertices to $\mathbf{o}$ and all the red ones to $(0, 1)$, leaving the positions of the blue vertices unchanged. Thus,

* the direction of a blue edge numbered by $i$ is in the support of $\omega(i, S^2)$ at a distance less than $\varepsilon$ from $X_i$,

* the direction of a green edge numbered by $g$ is at a distance less than $(\varepsilon + 4\varepsilon^{1/12N})$ from $X_g$, (as in the proof of Lemma 17.5),

* the direction of a purple edge numbered by $p$ is at a distance less than $(\varepsilon + 4\varepsilon^{1/12N})$ from $X_p$.

However, the directions of the edges of $\Gamma'_b$ cannot be in the image of $\prod_{e \in E(\Gamma)} p_{e,S^2}$ according to our conditions in the beginning of the proof of Proposition 17.7 (the $\varepsilon$ rescaling of $y$ does not change the image). Therefore, the support of $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e)))$ does not intersect the open subsets of the covering that use the blue color. It is now easy to conclude.

□

For an integer $N$, $Z_{\le N}$ denotes the truncation of $Z$ valued in $\mathcal{A}_{\le N}(\mathcal{L}) = \prod_{j=0}^{N} \mathcal{A}_j(\mathcal{L})$.

**Theorem 17.9.** Let $N$ be a natural number, and let $(\omega_{i,S^2})_{i \in \underline{3N}}$ be a family of volume-one forms of $S^2$. For any two composable tangles $(C_1, L_1, \tau_1)$ and
$(C_2, L_2, \tau_2)$ in parallelized rational homology cylinders, we have

$$Z(C_1C_2, L_1L_2, \tau_1\tau_2, .., (\omega_{i,S^2})_{i \in \underline{3N}}) = (Z(C_1, L_1, \tau_1, .., (\omega_{i,S^2}))Z(C_2, L_2, \tau_2, .., (\omega_{i,S^2})))_{\sqcup}$$

with the notation of Theorem 16.9 and Definition 16.10.

For any two composable $J_{bb,tt}$-oriented $q$-tangles $T_1$ and $T_2$, we also have

$$\mathcal{Z}^f(T_1T_2, .., (\omega_{i,S^2})_{i \in \underline{3N}}) = (\mathcal{Z}^f(T_1, .., (\omega_{i,S^2}))\mathcal{Z}^f(T_2, .., (\omega_{i,S^2})))_{\sqcup},$$

with the notation of Definition 16.44, and

$$\mathcal{Z}^f(T_1)\mathcal{Z}^f(T_2) = \mathcal{Z}^f(T_1T_2).$$

PROOF: Let us prove the first assertion. Apply Theorem 16.9, with

$$\tilde{\omega}(i, 1, S^2) = \omega_{i,S^2} \quad \text{and} \quad \tilde{\omega}(i, 0, S^2) = \omega(i, S^2)$$

with the form $\omega(i, S^2)$ of Proposition 17.7. We get

$$Z_{\le N}(C_1C_2, L_1L_2, \tau_1\tau_2, .., (\omega_{i,S^2})) = \left( \left( \prod_{j \in I} \widetilde{\mathrm{hol}}_{[0,1]} (\eta(., p_\tau(U^+K_j))) \right)_{\#j} \right. \\ \left. \qquad \widetilde{\mathrm{hol}}_{[1,0] \times y_1^-} (\eta_{B^-,..}) Z(C_1C_2, L_1L_2, \tau_1\tau_2, .., (\omega(i, S^2))) \widetilde{\mathrm{hol}}_{[0,1] \times y_2^+} (\eta_{B^+,..}) \right)_{\sqcup},$$

where $y_i^-$ (resp. $y_i^+$) represents the bottom (resp. top) configuration of $L_i$, with $y_2^- = y_1^+$ and

$$Z(C_1C_2, L_1L_2, \tau_1\tau_2, .., (\omega(i, S^2))) = (Z(C_1, L_1, \tau_1, .., (\omega(i, S^2)))Z(C_2, L_2, \tau_2, .., (\omega(i, S^2))))_{\sqcup}.$$

A neutral factor $(\widetilde{\mathrm{hol}}_{[0,1] \times y_1^+} (\eta_{B_1^+,..}) \widetilde{\mathrm{hol}}_{[1,0] \times y_2^-} (\eta_{B_2^-,..}))_{\sqcup}$ can be inserted in the middle. So the first equality of the statement becomes clear, up to the behavior of the factors $\widetilde{\mathrm{hol}}_{[0,1]}(\eta(., p_\tau(U^+K_j)))$ of Definition 16.6. For these factors, note that a component $K$ of $L_1L_2$ consists of a bunch of components $K_k$ from $L_1$ and $L_2$, for $k$ in a finite set $E$, and that $\eta(A, p_\tau(U^+K))$ is the sum of the corresponding $\eta(A, p_\tau(U^+K_k))$. Lemma 16.8 ensures that we have

$$\widetilde{\mathrm{hol}}_{[0,1]}\left(\sum_{k \in E} \eta(., p_{\tau}(U^{+}K_k))\right) = \left(\prod_{k \in E} \widetilde{\mathrm{hol}}_{[0,1]}\eta(., p_{\tau}(U^{+}K_k))\right)_{\sqcup}$$
in the commutative algebra $\check{\mathcal{A}}(\mathbb{R})$. This finishes the proof of the first assertion of Theorem 17.9.

Orient $L_1$ and $L_2$ in a compatible way. Recall $p_1(\tau_1\tau_2) = p_1(\tau_1) + p_1(\tau_2)$. Also recall the associativity of the product $(\cdot)_\sqcup$. According to the first assertion applied to straight tangles with the induced parallelization, if $L_1$ and $L_2$ are framed by parallels $L_{1\parallel}$ and $L_{2\parallel}$ induced by parallelizations $\tau_k$ such that $p_{\tau_k}(U^+L_k) \subset S_{WE}$ for $k \in \mathbb{Z}$, then we have

$$ \mathcal{Z}^f(C_1C_2, L_1L_2, (L_1L_2)_{\parallel}, \dots, (\omega_{i,S^2})_{i \in \mathcal{N}}) = (\mathcal{Z}^f(C_1, L_1, L_{1\parallel}, \dots, (\omega_{i,S^2}))\mathcal{Z}^f(C_2, L_2, L_{2\parallel}, \dots, (\omega_{i,S^2})))_{\sqcup}. $$

This generalizes to any pair $((L_1, L_{1\parallel}), (L_2, L_{2\parallel}))$ of parallelized $J_{bb,tt}$-oriented tangles with the invariant $\mathcal{Z}^f$ of framed tangles of Definition 16.17, as follows. When $(L_1, L_{1\parallel})$ is not representable as a straight tangle with respect to a parallelization, then $(L_1, L_{1\parallel+1})$ is, where $L_{1\parallel+1}$ is the parallel of $L_1$ such that $(L_{1\parallel+1} - L_{1\parallel})$ is homologous to a positive meridian of $L_1$ in a tubular neighborhood of $L_1$ deprived of $L_1$. Thus, the known behavior of $\mathcal{Z}^f$ under such a framing change yields the second equality of the statement when $T_1$ and $T_2$ have injective bottom and top configurations. According to Definitions 16.44 and 16.42, the second equality is also true when $T_1$ and $T_2$ are $q$-braids, thanks to the multiplicativity of $\text{hol}(\cdot)$ with respect to the product of Definition 16.10 in Lemma 16.34. Thus, Definition 16.44 of $\mathcal{Z}^f$ implies the second equality for general $q$-tangles. The third equality is a direct consequence of the second one when $\omega_{i,S^2} = \omega_{S^2}$ for all $i$. $\square$

## 17.3 Insertion of a tangle in a trivial $q$-braid.

In this section, we prove the following result, which is the cabling property of Theorem 13.12 generalized to all variants of the invariant $\mathcal{Z}^f$ of Definition 16.44.

**Proposition 17.10.** 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 $\mathcal{A}((B \times \mathbb{R})(\mathcal{L}/K))$.
Furthermore, if $L$ is $J_{bb,tt}$-oriented, for any $N \in \mathbb{N}$, for any subset $A$ of $\underline{3N}$ whose cardinality is a multiple of 3, and for any family of volume-one forms $(\omega_{i,S^2})_{i \in \underline{3N}}$, $\mathcal{Z}^f((y \times [0, 1]) (L/K)$, $A$, $(\omega_{i,S^2})_{i \in \underline{3N}}$ is obtained from $\mathcal{Z}^f(L, A$, $(\omega_{i,S^2})_{i \in \underline{3N}}$) by the natural injection from $\mathcal{A}(\mathcal{L})$ to $\mathcal{A}((B \times \mathbb{R})(\mathcal{L}/K))$.

We first prove the following particular case of Proposition 17.10.

**Lemma 17.11.** Under the hypotheses of Proposition 17.10, furthermore assume that $y \in \check{S}_B(\mathbb{C})$ and $L$ is a $J_{bb,tt}$-oriented framed tangle represented by a tangle embedding

$$L: \mathcal{L} \hookrightarrow \mathcal{C}$$

with injective bottom and top configurations. Let $N$ be a positive integer. Then there exists a family $(\omega(i, S^2))_{i \in \underline{3N}}$ of volume-one forms of $S^2$ such that

$$\mathcal{Z}^f((y \times [0, 1]) (L/K), A, (\omega(i, S^2))_{i \in \underline{3N}})$$

is obtained from $\mathcal{Z}^f(L, A$, $(\omega(i, S^2))_{i \in \underline{3N}}$) by the natural injection from $\mathcal{A}(\mathcal{L})$ to $\mathcal{A}((B \times \mathbb{R})(\mathcal{L}/K))$ for any subset $A$ of $\underline{3N}$ whose cardinality is a multiple of 3.

**PROOF:** Without loss of generality, translate and rescale $y$ so that $K = \{0\} \times [0, 1]$. Let $\eta \in ]0, 1[$ be the distance between $K$ and the other strands of $y \times [0, 1]$. Because of the known variation of $\mathcal{Z}^f$ under framing changes, there is no loss of generality in assuming that $L$ is straight with respect to a parallelization $\tau$, which we do. Let $y_1: B \hookrightarrow D_1$ be a planar configuration whose image contains $y$ and the images of the bottom and top configurations of $L$. Let $B(X_1, X_2, \dots, X_{3N})$ be a ball centered at $(X_1, X_2, \dots, X_{3N})$ of radius $24N\varepsilon^{\frac{1}{12N}}$ of $(S_H^2)^{3N}$ sitting in the subset $\mathcal{O}(N, y_1)$ of Lemma 17.3, with $\varepsilon \in ]0, \frac{1}{20^{12N}}[$ as in Lemma 17.5. For $i \in \underline{3N}$, let $\omega(i, S^2)$ be a volume-one form on $S^2$ supported on a disk $D(X_i, \varepsilon)$ of $S_H^2$. Define $\omega_1(i)$ on $C_2(R(\mathcal{C}))$ so that $\omega_1(i)$ coincides with $p_\tau^*(\omega(i, S^2))$ on $D(p_\tau)$, as in Lemma 16.15. Perform a global homothety $m_{1,\eta\varepsilon}$ of $\check{R}(\mathcal{C})$, of ratio $\eta\varepsilon$, where $D_1 \times [0, 1]$ is consequently changed to $D_{\eta\varepsilon} \times [0, \eta\varepsilon]$. Call $(\eta\varepsilon\mathcal{C}, \eta\varepsilon L)$ the intersection of the image of the long tangle $(\mathcal{C}, L)$ by this homothety with the part that replaces $D_1 \times [0, 1]$, which is now standard outside $D_{\eta\varepsilon} \times [0, 1]$. Use forms $(m_{1,\eta\varepsilon}^{-1})^*(\omega_1(i))$ for $(\eta\varepsilon\mathcal{C}, \eta\varepsilon L)$. So we have

$$\mathcal{Z}^f(\eta\varepsilon\mathcal{C}, \eta\varepsilon L, m_{1,\eta\varepsilon}(\tau), \dots, (\omega(i, S^2))) = \mathcal{Z}^f(\mathcal{C}, L, \tau, \dots, (\omega(i, S^2))).$$

Let $(y \times [0, 1]) (\eta\varepsilon L/K)$ be the tangle obtained from $(y \times [0, 1])$ by letting $R_{\eta\varepsilon, [0,1]}(\eta\varepsilon\mathcal{C})$ replace $D_{\eta\varepsilon} \times [0, 1]$. Graphs that do not involve vertices of
$R_{2\eta\epsilon,[-\eta\epsilon,2\eta\epsilon]}(\eta\epsilon\mathcal{C})$ cannot contribute to

$$ \mathcal{Z}^f(\eta\epsilon\mathcal{C}, (y \times [0, 1]) (\eta\epsilon L/K), A, (\omega(i, S^2))_{i \in \mathcal{B}_N}). $$

As in the proof of Lemma 17.5, the only contributing graphs are located in $\check{R}_{\eta\sqrt{\epsilon},[-\eta\sqrt{\epsilon},\eta\sqrt{\epsilon}]}(\eta\epsilon\mathcal{C})$.

We conclude that $\mathcal{Z}^f(\eta\epsilon\mathcal{C}, (y \times [0, 1]) (\eta\epsilon L/K), A, (\omega(i, S^2))_{i \in \mathcal{B}_N})$ is obtained from $\mathcal{Z}^f(\mathcal{C}, L, A, (\omega(i, S^2))_{i \in \mathcal{B}_N})$ by the natural injection from $\mathcal{A}(\mathcal{L})$ to $\mathcal{A}((B \times \mathbb{R})(\mathcal{L}/K))$. Since this is also true when $\eta$ is replaced by a smaller $\eta'$, this is also true when $\eta\epsilon L$ is replaced by a legal composition $\gamma^-(\eta\epsilon L)\gamma^+$ for braids $\gamma^-$ and $\gamma^+$ with constant projections in $\mathcal{S}_{B^\pm}(\mathcal{C}\mathcal{C})$, which respectively go from $\eta'y^-$ to $\eta y^-$, and from $\eta y^+$ to $\eta'y^+$ (up to adjusting the parallelizations), thanks to the isotopy invariance of $\mathcal{Z}^f$. Therefore, this is also true at the limit, when $\eta'$ tends to zero, thanks to Lemma 16.33. See Definition 16.44. $\square$

**Corollary 17.12.** Proposition 17.10 is true when $y \in \check{S}_B(\mathcal{C})$.

**PROOF:** Recall Theorem 16.45, which expresses the variation of

$$ \mathcal{Z}^f((y \times [0, 1]) (L/K), A, (\omega_{i,S^2})_{i \in \mathcal{B}_N}) $$

when $(\omega_{i,S^2})_{i \in \mathcal{B}_N}$ varies, for $q$-tangles. This variation is given by the insertion of factors on components going from bottom to bottom or from top to top, which are identical in both sides of the implicit equality to be proved, and $D$-holonomies for the bottom and top configurations, for $D \subseteq A$. The $D$-holonomies satisfy the duplication property of Proposition 16.43. The $D$-holonomies of the bottom and top configurations of $L$ contribute in the same way to both sides of the equality. The $D$-holonomies of the bottom and top configurations of $y \times [0, 1]$ are inverse to each other. After the insertion, they are duplicated both at the top and at the bottom on possibly different numbers of strands. Let $B^+$ (resp. $B^-$) be the set of upper (resp. lower) $\infty$-components of $L$. Lemma 6.23 ensures that for any diagram $\Gamma$ on $\mathcal{L}$ and for any duplication $\pi(B^+ \times K^+)^*$ (resp. $\pi(B^- \times K^-)^*$) of the upper part $K^+$ (resp. lower part $K^-$) of the long strand of $K$ by $B^+ \times [1, +\infty[$ (resp. $B^- \times ]-\infty, 0]$) of a diagram $\Gamma'$ on $B \times \mathbb{R}$, we have

$$ \Gamma\pi(B^+ \times K^+)^*(\Gamma') = \pi(B^- \times K^-)^*(\Gamma')\Gamma $$

in $\mathcal{A}((B \times \mathbb{R})(\mathcal{L}/K))$. Thus, the “holonomies” $\pi(B^- \times K^-)^*\widetilde{\mathrm{hol}}_{[1,0] \times y}(\eta_{B,\cdot})$ and $\pi(B^+ \times K^+)^*\widetilde{\mathrm{hol}}_{[0,1] \times y}(\eta_{B,\cdot})$ cancel, and Proposition 17.10 is true when
$y \in \check{S}_B(\mathbb{C})$ as soon as the bottom and top configurations of $L$ may be represented by injective configurations. When $L = T(\gamma^{-})(\mathcal{C}, L, L_{\parallel})T(\gamma^{+})$ is a general $q$-tangle and $\gamma^{-}$ and $\gamma^{+}$ are paths of configurations, $(y \times [0, 1]) (\frac{L}{K})$ is equal to

$$ (y \times [0, 1]) \left( \frac{T(\gamma^{-})}{K} \right) (y \times [0, 1]) \left( \frac{(\mathcal{C}, L, L_{\parallel})}{K} \right) (y \times [0, 1]) \left( \frac{T(\gamma^{+})}{K} \right). $$

So the result follows using the Functionality theorem 17.9 and the cabling theorem for $q$-braids (Proposition 16.43). $\square$

**PROOF OF PROPOSITION 17.10:** Corollary 17.12 leaves us with the case in which $y$ is a limit configuration. To treat this case, pick a path $\gamma: [0, 1] \to S_B(\mathbb{C})$ such that $\gamma(1) = y$ and $\gamma([0, 1]) \subset \check{S}_B(\mathbb{C})$, view $y \times [0, 1]$ as the path composition $\bar{\gamma}\gamma(0)\gamma$ where $\gamma(0)$ is thought of as a constant map. If $y^{-}$ and $y^{+}$ respectively denote the bottom and top configurations of $L$ and if $K^{-}$ and $K^{+}$ denote the strand of $K$ in $\bar{\gamma}$ and in $\gamma$, respectively, then $(y \times [0, 1]) (\frac{L}{K})$ may be expressed as

$$ T(\bar{\gamma}) \left( \frac{y^{-} \times [0, 1]}{K^{-}} \right) (\gamma(0) \times [0, 1]) \left( \frac{L}{K} \right) T(\gamma) \left( \frac{y^{+} \times [0, 1]}{K^{+}} \right). $$

Use the functoriality theorem 17.9, Corollary 17.12, the cabling theorem for $q$-braids (Proposition 16.43), and the commutation argument in the above proof to conclude. $\square$

## 17.4 Duplication property

We are about to show how $\mathcal{Z}^f$ and all its variants behave under a general parallel duplication of a component going from bottom to top in a tangle.

**Proposition 17.13.** Let $K$ be a component going from bottom to top or from top to bottom in a $q$-tangle $L$ in a rational homology cylinder $\mathcal{C}$. Let $y$ be an element of $S_B(\mathbb{C})$ for a finite set $B$. Let $L(y \times K)$ be the tangle obtained by duplicating $K$ as in Section 13.1. Then we have

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

with a natural extension of Notation 6.31. Furthermore, for any $N \in \mathbb{N}$, for any family $(\omega_{i,S^2})_{i \in \underline{3N}}$ of volume-one forms of $S^2$, and for any subset $A$ of $\underline{3N}$ whose cardinality is a multiple of 3, we have

$$ \mathcal{Z}^f(\mathcal{C}, L(y \times K), A, (\omega_{i,S^2})_{i \in \underline{3N}}) = \pi(B \times K)^* \mathcal{Z}^f(\mathcal{C}, L, A, (\omega_{i,S^2})_{i \in \underline{3N}}) $$

for any $J_{bb,tt}$-orientation of $L$.
In order to prove this proposition, we are going to prove the following lemmas.

**Lemma 17.14.** Let $L: \mathcal{L} \hookrightarrow \mathcal{C}$ be a straight tangle in a parallelized rational homology cylinder $(\mathcal{C}, \tau)$. Let $K$ be a component of $L$ going from bottom to top. Let $y$ be an element of $\dot{S}_B(\mathbb{C})$ for a finite set $B$. Let $N \in \mathbb{N}$. There exists a family of volume-one forms $(\omega(i, S^2))_{i \in \underline{3N}}$ such that

$$Z(C, L(y \times K), \tau, A, (\omega(i, S^2))_{i \in \underline{3N}}) = \pi(B \times K)^* Z(C, L, \tau, A, (\omega(i, S^2))_{i \in \underline{3N}})$$
for any subset $A$ of $\underline{3N}$ whose cardinality is a multiple of 3.

**Lemma 17.15.** Lemma 17.14 implies Proposition 17.13.

**PROOF:** The known behavior of $Z$ under strand orientation changes for components going from bottom to top of Lemma 16.46 allows us to reduce the proof to the case in which $K$ goes from bottom to top. Lemma 17.14, Theorem 16.45, and Proposition 16.43 imply that Proposition 17.13 holds when $L$ is a straight tangle (with injective bottom and top configurations) and when $y \in \dot{S}_B(\mathbb{C})$. Then the duplication property for braids of Proposition 16.43 and the functoriality imply that Proposition 17.13 holds if $y \in \dot{S}_B(\mathbb{C})$ for any $J_{bb,tt}$-oriented $q$-tangle $L$ for which $(K, K_\parallel)$ can be represented by a straight knot with respect to a parallelization $\tau$ of $\mathcal{C}$ and its associated parallel. Therefore, Proposition 17.13 is also true if $y \in \mathcal{S}_B(\mathbb{C})$ by iterating the duplication process as soon as $(K, K_\parallel)$ is representable by a straight knot. In particular, it is true when $K$ is a strand of a trivial braid whose framing has been changed so that $lk(K, K_\parallel) = 2$. (Recall Lemma 7.39 and Proposition 16.13.) Thanks to the functoriality of $\mathcal{Z}^f$, since an element whose degree 0 part is 1 is determined by its square, Proposition 17.13 is also true when $K$ is a strand of a trivial braid whose framing has been changed so that $lk(K, K_\parallel) = 1$. If our general $(K, K_\parallel)$ is not representable, then Proposition 17.13 is true when $L$ is composed by a trivial braid such that the framing of the strand $I$ that extends $K$ is changed so that $lk(I, I_\parallel) = -1$. So it is also true for $L$. $\square$

Let us introduce some notation for the proof of Lemma 17.14. Choose a tubular neighborhood

$$N_{\eta_0}(K) = D_{\eta_0} \times \mathbb{R}_K$$

of $K = \{0\} \times \mathbb{R}_K$ in $R(\mathcal{C}) \setminus (L(\mathcal{L}) \setminus K)$ for some $\eta_0$ such that $0 < 10\eta_0 < 1$, where $D_{\eta_0}$ denotes the disk of radius $\eta_0$ centered at 0 in $\mathbb{C}$. Assume that the trivialization $\tau$ maps $(d \in D_{\eta_0}, k \in \mathbb{R}_K, e_1 = \vec{N})$ to an oriented tangent vector to $d \times K$, and that $\tau$ maps $(d, k, (e_2, e_3))$ to the standard frame $(1, i)$ of $D_{\eta_0}(\times k) \subset \mathbb{C}$. Pick a representative $y$ of $y$ in $\tilde{C}_B[D_{1/2}]$. For $\eta \in ]0, \eta_0]$,
let $L(\eta^2 y \times K)$ denote the tangle obtained from $L$ by replacing $\{0\} \times \mathbb{R}_K$ by $\eta^2 y \times \mathbb{R}_K$ in $D_{\eta_0} \times \mathbb{R}_K$.

Let us now reduce the proof of Lemma 17.14 to the proof of the following lemma.

**Lemma 17.16.** There exist $\eta_1 \in ]0, \eta_0]$ and volume-one forms $(\omega(i, S^2))$ of $S^2$ for $i \in \underline{3N}$ such that

$$Z(C, L(\eta^2 y \times K), \tau, A, (\omega(i, S^2))_{i \in \underline{3N}}) = \pi(B \times K)^* Z(C, L, \tau, A, (\omega(i, S^2))_{i \in \underline{3N}})$$

for any $\eta \in ]0, \eta_1]$ and any subset $A$ of $\underline{3N}$ whose cardinality is a multiple of three.

**Lemma 17.17.** Lemma 17.16 implies Lemma 17.14.

**PROOF:** Definition 16.44 and Lemma 16.33 allow us to write

$$ \begin{align*} \mathcal{Z}^f(C, L(y \times K), A, (\omega(i, S^2))_{i \in \underline{3N}}) & = \lim_{\eta \to 0} \mathcal{Z}^f(C, L(\eta^2 y \times K), A, (\omega(i, S^2))_{i \in \underline{3N}}) \\ & = \pi(B \times K)^* \mathcal{Z}^f(C, L, A, (\omega(i, S^2))_{i \in \underline{3N}}). \end{align*} $$

□

To prove Lemma 17.16, we need some preliminary lemmas, which involve the following new type of Jacobi diagram. A special Jacobi $\underline{3N}$-diagram on $B \times \mathbb{R}$ is a connected graph $\Gamma_s$ without looped edges with univalent vertices, trivalent vertices, and one bivalent vertex, equipped with an injection $j_E$ from its set $E(\Gamma_s)$ of edges into $\underline{3N}$ and with an isotopy class of injections $j_{\Gamma_s}$ from its set $U(\Gamma_s)$ of univalent vertices into $B \times \mathbb{R}$. The space of these diagrams is denoted by $\mathcal{D}_{\underline{3N}}^{e, \text{special}}(B \times \mathbb{R})$. For a special Jacobi $\underline{3N}$-diagram $\Gamma_s$ with univalent vertices on at least two strands, the space of configurations of $\Gamma_s$ with respect to $y$ is the space $\check{\mathcal{V}}(y, \Gamma_s)$ of injections of the set $V(\Gamma_s)$ of vertices of $\Gamma_s$ into $\mathbb{C} \times \mathbb{R}$ whose restriction to $U(\Gamma_s)$ is the composition of $y \times 1_{\mathbb{R}}$ with an injection from $U(\Gamma_s)$ into $B \times \mathbb{R}$ in the isotopy class $[j_{\Gamma_s}]$, up to vertical translation, for our representative $y \in \check{C}_B[D_{1/2}]$. This space $\check{\mathcal{V}}(y, \Gamma_s)$ is similar to former spaces $\check{\mathcal{V}}(y, \Gamma)$ of Section 15.2 and is compactified as in Chapter 15. See also Lemma 16.27. Its compactification is its closure in $S_{V(\Gamma)}(\mathbb{R}^3)$. If $\Gamma_s$ has no univalent vertices or univalent vertices on one strand, then configurations are also considered up to dilation, and the compactification is again the closure in $S_{V(\Gamma)}(\mathbb{R}^3)$. The configurations of $\check{\mathcal{V}}(y, \Gamma_s)$ are normalized so that a vertex of $\Gamma_s$ is sent to $D_{1/2} \times \{0\}$.
**Lemma 17.18.** Let $B_\infty$ be a finite set. Let $y_\infty: B_\infty \hookrightarrow D_1$ be a planar configuration. Let $N$ be a natural number. The set $O_s(N, y_\infty)$ of points $(X_i)_{i \in \underline{3N}}$ of $(S_H^2)^{3N}$ that are regular values of

* the maps $g(\Gamma)$ of Lemma 17.3 associated to $\underline{3N}$-numbered Jacobi diagrams $\Gamma$ on $\sqcup_{b \in B_\infty} \mathbb{R}_b$ and to the configuration $y_\infty$ and

* similar maps $g(\Gamma_s)$ associated to special $\underline{3N}$-numbered Jacobi diagrams $\Gamma_s$ (as above with one bivalent vertex) on $\sqcup_{b \in B_\infty} \mathbb{R}_b$ and to the configuration $y_\infty$

is a dense open subset of $(S_H^2)^{3N}$.

**PROOF:** The arguments of Lemma 17.3 allow us to prove that the images of the above maps $g(\Gamma)$ and of the maps $g(\Gamma_s)$, when the univalent vertices of $\Gamma_s$ are on one strand or when $\Gamma_s$ has no univalent vertices, are compact semi-algebraic subsets of $(S_H^2)^{3N}$ of codimension at least one. The complement of the union of these images is therefore open and dense.

For a special $\Gamma_s$ with univalent vertices on at least 2 strands, the images under $g(\Gamma_s)$ of the boundary of the configuration space and of the parts where $g(\Gamma_s)$ is not a submersion are also compact semi-algebraic subsets of $(S_H^2)^{3N}$ of codimension at least one. $\square$

**Lemma 17.19.** Assume that the configuration $y_\infty$ of Lemma 17.18 contains the configuration $y: B \hookrightarrow D_1$, the bottom configuration $y_-$ of $L$, the top configuration $y_+$ of $L$, and 0 as subconfigurations.

Let $(X_i)_{i \in \underline{3N}}$ be the center of a tiny ball of radius $\epsilon_0 > 0$ in the set $O_s(N, y_\infty)$ defined in Lemma 17.18. There exist $\epsilon_1 \in ]0, \frac{\epsilon_0}{\sqrt{3N}}[$ and $M_1 \in ]1, +\infty[$ such that for any family $(\omega(i, S^2))_{i \in \underline{3N}}$ of volume-one forms of $S^2$ supported inside disks $D(X_i, \epsilon)$ of $S^2$ centered at $X_i$ of radius $\epsilon \in ]0, \epsilon_1]$,

* for any Jacobi diagram $\Gamma \in \mathcal{D}_{\underline{3N}}^e(B_\infty \times \mathbb{R})$ and for any special Jacobi diagram $\Gamma \in \mathcal{D}_{\underline{3N}}^{e, \text{special}}(B \times \mathbb{R})$ with univalent vertices on at most one strand, the support of $\bigwedge_{e \in E(\Gamma)} (p_{S^2} \circ p_e)^* (\omega(j_E(e), S^2))$ in $C(\mathbb{R}^3, y_\infty \times K; \Gamma)$ is empty,

* for any special Jacobi diagram $\Gamma \in \mathcal{D}_{\underline{3N}}^{e, \text{special}}(B \times \mathbb{R})$,

- the support of $\bigwedge_{e \in E(\Gamma)} (p_{S^2} \circ p_e)^* (\omega(j_E(e), S^2))$ in the space $\check{\mathcal{V}}(y, \Gamma)$ introduced before Lemma 17.18 is contained in disjoint open subsets where $\prod_{e \in E(\Gamma)} (p_{S^2} \circ p_e)$ is a diffeomorphism onto

$$ \prod_{e \in E(\Gamma)} \check{D}(X_{j_E(e)}, \epsilon) $$
and the distance of the images of a vertex under two configurations
is at most $M_1\epsilon$,<fnref n="3" /> and

- the images of the vertices under the configurations of the support
are contained in $D_{M_1} \times [-M_1, M_1]$.

PROOF: The definition of $O_s(N, y_\infty)$ in Lemma 17.18 and the hypotheses on $(X_i)_{i \in \mathbb{3N}}$ in Lemma 17.20 guarantee that the first assertion is satisfied for all $\epsilon_1 \in ]0, \frac{\epsilon_0}{\sqrt{3N}}[$. Proceed as in Section 11.4, to reduce the support to disjoint subsets, where $\prod_{e \in E(\Gamma)} (p_{S^2} \circ p_e)$ is a diffeomorphism onto $\prod_{e \in E(\Gamma)} \dot{D}(X_{j_E(e)}, \epsilon_1)$. (It is simpler here.) Then there exists an $M$ such that the distance of the images of a vertex under two configurations is at most $M\epsilon$ on the preimage of $\prod_{e \in E(\Gamma)} \dot{D}(X_{j_E(e)}, \epsilon)$ under such a diffeomorphism, for any $\epsilon \in ]0, \epsilon_1]$. $\square$

Choose a Riemannian metric on $\tilde{R}(\mathcal{C})$, which coincides with the standard metric of $\mathbb{R}^3$ outside $\tilde{R}_{1,[0,1]}(\mathcal{C})$, and assume that this Riemannian metric restricts as the natural product metric on $N_{\eta_0}(K)$, locally. (Reduce $\eta_0$ if necessary.)

Let $C_{2,\le 10\eta_0}(N_{\eta_0}(K))$ denote the closure in $C_2(R(\mathcal{C}))$ of the space of pairs of points $(x_1, x_2) \in N_{\eta_0}(K)^2$ at a distance less than $10\eta_0$ from each other. Naturally extend $p_\tau$ to $C_{2,\le 10\eta_0}(N_{\eta_0}(K))$ by viewing $D_{\eta_0} \times \mathbb{R}_K$ as a subspace of $\mathbb{R}^3$, locally, with the usual formula $p_\tau = \frac{1}{\|x_2-x_1\|}(x_2-x_1)$.

Let $\Gamma \in \mathcal{D}_{\mathbb{3N}}^e(\mathcal{L})$. Set $U_K(\Gamma) = j_\Gamma^{-1}(\mathbb{R}_K)$, where $\mathbb{R}_K$ is viewed as the domain of $K$. For $\eta \in ]0, \eta_0]$, let $C(R(\mathcal{C}), L, \eta; \Gamma)$ be the configuration space obtained from $C(R(\mathcal{C}), L; \Gamma)$ by replacing the condition that the restriction of the configurations to the set $U_K(\Gamma)$ of univalent vertices on $\mathbb{R}_K$ factors through $K \circ j_\Gamma$, for a $\Gamma$-compatible injection $j_\Gamma$ from $U(\Gamma)$ to $\mathcal{L}$, with the condition that $U_K(\Gamma)$ is mapped to the interior of $N_\eta(K)$. (In other words, the conditions on the restriction of a configuration $c$ to $U(\Gamma)$ now only impose that $c(U_K(\Gamma)) \subset \dot{N}_\eta(K)$, and that $c|_{U(\Gamma)\setminus U_K(\Gamma)}$ may be expressed as $L \circ j_\Gamma|_{U(\Gamma)\setminus U_K(\Gamma)}$ for some $\Gamma$-compatible $j_\Gamma$.) There is a natural projection $p_K$ from this configuration space $C(R(\mathcal{C}), L, \eta; \Gamma)$ to $\dot{D}_\eta^{U_K(\Gamma)}$, and $C(R(\mathcal{C}), L; \Gamma)$ is contained in the preimage $p_K^{-1}((0)^{U_K(\Gamma)})$ in $C(R(\mathcal{C}), L, \eta; \Gamma)$. (This preimage also contains $C(R(\mathcal{C}), L; \tilde{\Gamma})$ for Jacobi diagrams $\tilde{\Gamma}$ different from $\Gamma$ because the linear order of the vertices of $U_K(\Gamma)$ is not induced by $j_\Gamma$.)

**Lemma 17.20.** Let $(X_i)_{i \in \mathbb{3N}}$ and $\epsilon_1$ be as in Lemma 17.19. There exist propagating chains $P(i)$ of $(\tilde{C}_2(R(\mathcal{C})), \tau)$ and $\epsilon_1$-dual propagating forms $\omega(i)$

<footnote n="3">³In $\tilde{V}(y, \Gamma)$, configurations are normalized so that univalent vertices go to $y(B) \times \mathbb{R}$ for our fixed representative of $y$ and one vertex goes to $y(B) \times \{0\}$. The distance is the Euclidean distance of $\mathbb{R}^3$.</footnote>
of $(C_2(R(C)), \tau)$, for $i \in \underline{3N}$, and $\eta_2 \in [0, \eta_0]$ such that

* we have $P(i) \cap D(p_\tau) = p_\tau^{-1}(X_i) \cap D(p_\tau)$ for any $i \in \underline{3N}$,

* we have $P(i) \cap C_{2, \le 10\eta_2}(N_{\eta_2}(K)) = p_\tau^{-1}(X_i) \cap C_{2, \le 10\eta_2}(N_{\eta_2}(K))$, for the above natural extension of $p_\tau$ on $C_{2, \le 10\eta_2}(N_{\eta_2}(K))$,

* the $P(i)$ are in general 3N-position with respect to $(\check{R}(\mathcal{C}), L, \tau)$ (again as in Definition 11.3), and

* $\omega(i)$ restricts to $D(p_\tau) \cup C_{2, \le 10\eta_2}(N_{\eta_2}(K))$ as $p_\tau^*(\omega(i, S^2))$ for some form $\omega(i, S^2)$ as in Lemma 17.19.

Let $\Gamma \in \mathcal{D}_{\underline{3N}}^e(\mathcal{L})$.

For $(\omega(i))_{i \in \underline{3N}}$ as above and $\eta \in [0, \eta_2]$, let $\text{Supp}(\Gamma, \eta; (\omega(i)))$ denote the support of $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e)))$ in $C(R(\mathcal{C}), L, \eta; \Gamma)$.

For $(P(i))_{i \in \underline{3N}}$ as above, for any configuration $c$ in the discrete set

$$I_S(L, \Gamma, (P(i))) = C(R(\mathcal{C}), L; \Gamma) \cap \bigcap_{e \in E(\Gamma)} p_e^{-1}(P(j_E(e))),$$

and for any edge $e$ of $E(\Gamma)$, $p_e(c)$ is in the interior of a 4-cell $\Delta_e(c)$ of $P(j_E(e))$, and $C_2(R(\mathcal{C}))$ is diffeomorphic to $D_{\varepsilon_1} \times \Delta_e(c)$ near $p_e(c)$, where $D_{\varepsilon_1}$ is the local fiber of a tubular neighborhood of $\Delta_e(c)$. Let $p_{c,e}$ be the associated projection onto $D_{\varepsilon_1}$ in a neighborhood of $c$ in $C(R(\mathcal{C}), L, \eta; \Gamma)$. Without loss of generality, assume that $\omega(j_E(e))$ may be expressed as $p_{c,e}^*(\omega_\varepsilon(c, e))$ locally, for a 2-form $\omega_\varepsilon(c, e)$ supported on $D_\varepsilon$, such that $\int_{D_\varepsilon} \omega_\varepsilon(c, e)$ is the rational coefficient of the above 4-cell $\Delta_e(c)$ in $P(j_E(e))$.

With this notation, there exist $\varepsilon \in ]0, \varepsilon_1]$, propagating chains $P(i)$, $\varepsilon$-dual propagating forms $\omega(i)$ as above, $\eta_3 \in ]0, \eta_2]$, and $M_2 > 1$ such that for any $\eta \in ]0, \eta_3]$ and for any Jacobi diagram $\Gamma \in \mathcal{D}_{\underline{3N}}^e(\mathcal{L})$,

* $\text{Supp}(\Gamma, \eta; (\omega(i)))$ is contained in disjoint submanifolds, indexed by the configurations $c$ of $I_S(\Gamma, (P(i)))$, and which contain those, where $p_K \times (p_E = \prod_{e \in E(\Gamma)} p_{c,e})$ is a diffeomorphism onto $\check{D}_\eta^{U_K(\Gamma)} \times \check{D}_\varepsilon^{E(\Gamma)}$, and where the distance of the images of a vertex under two configurations of $(p_K \times p_E)^{-1}(\check{D}_\eta^{U_K(\Gamma)} \times \{W\})$ is at most $M_2\eta$ for any $W \in \check{D}_\varepsilon^{E(\Gamma)}$,

* the involved configurations map vertices of $V(\Gamma) \setminus U_K(\Gamma)$ at a distance greater than $9M_2\eta$ from $K$, and

* they map two distinct vertices of $U_K(\Gamma)$ at a distance greater than $9M_2\eta$ from each other.
PROOF: The existence of the $P(i)$ in general $3N$-position with prescribed behavior near the boundary can be proved as in Section 11.3. Fix a graph $\Gamma \in \mathcal{D}_{3N}^e(\mathcal{L})$. Once the $P(i)$ are in general $3N$-position, for such a given graph $\Gamma$, $I_S(\Gamma, (P(i)))$ consists of a finite number of isolated intersection points at which trivalent vertices cannot be on $K$. Indeed, this would correspond to a degenerate configuration for a graph in which 3 univalent vertices replace the trivalent vertex on $K$.

The existence of a family $(\omega(i))_{i \in \mathcal{N}}$ of propagating forms of $R(\mathcal{C})$, $\varepsilon_1$-dual to $P(i)$, which may be expressed as $p_\tau^*(\omega(i, S^2))$ on $D(p_\tau) \cup C_{2, \le 10\eta_2}(N_{\eta_2}(K))$, with respect to a family $(\omega(i, S^2))_{i \in \mathcal{N}}$ of 2-forms of volume 1 supported inside a disk $D(X_i, \varepsilon)$, as in Lemma 17.19, can be proved as in Section 11.4.

Let $c \in I_S(\Gamma, (P(i)))$. Transversality implies that the restriction to a neighborhood of $c$ in $C(R(\mathcal{C}), L; \Gamma)$ of $p_E$ is a submersion, with the notation of the statement. This implies that the restriction of $p_K \times p_E$ to a neighborhood of $c$ in $C(R(\mathcal{C}), L, \eta; \Gamma)$ is a submersion, too, so that it is a local diffeomorphism in a neighborhood of $c$ in $C(R(\mathcal{C}), L, \eta; \Gamma)$. After reducing $\eta$ and $\varepsilon$, we obtain a neighborhood $N(c)$ of $c$ in $C(R(\mathcal{C}), L, \eta; \Gamma)$ such that $p_K \times p_E$ is a diffeomorphism from $N(c)$ onto $\tilde{D}_\eta^{U_K(\Gamma)} \times \tilde{D}_\varepsilon^{E(\Gamma)}$ for all $c \in I_S(\Gamma, (P(i)))$. The compact intersection of $C(R(\mathcal{C}), L; \Gamma)$ with the complement of $\cup_{c \in I_S(\Gamma, (P(i)))} N(c)$ is mapped outside $\prod_{e \in E(\Gamma)} P(j_E(e))$ by $\prod_{e \in E(\Gamma)} p_e$. Therefore, its image avoids a neighborhood of $\prod P(j_E(e))$. Reducing $\varepsilon$ allows us to assume that it avoids the compact closure of the support of $\bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e)))$. We may now reduce $\eta$ so that $C(R(\mathcal{C}), L, \eta; \Gamma)$ does not meet this compact closure outside $\cup_{c \in I_S(\Gamma, (P(i)))} N(c)$. This can be achieved simultaneously for all the finitely many considered $\Gamma$.

Now, the Jacobians of the corresponding inverse local diffeomorphisms (viewed as maps from $D_\eta^{U_K(\Gamma)} \times \prod_{e \in E(\Gamma)} D_\varepsilon$ to $R(\mathcal{C})^{V(\Gamma)} \setminus \text{diag})$ are bounded (after reducing $\varepsilon$ and $\eta$ if necessary).

So we get a $M_2$ for which the distance of the images of a vertex under two configurations of any $(p_K \times p_E)^{-1}(\tilde{D}_\eta^{U_K(\Gamma)} \times \{W\})$ in a connected component of $\text{Supp}(\Gamma, \eta; (\omega(i)))$ is at most $M_2\eta$, for all the finitely many considered $\Gamma$. It is easy to reduce $\eta_3$ so that the last two conditions are satisfied with our given $M_2$. $\square$

$$ \text{Set } \mathcal{D}_{\le N, 3N}^e(\cdot) = \cup_{k=0}^N \mathcal{D}_{k, 3N}^e(\cdot). \text{ For } \Gamma \in \mathcal{D}_{\le N, 3N}^e(\mathcal{L}(B \times \mathbb{R}_K)), \text{ let} \\ I_S(L(\eta^2 y \times K), \Gamma, (P(i))_{i \in \mathcal{N}}) $$

denote the set of configurations $c$ of

$$ C(R(\mathcal{C}), L(\eta^2 y \times K); \Gamma) \bigcap \bigcap_{e \in E(\Gamma)} p_e^{-1}(P(j_E(e))) $$
with respect to the propagating chains $P(i)$.

The following crucial lemma justifies the introduction of special Jacobi diagrams.

**Lemma 17.21.** Let $\eta_1$ be the minimum in the set $\{\eta_3, \frac{1}{(2N-1)8NM_1}, \frac{1}{100NM_2}\}$ of positive numbers introduced in Lemmas 17.19 and 17.20. For any family $(P(i))_{i \in \underline{3N}}$ of propagating chains as in Lemma 17.20, for any

$$ \Gamma_B \in \mathcal{D}_{\le N, \underline{3N}}^e (\mathcal{L}(B \times \mathbb{R}_K)), $$

for any configuration $c_{\eta_1}$ of the set $I_S(L(\eta_1^2 y \times K), \Gamma_B, (P(i))_{i \in \underline{3N}})$, there exists a continuous map

$$ \begin{array}{rcl} ]0, \eta_1] & \to & C_V(\Gamma_B)(R(\mathcal{C})) \\ \eta & \mapsto & c_\eta \end{array} $$

such that, for any $\eta \in ]0, \eta_1]$, $c_\eta$ belongs to $I_S(L(\eta^2 y \times K), \Gamma_B, (P(i))_{i \in \underline{3N}})$ and the graph $\Gamma_B$ configured by $c_\eta$ is the union of

* (small red) special Jacobi diagrams $\Gamma_s$ on $B \times \mathbb{R}_K$ of diameter less than $10\eta$ configured on $\eta^2 y \times K$ in $N_\eta(K)$ (with univalent vertices on at least two strands of $B \times \mathbb{R}_K$), and

* a uni-trivalent (blue and purple) graph $\Gamma$ on the domain $\mathcal{L}$ of $L$ configured so that its univalent vertices are

- either univalent vertices of $\Gamma_B$ on $(L \setminus K) \cup (\eta^2 y \times K)$

- or trivalent vertices of $\Gamma_B$ attached to a bivalent vertex of a (small red) special graph $\Gamma_s$, as in Figure 17.1.

<figure><img src="image_137.png" /><figcaption>Figure 17.1: A configured diagram $\Gamma_B$ with its small red vertices, its small thin red edges, its big blue vertices (the rightmost four), and its thick blue edges (the rightmost three)</figcaption></figure>

Furthermore, the configuration $c_\eta$ arises as a transverse intersection point, and the intersections that involve at least one (red) special Jacobi diagram
cancel algebraically. For a fixed $\eta$, the remaining configurations are in natural one-to-one correspondence—independent of $\eta$—with triples $(\Gamma, f, c')$, where $\Gamma \in \mathcal{D}_{\le N, 3N}^e(\mathcal{L})$, $U_K(\Gamma)$ is the set of univalent vertices of $\Gamma_B$ on $B \times \mathbb{R}_K$, $f \in B^{U_K(\Gamma)}$, and $c' \in I_S(L, \Gamma, (P(i))_{i \in \underline{3N}})$. The inverse of this natural one-to-one correspondence maps $(\Gamma, f, c')$ to a pair $(\Gamma_f, c_\eta)$, where

* $\Gamma_f$ is a Jacobi diagram on $\mathcal{L}(B \times \mathbb{R}_K)$ obtained from $\Gamma$ by changing the (isotopy class of the) injection from $U_K(\Gamma)$ to $\mathbb{R}_K$ to the injection from $U_K(\Gamma)$ to $B \times \mathbb{R}_K$ that maps a vertex $u$ to $f(u) \times \mathbb{R}_K$ so that the order of vertices on each strand of $B \times \mathbb{R}_K$ is induced by their former order on $\mathbb{R}_K$,

* $c_\eta$ belongs to $I_S(L(\eta^2 y \times K), \Gamma_f, (P(i))_{i \in \underline{3N}})$,

* $d(c_\eta(v), c'(v))$ is smaller than $\eta$ for any vertex $v$ of $\Gamma$,

and the sign of the algebraic intersection at $c_\eta$ is the same as the sign of the algebraic intersection at $c'$, with respect to consistent vertex-orientations of $\Gamma$ and $\Gamma_f$.

Furthermore, for any $\eta \in ]0, \eta_1]$, the support $\text{Supp}(L(\eta^2 y \times K), \Gamma_B; (\omega(i)))$ of $\bigwedge_{e \in E(\Gamma_B)} p_e^*(\omega(j_E(e)))$ in $C(R(\mathcal{C}), L(\eta^2 y \times K); \Gamma_B)$ consists of disjoint neighborhoods $N_{L(\eta^2 y \times K)}(c_\eta)$ of configurations $c_\eta$ as above, where projections $p_{c_\eta, e}$ as in the statement of Lemma 17.20 make sense, and such that the restriction of $\prod_{e \in E(\Gamma_B)} p_{c_\eta, e}$ to $N_{L(\eta^2 y \times K)}(c_\eta)$ is a diffeomorphism onto $\tilde{D}_\varepsilon^{E(\Gamma_B)}$.

**PROOF:** Let $\Gamma_B \in \mathcal{D}_{\le N, 3N}^e(\mathcal{L}(B \times \mathbb{R}_K))$. Instead of starting with a configuration $c_{\eta_1} \in I_S(L(\eta_1^2 y \times K), \Gamma_B, (P(i))_{i \in \underline{3N}})$, as in the statement, we consider a configuration $c$ in $\text{Supp}(L(\eta_4^2 y \times K), \Gamma_B; (\omega(i)))$, for some $\eta_4 \in ]0, \eta_1]$. View $\Gamma_B$ as a graph configured by $c$. So its vertices become elements of $R(\mathcal{C})$. Color the vertices of $\Gamma_B$ in $N_{\eta_1^2}(K)$ with (red, 1). Next color by (red, 2) its uncolored vertices at a distance less than $4\eta_1^2 M_1$ from a vertex colored by (red, 1). For $k \ge 2$, inductively color the still uncolored vertices that are at a distance less than $4\eta_1^2 M_1$ from a vertex $v$ colored by (red, $k$), by (red, $k+1$).

Define the map

$$ 
\begin{array}{rcl}
r: & \check{R}(\mathcal{C}) & \to [0, \eta_1] \\
& (z_D, t) \in D_{\eta_1} \times \mathbb{R}_K & \mapsto |z_D| \\
& x \in R(\mathcal{C}) \setminus (\check{D}_{\eta_1} \times \mathbb{R}_K) & \mapsto \eta_1.
\end{array}
$$

Note that a vertex $v$ colored by (red, $k$) with $k \ge 2$ satisfies $\eta_1^2 \le r(v) \le 4k\eta_1^2 M_1$, by induction. So we have $r(v) \le 8N\eta_1^2 M_1 \le \eta_1$ for such a vertex.

Color the vertices that are still uncolored after this algorithm blue. Color the edges between two blue vertices blue. Color the edges between a red
vertex and a blue one purple. Also color the edges between red vertices at a
distance greater than $8N\eta_1^2 M_1$ purple. Color the remaining edges between
two red vertices red.

Remove the red edges between red vertices, and the red vertices that
do not belong to a purple edge from $\Gamma_B$. Blow up the obtained graph $\tilde{\Gamma}$
at red vertices that belong to at least two purple edges, so that these red
vertices become univalent. A red vertex that belongs to $r$ purple edges is
transformed into $r$ red vertices during this process. Let $\Gamma'$ be the obtained
configured uni-trivalent graph with blue and purple edges. Its red vertices are
in $N_{\eta_1}(K)$. Let $U_K(\Gamma')$ denote the set of red vertices of $\Gamma'$. The restriction of $c$
to $V(\Gamma')$ is in $C(R(C), L, \eta_1; \Gamma') \cap \text{Supp}(\bigwedge_{e \in E(\Gamma')} p_e^*(\omega(j_E(e)))))$. So, according
to Lemma 17.20, it is in one of the disjoint submanifolds where $p_K \times p_E$
restricts as a diffeomorphism onto $\tilde{D}_{\eta_1}^{U_K(\Gamma')} \times \tilde{D}_{\varepsilon}^{E(\Gamma')}$, with $p_E = \prod_{e \in E(\Gamma')} p_{c,e}$,
and where the distance of the images of a vertex under two configurations of
$(p_K \times p_E)^{-1}(\tilde{D}_{\eta}^{U_K(\Gamma')} \times \{W\})$ is at most $M_2\eta$ for any $W \in \tilde{D}_{\varepsilon}^{E(\Gamma')}$. Set $W_0 =
p_E(c)$. (For $c \in I_S(L(\eta_1^2 y \times K), \Gamma_B, (P(i)))$, we have $W_0 = 0 = (0)_{c \in E(\Gamma')}$.)
Then $c' = (p_K \times p_E)^{-1}((0)_{U_K(\Gamma')}, W_0)$ is a configuration of a graph $\Gamma$ on $\mathcal{L}$,
obtained from $\Gamma'$ by adding the data of an isotopy of injections of $U_K(\Gamma')$
into $\mathbb{R}_K$, where $U_K(\Gamma) = U_K(\Gamma')$. According to Lemma 17.20, no collision of
vertices of $\Gamma$ can occur. So the red vertices of $\Gamma'$ were univalent in $\tilde{\Gamma}$ (there
was no need to blow them up) and $\tilde{\Gamma} = \Gamma'$.

Furthermore, $c'$ maps two red vertices at a distance at least $9M_2\eta_1$ from
each other. In particular, two red vertices of $\tilde{\Gamma}$ are at a distance at least
$7M_2\eta_1$ from each other, with respect to $c$.

The univalent vertices of $\tilde{\Gamma}$ are either univalent vertices of $\Gamma_B$, sent to
$\eta_1^2 y \times K$ or to $L \setminus K$ by $c$, or trivalent vertices of $\Gamma_B$, which belong to a
bivalent vertex of a red subgraph of $\Gamma_B$. Let $\Gamma_R$ be the subgraph of $\Gamma_B$
consisting of its red vertices and of its red edges.

Let $\Gamma_{R,1}$ be a connected component of $\Gamma_R$ such that $\Gamma_{R,1}$ is not reduced to a univalent vertex. Since two vertices of $\Gamma_{R,1}$ are at a distance at most $(2N-1)8N\eta_1^2 M_1 \le \eta_1$ from each other, there is a most one red vertex of $\tilde{\Gamma}$ in $\Gamma_{R,1}$, and $c$ sends $\Gamma_{R,1}$ to a part of $N_{\eta_1}(K)$ identified with a part of $\mathbb{R}^3$ of diameter less than $10\eta_1$. So such a $\Gamma_{R,1}$ configured by $c$ may be viewed as a graph with straight edges, directed by the $X_i$ if $c \in I_S(L(\eta_1^2 y \times K), \Gamma_B, (P(i)))$, and by the $\tilde{W}_i \in D(X_i, \varepsilon)$ in general. In particular, Lemma 17.19 implies that $\Gamma_{R,1}$ must be a configured special Jacobi diagram in $D_{\eta_1^2 M_1} \times [x - \eta_1^2 M_1, x + \eta_1^2 M_1]$. Its configuration $\eta_4^2 c_{R,1}$ is determined up to translation along $\mathbb{R}_K$. The projection of the bivalent vertex $\alpha_1$ of $\Gamma_{R,1}$ to $D_{\eta_1^2 M_1}$ is $p_C(\eta_4^2 c_{R,1}(\alpha_1))$.

Let $\mathcal{A}$ denote the set of bivalent vertices of $\Gamma_R$. Write the corresponding
configured special Jacobi diagrams $(\Gamma_{R,\alpha}, c_{R,\alpha})_{\alpha \in \mathcal{A}}$. Let $U_K(\Gamma_B) = j_{\Gamma_B}^{-1}(B \times \mathbb{R}_K)$. Note the natural inclusions $U_K(\Gamma) \subseteq U_K(\Gamma_B) \sqcup \mathcal{A}$ and $\mathcal{A} \subseteq U_K(\Gamma)$. Let $f_{\Gamma_B}: U_K(\Gamma_B) \to B$ be the map that sends $u \in j_{\Gamma_B}^{-1}(\{b\} \times \mathbb{R}_K)$ to $b$. Let $f$ denote the restriction of $f_{\Gamma_B}$ to $U_K(\Gamma) \setminus \mathcal{A}$.

So far, our analysis allows us to associate

$$ \Phi(\Gamma_B, c) = (\Gamma, c', \mathcal{A} \subset U_K(\Gamma), f: U_K(\Gamma) \setminus \mathcal{A} \to B, (\Gamma_{R,\alpha}, c_{R,\alpha})_{\alpha \in \mathcal{A}}) $$

to our configured graph $(\Gamma_B, c)$ as above, where

* $\Gamma \in \mathcal{D}_{\le N, 3N}^e(\mathcal{L})$,

* $c' \in \text{Supp}(L, \Gamma; (\omega(i))) = C(R(\mathcal{C}), L; \Gamma) \cap \text{Supp}(\bigwedge_{e \in E(\Gamma)} p_e^*(\omega(j_E(e))))$,

* $p_E(c') = p_E(c) = W_0$,

* $c' \in I_S(L, \Gamma, (P(i)))$ when $c \in I_S(L(\eta_4^2 y \times K), \Gamma_B, (P(i)))$,

* the $\Gamma_{R,\alpha}$ are special Jacobi diagrams on $B \times \mathbb{R}_K$ whose disjoint union is numbered in $\underline{3N} \setminus j_E(E(\Gamma))$,

* the $c_{R,\alpha}$ are configurations of these diagrams with respect to $y \times \mathbb{R}$ in

$$ \bigcap_{e \in E(\Gamma_{R,\alpha})} (p_{S^2} \circ p_e^{-1})(\tilde{W}_{j_E(e)}), $$

(or in $\bigcap_{e \in E(\Gamma_{R,\alpha})} (p_{S^2} \circ p_e^{-1})(X_{j_E(e)})$ when $c \in I_S(L(\eta_4^2 y \times K), \Gamma_B, (P(i)))$).

Let us now show how to reconstruct $(\Gamma_B, c)$ from the data

$$ (\Gamma, c', \mathcal{A} \subset U_K(\Gamma), f: U_K(\Gamma) \setminus \mathcal{A} \to B, (\Gamma_{R,\alpha}, c_{R,\alpha})_{\alpha \in \mathcal{A}}). $$

The graph $\Gamma_B$ is obtained from $\Gamma$ and the $\Gamma_{R,\alpha}$ by gluing $\Gamma$ and the $\Gamma_{R,\alpha}$ at the vertices $\alpha$. The components of its univalent vertices are determined by those of the univalent vertices of $\Gamma_{R,\alpha}$ and of $\Gamma$ outside $\mathcal{A}$, and by $f$. Their order on a strand of $B \times \mathbb{R}_K$ is the restriction to the vertices of such a strand of the order induced on $U_K(\Gamma_B)$ by letting the ordered set of the univalent vertices of $\Gamma_{R,\alpha}$ replace $\alpha$, for every $\alpha \in \mathcal{A}$, in the ordered set of the univalent vertices of $\Gamma$.

Below, we construct a smooth embedding

$$ \mathbf{c}: ]0, \eta_1] \times \mathring{D}_{\varepsilon}^{E(\Gamma_B)} \hookrightarrow C_{V(\Gamma_B)}(R(\mathcal{C})) $$

such that $c(\eta, W_B)$ belongs to $C(R(\mathcal{C}), L(\eta^2 y \times K); \Gamma_B)$ for any $(\eta, W_B) \in ]0, \eta_1] \times \mathring{D}_{\varepsilon}^{E(\Gamma_B)}$ and our configuration $c$ is equal to $\mathbf{c}(\eta_4, W_{B,0})$.
Lemma 17.19 guarantees that $p_{E,\alpha} = \prod_{e \in E(\Gamma_{R,\alpha})} (p_{S^2} \circ p_e)$ is a diffeomorphism from a neighborhood of $c_{R,\alpha}$ in the configuration space of configurations of $\Gamma_{R,\alpha}$ on $y \times \mathbb{R}_K$ up to translation along $\mathbb{R}_K$ onto $\prod_{e \in E(\Gamma_{R,\alpha})} \mathring{D}(X_{j_{E}(e)}, \varepsilon)$, for every $\alpha \in \mathcal{A}$. Use an implicit diffeomorphism $\phi_e: D(X_{j_{E}(e)}, \varepsilon) \to D_{\varepsilon}$ to identify $D(X_{j_{E}(e)}, \varepsilon)$ with $D_{\varepsilon}$. Set $W_{\alpha,0} (= (\phi_e \circ p_{S^2} \circ p_e(c_{R,\alpha}))_{e \in E(\Gamma_{R,\alpha})}) = p_{E,\alpha}(c_{R,\alpha})$. Lemma 17.20 provides a neighborhood of $c'$ in $C(R(\mathcal{C}), L, \eta_1; \Gamma)$, where $p_K \times p_E$ is a diffeomorphism onto $\mathring{D}_{\eta_1}^{U_K(\Gamma)} \times \mathring{D}_{\varepsilon}^{E(\Gamma)}$. Recall $p_E(c) = W_0$. These diffeomorphisms assemble to form a diffeomorphism $\Psi_{\eta}$ from a neighborhood of $c(\eta, W)$ in $C(R(\mathcal{C}), L(\eta^2 y \times K); \Gamma_B)$ to $\mathring{D}_{\varepsilon}^{E(\Gamma_B)}$, whose inverse $\Psi_{\eta}^{-1} = c(\eta, \cdot)$ is described below. Write $W_B = (W_e)_{e \in E(\Gamma_B)} = ((W_{\alpha})_{\alpha \in \mathcal{A}}, W)$, with $W_{\alpha} = (W_e)_{e \in E(\Gamma_{R,\alpha})}$ and $W = (W_e)_{e \in E(\Gamma)}$. $\Psi_{\eta}^{-1}(W)$ is constructed from representatives $\eta^2 c_{R,\alpha}(\alpha)$ (where the vertical translation parameter is fixed) of the $\eta^2 p_{E,\alpha}^{-1}(W_{\alpha})$ by assembling them with

$$c_1(\eta, W) = (p_K \times p_E)^{-1} \left( \left( \left( \eta^2 y(f(u)) \right)_{u \in U_K(\Gamma) \setminus \mathcal{A}}, \left( p_{\mathcal{C}}(\eta^2 c_{R,\alpha}(\alpha)) \right)_{\alpha \in \mathcal{A}} \right), W \right)$$

so that the height (projection onto $\mathbb{R}_K$) $p_{\mathbb{R}}(\eta^2 c_{R,\alpha}(\alpha))$ of $\alpha$ in $\eta^2 c_{R,\alpha}(\alpha)$ coincides with $p_{\mathbb{R}}(c_1(\eta, W)(\alpha))$.

This ensures that $c(\eta, W_B) = \Psi_{\eta}^{-1}(W_B)$ arises as a transverse intersection, for any $W_B$. This is true when $W_B = 0$, in particular. So it is true for the configuration $c_{\eta_1}$ of the statement, which may be expressed as $c(\eta_1, 0)$. The family $c_{\eta}$ is the continuous family $\Psi_{\eta}^{-1}(0)$, in this case. The sign of the corresponding intersection is the sign of the Jacobian of $\Psi_{\eta}$. Since we started with an arbitrary configuration in $\text{Supp}(L(\eta_4^2 y \times K), \Gamma_B; (\omega(i)))$ for some $\eta_4 \in [0, \eta_1]$, the above arguments also prove the final assertion of the lemma. Let us finally focus on the claimed algebraic cancellation. From now on, we set $W = 0$.

Assume that $\mathcal{A} \neq \emptyset$. Every $\alpha \in \mathcal{A}$ is contained in one edge $e(\alpha)$ of $\Gamma$. Choose $\alpha_0$ in $\mathcal{A}$ such that $j_E(e(\alpha_0))$ is minimal. Let $s(\Gamma_{R,\alpha_0})$ be obtained from $\Gamma_{R,\alpha_0}$ by exchanging the labels of the two edges $e_1$ and $e_2$ of $\Gamma_{R,\alpha_0}$ that contain $\alpha_0$, and by reversing their orientations if they both come from $\alpha_0$ or go to $\alpha_0$ as in Lemma 9.11. Let $s_{\alpha_0}(\Gamma_B)$ be obtained from $\Gamma_B$ by performing the same changes. Let $s(c_{R,\alpha_0})$ be obtained from $c_{R,\alpha_0}$ by changing the position of $c_{R,\alpha_0}(\alpha_0)$ by a central symmetry with respect to the middle of the two other ends of $e_1$ and $e_2$. The intersection point associated to the configured graph

$$(s_{\alpha_0}(\Gamma_B), c_2) = \Phi^{-1}(\Gamma, c', \mathcal{A}, f, (\Gamma_{R,\alpha}, c_{R,\alpha})_{\alpha \in \mathcal{A} \setminus \{\alpha_0\}}, s(\Gamma_{R,\alpha_0}), s(c_{R,\alpha_0}), 0)$$

and the intersection point associated to $(\Gamma_B, c)$ cancel algebraically, as in
Lemma 9.11. (Our process defines an involution on the configured graphs $(\Gamma_B, c)$ with $\mathcal{A} \neq \emptyset$ such that a configured graph and its image cancel.)

Therefore, the configured graphs $(\Gamma_B, c)$ that contribute to the intersection are the graphs for which $\mathcal{A} = \emptyset$. They are obtained from some $\Gamma \in \mathcal{D}_{\le N, 3N}^e(\mathcal{L})$, some $c'$, and some $f: U_K(\Gamma) \to B$ as in the statement. $\square$

PROOF OF LEMMA 17.16: Lemma 17.21 implies that

* for any $\Gamma \in \mathcal{D}_{\le N, 3N}^e(\mathcal{L})$, $I(C, L, \Gamma, o(\Gamma), (\omega(i))_{i \in \mathcal{N}})$ is the algebraic intersection $I(C, L, \Gamma, o(\Gamma), (P(i))_{i \in \mathcal{N}})$ of the preimages of the propagating chains $P(i)$ in $C(R(C), L; \Gamma)$ with respect to $\Gamma$,

* for any $\eta \in ]0, \eta_1]$ and for any $\Gamma_B \in \mathcal{D}_{\le N, 3N}^e(\mathcal{L}(B \times \mathbb{R}_K))$, we have

$$I(C, L(\eta^2 y \times K), \Gamma_B, o(\Gamma_B), (\omega(i))) = I(C, L(\eta^2 y \times K), \Gamma_B, o(\Gamma_B), (P(i))),$$

and

* for any $\eta \in ]0, \eta_1]$ and for any subset $A$ of $\underline{3N}$ with cardinality $3k$, we have

$$\sum_{\Gamma \in \mathcal{D}_{k,A}^e(\mathcal{L})} \zeta_{\Gamma} I(C, L, \Gamma, (P(i))_{i \in A}) \pi(B \times K)^* ([\Gamma]) = \sum_{\Gamma_B \in \mathcal{D}_{k,A}^e(\mathcal{L}(B \times \mathbb{R}_K))} \zeta_{\Gamma_B} I(C, L(\eta^2 y \times K), \Gamma_B, (P(i))_{i \in A}) [\Gamma_B].$$

$\square$

Now, both the second duplication property and the first duplication property for components going from bottom to top or from top to bottom of Theorem 13.12 are proved. Below, we prove the first duplication property, more generally, in the doubling case.

**Lemma 17.22.** Let $\nu$ be the element of $\mathcal{A}(\mathbb{R})$ obtained from

$$\mathcal{Z}^f \left( \begin{smallmatrix} (\begin{smallmatrix} \bullet & \bullet \\ \bullet & \bullet \end{smallmatrix}) & \bullet \\ \begin{smallmatrix} \bullet & \bullet \\ \bullet & \bullet \end{smallmatrix} & \begin{smallmatrix} \bullet & \bullet \\ \bullet & \bullet \end{smallmatrix} \end{smallmatrix} \right) \in \mathcal{A} \left( \begin{smallmatrix} \uparrow 1 & \uparrow 2 & \uparrow 3 \end{smallmatrix} \right)$$

by inserting $\uparrow 1 \uparrow 2 \uparrow 3$ in $\mathbb{R}$ as indicated by the picture $\xrightarrow{1} \xrightarrow{2} \xrightarrow{3}$. Let $\nu^{-\frac{1}{2}}$ be the unique element of $\mathcal{A}(\mathbb{R})$ whose degree 0 part is 1 such that $(\nu^{-\frac{1}{2}})^2 \nu$ equals 1. Then we have

$$\mathcal{Z}^f(\bullet) = \mathcal{Z}^f(\bullet) = \nu^{-\frac{1}{2}}.$$
according to Lemma 6.33, so that we get

$$ \mathcal{Z}^f(2 \times \text{Circ})^2 \pi(2 \times \mathbb{R})^*(\nu) = 1 $$

in the algebra $\mathcal{A}(\mathbb{R} \sqcup \mathbb{R})$. Since $\pi(2 \times \mathbb{R})^*$ is an algebra morphism from $\mathcal{A}(\mathbb{R})$ to $\mathcal{A}(\mathbb{R} \sqcup \mathbb{R})$, Lemma 17.22 implies

$$ \pi(2 \times \mathbb{R})^* (\mathcal{Z}^f(\text{Circ})^2) \pi(2 \times \mathbb{R})^*(\nu) = 1 $$

in the algebra $\mathcal{A}(\mathbb{R} \sqcup \mathbb{R})$. Since the multiplication by an element whose degree 0 part is 1 is injective, and since an element whose degree 0 part is 1 is determined by its square, we get $\mathcal{Z}^f(2 \times \text{Circ}) = \pi(2 \times \mathbb{R})^*(\mathcal{Z}^f(\text{Circ}))$ as desired. $\square$

**Proposition 17.24.** Let $K$ be a component of a $q$-tangle $L$ in a rational homology cylinder $C$. Let $L(2 \times K)$ be the tangle obtained by duplicating $K$ as in Section 13.1. Then we have

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

**PROOF:** A tangle $L$, with a strand $K$ going from bottom to bottom, can be written as a composition

$$ L_1 L_2 = \text{Circ} \parallel L_1 \parallel \text{Circ} $$

of some tangle $L_1$, with a cabling $L_2$ of a trivial braid by the replacement of a strand by $\text{Circ}$, where $K$ is the concatenation of one strand of $L_1$ going from bottom to top, $\text{Circ}$, and another strand of $L_1$, which goes from top to bottom. The statement for such a pair follows from Lemma 17.23, Proposition 17.10, and Proposition 17.13, using functoriality. The case in which $K$ goes from top to top can be treated similarly, by sending $\text{Circ}$ below. So the proposition is proved. $\square$

Theorem 13.12 is now proved. $\square$

**Remark 17.25.** Theorem 16.45 and Proposition 16.43 do not allow me to prove

$$ \mathcal{Z}^f(C, L(2 \times K), A, (\omega(i, S^2))_{i \in \overline{3N}}) = \pi(2 \times K)^* \mathcal{Z}^f(C, L, A, (\omega(i, S^2))_{i \in \overline{3N}}) $$

for a $J_{bb,tt}$-oriented $q$-tangle $L$, an integer $N$, a subset $A$ of $\overline{3N}$ whose cardinality is a multiple of 3, and a family of volume-one forms $(\omega(i, S^2))_{i \in \overline{3N}}$. Indeed, I can see no reason to believe that $\pi(2 \times \mathbb{R})^*(\widetilde{\text{hol}}_{[0,t]}(\eta(., S_{WE})))$ is the product of twice $\widetilde{\text{hol}}_{[0,t]}(\eta(., S_{WE}))$ on the two strands of $2 \times \mathbb{R}$. Unfortunately, as noticed in Remark 16.48, I do not know how to get rid of our noncanonical normalization of $\mathcal{Z}^f(C, L, A, (\omega(i, S^2))_{i \in \overline{3N}})$ and of the corresponding factors $\widetilde{\text{hol}}_{[0,t]}(\eta(., S_{WE}))$, which might not behave well under duplication.
## 17.5 Behavior of $\mathcal{Z}^f$ with respect to the co-product

The behavior of $\mathcal{Z}^f$ with respect to the coproduct described in Theorem 13.12 is justified after the statement of Theorem 13.12. Proposition 17.27 below shows how this behavior generalizes to the variants of $\mathcal{Z}^f$.

Before stating and proving it, let us prove that the compatibility between product and coproduct implies the following preliminary lemma.

**Lemma 17.26.** *Say that a map $F$ from the set $\mathcal{P}_{(3)}(\underline{3N})$ of subsets of $\underline{3N}$ whose cardinalities are multiple of 3 to a space of Jacobi diagrams is group-like if*

$$ \Delta(F(B)) = \sum_{(B_1, B_2) \in P_2(B)} \frac{|B_1|! |B_2|!}{|B|!} F(B_1) \otimes F(B_2) $$

for any element $B$ of $\mathcal{P}_{(3)}(\underline{3N})$.

Let $F$ and $G$ be two maps from $\mathcal{P}_{(3)}(\underline{3N})$ to spaces $A_F$ and $A_G$ of Jacobi diagrams, such that there is a product from $A_F \times A_G$ to a space of Jacobi diagrams $A_{FG}$.<fnref n="4" /> If $F$ and $G$ are group-like, then $(FG)_{\sqcup}$ is group-like, too.

**PROOF:** Let $A \in \mathcal{P}_{(3)}(\underline{3N})$. We have

$$ \begin{aligned} \Delta((FG)_{\sqcup}(A)) &= \sum_{(B,C) \in P_2(A)} \frac{|B|! |C|!}{|A|!} \Delta(F(B)) \Delta(G(C)) \\ &= \sum_{(B_1, B_2, C_1, C_2) \in P_4(A)} \frac{|B_1|! |B_2|! |C_1|! |C_2|!}{|A|!} (F(B_1)G(C_1) \otimes F(B_2)G(C_2)), \end{aligned} $$

with

$$ (FG)_{\sqcup}(A_1) = \sum_{(B_1, C_1) \in P_2(A_1)} \frac{|B_1|! |C_1|!}{|A_1|!} F(B_1)G(C_1). $$

So we get

$$ \Delta((FG)_{\sqcup}(A)) = \sum_{(A_1, A_2) \in P_2(A)} \frac{|A_1|! |A_2|!}{|A|!} (FG)_{\sqcup}(A_1) \otimes (FG)_{\sqcup}(A_2). $$

□

<footnote n="4">⁴This assumption guarantees that the product $(FG)_{\sqcup}$ of Definition 16.10 makes sense.</footnote>
**Proposition 17.27.** For any $q$-tangle $L$ in a rational homology cylinder $C$, for any $N \in \mathbb{N}$, for any subset $A$ of cardinality $3k$ of $\underline{3N}$, and for any family $(\omega(i, S^2))_{i \in \underline{3N}}$ of volume-one forms of $S^2$, we have

$$ \Delta_k \left( \mathcal{Z}^f(C, L, A, (\omega(i, S^2))_{i \in \underline{3N}}) \right) = \sum_{j=0}^{k} \sum_{\substack{A_1 \subset A, \\ |A_1|=3j, \\ A_2 = A \setminus A_1}} \frac{|A_1|! |A_2|!}{|A|!} \mathcal{Z}^f(C, L, A_1, (\omega(i, S^2))) \otimes \mathcal{Z}^f(C, L, A_2, (\omega(i, S^2))) $$

with the coproduct maps $\Delta_n$ defined in Section 6.5.

**PROOF:** Observe that the statement of Proposition 17.27 is valid for $q$-tangles that can be represented as straight tangles with respect to a parallelization $\tau$ such that $p_1(\tau) = 0$, by Theorem 16.16. The coefficients are treated as in Lemma 16.2.

Say that a map $F$ from $\mathcal{P}_{(3)}(\underline{3N})$ to a space $\mathcal{A}_F$ of Jacobi diagrams is *cardinality-determined* if it maps any element $A$ of $\mathcal{P}_{(3)}(\underline{3N})$ to a degree $|A|/3$ element $F_{|A|/3}$ which depends only on the cardinality of $A$. The truncation $F_{\le N} = (F_k)_{k \in \mathbb{N}: k \le N}$ until degree $N$ of any element $(F_k)_{k \in \mathbb{N}}$ of $\mathcal{A}_F$ can be viewed as such a cardinality-determined map. Note that such a truncation of a group-like element is group-like in the sense of Lemma 17.26.

Let $T$ be a trivial $q$-braid (represented by a constant path) except for the framing of one of its strands $K$, which is $lk(K, K_\parallel) = 1$ instead of 0. Definition 16.17 implies $\mathcal{Z}^f(D_1 \times [0, 1], T, \cdot, (\omega(i, S^2))) = \exp(\alpha)\#_K[\emptyset]$. In particular, this expression does not depend on $(\omega(i, S^2))_{i \in \underline{3N}}$, and $\mathcal{Z}^f(D_1 \times [0, 1], T, \cdot, (\omega(i, S^2)))$ is group-like. Similarly, $\exp_{\le N}(-\frac{1}{4}p_1(\tau)\beta(\cdot))_{\sqcup}$ is group-like. Therefore, Lemma 17.26 allows us to conclude the proof of Proposition 17.27 for framed tangles with injective bottom and top configurations. Use Lemma 16.33 to conclude for general $q$-tangles. $\square$

**Proposition 17.28.** Let $(\omega(i, S^2))_{i \in \underline{3N}}$ be a family of volume-one forms of $S^2$. Let $C$ be a rational homology cylinder. Let $L$ be a $J_{bb,tt}$-oriented $q$-tangle of $C$. Let $\check{\mathcal{Z}}^f(C, L, \cdot, (\omega(i, S^2)))$ denote the projection of $\mathcal{Z}^f(C, L, \cdot, (\omega(i, S^2)))$ on $\check{\mathcal{A}}(\mathcal{L})$. Then we have

$$ \mathcal{Z}^f(C, L, \cdot, (\omega(i, S^2))_{i \in \underline{3N}}) = \left( \check{\mathcal{Z}}^f(C, L, \cdot, (\omega(i, S^2))_{i \in \underline{3N}}) \mathcal{Z}^f_{\le N}(C, \emptyset) \right)_{\sqcup} $$

**PROOF:** $\mathcal{Z}^f(C, \emptyset, \cdot, (\omega(i, S^2))_{i \in \underline{3N}}) = \mathcal{Z}(R(C))$ is independent of $(\omega(i, S^2))_{i \in \underline{3N}}$. See Theorems 7.40 and 16.16. $\square$
## 17.6 A proof of universality

In this section, we apply the properties of $\mathcal{Z}^f$ to study its variation under crossing changes and generalize Theorem 6.9.

Define a singular tangle representative with $n$ double points to be an oriented 1-dimensional manifold $L(\mathcal{L})$ immersed in $\mathcal{C}$ such that

* the boundary of $L(\mathcal{L})$ sits in the interior of $D_1 \times \{0, 1\}$,

* $L(\mathcal{L})$ meets a neighborhood $N(\partial(D_1 \times [0, 1]))$ as vertical segments, and

* the only singular points of the immersion are $n$ double points $\times$, for which the directions of the two meeting branches generate a tangent plane.

Define a singular tangle with $n$ double points to be an equivalence class of such representatives under the equivalence relation defined as in the nonsingular case in Definition 12.15, by adding the adjective singular.

Extend the invariant $\mathcal{Z}$ to unframed singular tangles by the local rule

$$ \mathcal{Z}(\mathcal{X}) = \mathcal{Z}(\mathcal{X}) - \mathcal{Z}(\mathcal{X}). $$

This local rule relates the invariants $\mathcal{Z}$ of three singular tangles that coincide outside the represented ball and are as in the pictures in this ball.

Define the chord diagram $\Gamma_C(L)$ associated to an unframed singular tangle $L$ with $n$ double points to be the following diagram on the domain $\mathcal{L}$ of $L$. The diagram $\Gamma_C(L)$ has $2n$ vertices. Its vertices are univalent and located at the preimages of the double points. It has $n$ edges, one between each pair of preimages of a double point. These edges are called *chords*. (The chords are attached on the left-hand side of the oriented domain $\mathcal{L}$, when orientations of univalent vertices are needed, as in Definition 6.16.)

**Notation 17.29.** Respectively denote the images of $\mathcal{Z}$ and of $\tilde{p} \circ \mathcal{Z} = \tilde{\mathcal{Z}}$ under the quotient by the $1T$-relation by $\overline{\mathcal{Z}}$ and $\tilde{\mathcal{Z}}$. (So $\overline{\mathcal{Z}}(L)$ and $\tilde{\mathcal{Z}}(L)$ respectively belong to $\mathcal{A}(\mathcal{L})/(1T)$ and $\tilde{\mathcal{A}}(\mathcal{L})/(1T)$.)

In this section, we prove the following theorem, which is a generalization of Theorem 6.9 from knots to links and tangles.

**Theorem 17.30.** Let $n$ be a natural number. For any unframed singular tangle $L$ with $n$ double points in a rational homology cylinder $\mathcal{C}$, the expansion $\overline{\mathcal{Z}}_{\le n-1}(L)$ up to degree $n-1$ of $\overline{\mathcal{Z}}(L)$ vanishes, and its expansion $\overline{\mathcal{Z}}_{\le n}(L)$ up to degree $n$ is equal to

$$ \overline{\mathcal{Z}}_{\le n}(L) = [\Gamma_C(L)] \\ \text{in } \mathcal{A}(\mathcal{L})/(1T). \text{ So we have } \overline{\mathcal{Z}}_{\le n}(L) = [\Gamma_C(L)] \text{ in } \tilde{\mathcal{A}}(\mathcal{L})/(1T). $$
Theorem 6.9 and its proof, whose easiest part is presented in Section 6.2, generalize to k-component oriented links, with numbered components, to produce the following corollary to Theorem 17.30. The isomorphism of the corollary was first shown by Dror Bar-Natan and Maxim Kontsevich [BN95a].

**Corollary 17.31.** With the notation of Section 6.1, for any $(k, n) \in \mathbb{N}^2$, $\bar{\mathcal{Z}}_{\le n}$ induces an isomorphism from $\mathcal{F}_{n}(K_k; \mathbb{Q})/\mathcal{F}_{n+1}(K_k; \mathbb{Q})$ to $\check{\mathcal{A}}(\bigsqcup_{i=1}^k (S^1)_i)/(1T)$, where $(S^1)_i$ is the copy of $S^1$ associated to the $i^{th}$ component of a link.

□

In order to prove Theorem 17.30, we first define framed singular tangles and extend $\mathcal{Z}^f$ to these tangles. A *parallelization* of a singular tangle is an isotopy class of parallels as in the nonsingular case, with the same restrictions near the boundary, where the parallel of a neighborhood of a double point is on one side of the tangent plane of the double point $\times$. Recall that there are two ways of desingularizing $\times$, the *positive* one for which $\times$ is replaced by $\times$, and the *negative* one for which $\times$ is replaced by $\times$. In particular, every desingularization of such a singular tangle gets a natural parallelization from the parallelization of the singular tangle. Locally, the parallel of each branch is well-defined.

In general, we define the self-linking number of a component of a singular framed tangle as before, where the *components* of a singular framed tangle are in natural one-to-one correspondence with the components of its domain. Let $p$ be a double point for which both branches belong to a component $K_j$. Let $L(p, +)$ and $L(p, -)$ respectively denote the positive and negative desingularizations of $L$ at $p$. Then the self-linking numbers of $K_j$ in $L$ and in these two desingularizations are related by

$$lk((K_j, K_{j\parallel}) \subset L(p, +)) = lk((K_j, K_{j\parallel}) \subset L) + 1$$

$$\text{and } lk((K_j, K_{j\parallel}) \subset L(p, -)) = lk((K_j, K_{j\parallel}) \subset L) - 1.$$

We formally extend $\mathcal{Z}^f$ to (framed) singular $q$-tangles by the formula

$$\mathcal{Z}^f(\mathcal{X}) = \mathcal{Z}^f(\mathcal{X}) - \mathcal{Z}^f(\mathcal{X}),$$

where the parallels of the three tangles are supposed to be behind and to match on the boundary of the ball.

As an example, we have

$$\mathcal{Z}^f(\mathcal{X}) = \mathcal{Z}^f(\mathcal{X}) - \mathcal{Z}^f(\mathcal{X}) = \mathcal{Z}^f(\mathcal{X}) - \mathcal{Z}^f(\mathcal{X}),$$
where the endpoints of the tangles are assumed to lie on the real line. So Lemmas 12.18 and 12.19 yield

$$ \mathcal{Z}_{\le 1}^{f} \left( \begin{array}{c} \text{圈} \\ \text{圈} \end{array} \right) = \left[ \begin{array}{c|c} \text{圈} & \text{圈} \\ \text{圈} & \text{圈} \end{array} \right]. $$

Note that $\mathcal{Z}^f$ is now a functor on the category of singular $q$-tangles, which satisfies the cabling property and the duplication properties of Theorem 13.12 provided that the components involved in a double point are not duplicated.

**Proposition 17.32.** Let $n$ be a natural number. For any singular $q$-tangle $L$ with $n$ double points, we have

$$ \mathcal{Z}_{\le n}^{f}(L) = [\Gamma_C(L)]. $$

**PROOF:** In the proof below, we evaluate the lowest degree part that does not vanish in $\mathcal{Z}^f(L)$ for various singular $q$-tangles. Note that this part is unchanged when such a singular $q$-tangle is multiplied by a nonsingular $q$-tangle (except for the modification of the domains) since the lowest degree part that does not vanish for a nonsingular $q$-tangle is the class of the empty diagram. In particular, the lowest degree nonvanishing part is independent of the bottom and top configurations of our $q$-tangles, which will not be specified.

Applying the cabling property of $\mathcal{Z}^f$ to the following cable of a trivial braid with three strands yields

$$ \mathcal{Z}_{\le 1}^{f} \left( \begin{array}{c|c} \text{圈} & \text{圈} \end{array} \right) = \begin{array}{c} \text{圈} \\ \text{圈} \end{array}. $$

We thus get

$$ \mathcal{Z}_{\le 1}^{f} \left( \begin{array}{c} \text{圈} \\ \text{圈} \end{array} \right) = \begin{array}{c} \text{圈} \\ \text{圈} \end{array}. $$

by functoriality as desired for such a $q$-tangle.

Starting with a trivial braid and successively cabling some of its strands by replacing $n$ of them by $\begin{array}{c} \text{圈} \\ \text{圈} \end{array}$, we find

$$ \mathcal{Z}_{\le n}^{f} \left( \begin{array}{c|c} \text{圈} & \text{圈} \\ \text{圈} & \text{圈} \end{array} \right) = \begin{array}{c|c} \text{圈} & \text{圈} \\ \text{圈} & \text{圈} \end{array}. $$
(Here, we first apply the cabling property to the first strand of a trivial braid $\mathbf{1}_{n+k}$ and regard the resulting tangle $T_1$ as a product $\mathbf{1}_{n+k-1}T_1$, where $\mathbf{1}_{n+k-1}$ is a trivial braid. We next apply the cabling property to the first strand of $\mathbf{1}_{n+k-1}$. We keep going to find the obtained formula with the help of the functoriality property.)

Since every singular $q$-tangle with $n$ double points can be written as a product of a $q$-tangle as above and a nonsingular $q$-tangle by moving the double points below, the proposition follows. $\square$

We are ready to deduce Theorem 17.30 from Proposition 17.32.

**PROOF OF THEOREM 17.30:** Assume that the singular $q$-tangle $L$ with $n$ double points has $k$ components $K_i$ for $i = 1, \dots, k$.

The case $n = 0$ is obvious. Assume that $n = 1$. Let $s_i$ denote the self-linking number of $K_i$ in $L$. Let $L^+$ be the positive desingularization of $L$, and let $L^-$ be the negative desingularization of $L$. Let $s_i^+$ (resp. $s_i^-$) denote the self-linking number of the $i^{th}$ component in $L^+$ (resp. in $L^-$). Recall that $\bar{Z}$ is the image of $Z$ under the quotient by the $1T$-relation. Also recall

$$Z(L^+) = \prod_{j=1}^{k} (\exp(-s_j^+ \alpha) \#_j) Z^f(L^+).$$

If the two strands involved in the double point belong to two distinct components, then we have $s_j^+ = s_j^- = s_j^-$ for any $j \in \underline{k}$ and

$$Z(\times) - Z(\times) = \prod_{j=1}^{k} (\exp(-s_j\alpha)\#_j) (Z^f(\times) - Z^f(\times)).$$

So the result follows since the lowest degree part of any $\exp(-s_j\alpha)$ is the class of the empty diagram.

If the two strands involved in the double point belong to the same component $K_i$, then we have $s_i^+ = s_i^- + 2$ and

$$Z(\times) - Z(\times) = \prod_{j=1}^{k} (\exp(-s_j^+ \alpha) \#_j) (Z^f(\times) - Z^f(\times)) \\ + \left( \prod_{j=1}^{k} (\exp(-s_j^+ \alpha) \#_j) - \prod_{j=1}^{k} (\exp(-s_j^- \alpha) \#_j) \right) Z^f(\times),$$

where the $(\prod_{j=1}^k (\exp(-s_j^+ \alpha) \#_j) - \prod_{j=1}^k (\exp(-s_j^- \alpha) \#_j))$ “factor” begins with its degree one part, which is $-2\alpha_1 = -[\hat{\zeta}]$. We get

$$\bar{Z}_{\le 1}(\times) - \bar{Z}_{\le 1}(\times) = [\hat{\zeta}]$$
as desired. Note that this equality would be wrong if $\overline{\mathcal{Z}}$ were replaced by $\mathcal{Z}$, that is, without moding out by $1T$.

Let us now conclude the proof by induction on $n$. Assume that the result is known for singular $q$-tangles with less than $n$ double points. Let $P$ denote the set of double points of $L$. For $i \in \underline{k}$, let $P_i$ denote the set of double points for which both branches belong to $K_i$. For a subset $I$ of $P$, the $q$-tangle obtained from $L$ by performing negative desingularizations on double points of $I$ and positive ones on double points of $P \setminus I$ is denoted by $L_I$, and $s_{j,I}$ denotes the self-linking number of the component $K_{j,I}$ in $L_I$.

Then $\overline{\mathcal{Z}}(L)$ is equal to

$$ \sum_{I \subseteq P} (-1)^{|I|} \overline{\mathcal{Z}}(L_I) = \sum_{I \subseteq P} (-1)^{|I|} \prod_{j=1}^{k} (\exp(-s_{j,I}\alpha)\#_j) \mathcal{Z}^f(L_I) \mod 1T \\ = \sum_{I \subseteq P} (-1)^{|I|} \left( \prod_{j=1}^{k} (\exp(-s_{j,I}\alpha)\#_j) - \prod_{j=1}^{k} (\exp(-s_{j,\emptyset}\alpha)\#_j) \right) \mathcal{Z}^f(L_I) \\ + \prod_{j=1}^{k} (\exp(-s_{j,\emptyset}\alpha)\#_j) \sum_{I \subseteq P} (-1)^{|I|} \mathcal{Z}^f(L_I) \mod 1T. $$

The lowest degree term of the last line is $[\Gamma_C(L)]$. So it suffices to prove that the previous line does not contain terms of degree less than $n+1$ (mod $1T$). This previous line can be rewritten as

$$ T_2 = \sum_{I \subseteq P} (-1)^{|I|} \left( 1 - \prod_{j=1}^{k} \left( \exp\left( (s_{j,I} - s_{j,\emptyset})\alpha \right) \#_j \right) \right) \mathcal{Z}(L_I) \mod 1T, $$

with $s_{j,I} - s_{j,\emptyset} = -2|I \cap P_j| = \sum_{p \in I \cap P_j} (-2)$. So $T_2$ can be rewritten as

$$ T_2 = \sum_{I \subseteq P} (-1)^{|I|} \left( 1 - \prod_{j=1}^{k} \left( \prod_{p \in I \cap P_j} \exp(-2\alpha_p) \#_j \right) \right) \overline{\mathcal{Z}}(L_I), $$

where $\alpha_p$ is a copy of $\alpha$.

Let $F$ be the set of maps $f: \cup_{j=1}^k P_j \to \mathbb{N}$ such that $f(\cup_{j=1}^k P_j) \neq \{0\}$. Let $F(I)$ be the set of maps $f$ of $F$ such that $f(p) = 0$ for any $p \notin I$. Then we have

$$ 1 - \prod_{j=1}^{k} \left( \prod_{p \in I \cap P_j} \exp(-2\alpha_p) \#_j \right) = - \sum_{f \in F(I)} D_f $$

with

$$ D_f = \prod_{j=1}^{k} \left( \left( \prod_{p \in P_j} \frac{(-2\alpha)^{f(p)}}{f(p)!} \right) \#_j \right). $$
Now, $T_2$ can be rewritten as

$$ \sum_{f \in F} D_f \sum_{I: I \subseteq P, f \in F(I)} (-1)^{|I|+1} \bar{Z}(L_I). $$

Set $N(f) = \{p : p \in \bigcup_{j=1}^k P_j, f(p) \neq 0\}$. Let $L(N(f), -)$ denote the singular $q$-tangle with $(n - |N(f)|)$ double points obtained from $L$ by desingularizing the double points of $N(f)$ in a negative way. Since the condition $f \in F(I)$ is equivalent to the condition $N(f) \subseteq I$, we get

$$ \sum_{I: I \subseteq P, f \in F(I)} (-1)^{|I|+1} \bar{Z}(L_I) = (-1)^{|N(f)|+1} \bar{Z}(L(N(f), -)). $$

Note that $(-2\alpha)^{f(p)}$ is of degree at least 3 as soon as $f(p) \neq 0$ (when working modulo 1T since $\alpha_1 = \alpha_2 = 0$). So the degree of $D_f$ is at least $3|N(f)|$. By induction, the degree of $\bar{Z}(L(N(f), -))$ is $n - |N(f)|$. Therefore, the parts of $T_2$ of degree at most $n$ vanish. $\square$
<figure><img src="image_138.png" /></figure>
Part IV

Universality
<figure><img src="image_139.png" /></figure>
# Chapter 18

## The main universality statements and their corollaries

### 18.1 Universality with respect to Lagrangian-preserving surgeries

Let us recall some definitions quickly surveyed in the book introduction.

**Definition 18.1.** An integer (resp. rational) homology handlebody of genus $g$ is a compact oriented 3-manifold $A$ with the same integral (resp. rational) homology as the usual solid handlebody $H_g$ of Figure 1.1. The Lagrangian $\mathcal{L}_A$ of a compact 3-manifold $A$ is the kernel of the map induced by the inclusion from $H_1(\partial A; \mathbb{Q})$ to $H_1(A; \mathbb{Q})$.

**Exercise 18.2.** Show that if $A$ is a rational homology handlebody of genus $g$, then $\partial A$ is a connected genus $g$ surface. (See Appendix A.1, where some basic properties of homology are recalled.)

In Figure 1.1, the Lagrangian of $H_g$ is freely generated by the classes of the curves $a_i$.

**Definition 18.3.** An integral (resp. rational) Lagrangian-Preserving (or LP) surgery $(A'/A)$ is the replacement of an integer (resp. rational) homology handlebody $A$, embedded in the interior of a 3-manifold $M$, by another such $A'$, whose boundary $\partial A'$ is identified with $\partial A$ by an orientation-preserving diffeomorphism sending $\mathcal{L}_A$ to $\mathcal{L}_{A'}$. The manifold $M(A'/A)$ obtained by such an LP-surgery is

$$M(A'/A) = (M \setminus \mathrm{Int}(A)) \cup_{\partial A} A'$$

(This only defines the topological structure of $M(A'/A)$, but $M(A'/A)$ is equipped with its unique smooth structure.)
We present an interesting example of an integral Lagrangian-preserving surgery in Subsection 18.3.1. The Matveev Borromean surgery of Section 18.4 is another example of integral LP-surgery.

**Lemma 18.4.** If $(A'/A)$ is an integral (resp. rational) LP-surgery, then the homology of $M(A'/A)$ with $\mathbb{Z}$-coefficients (resp. with $\mathbb{Q}$-coefficients) is canonically isomorphic to $H_*(M; \mathbb{Z})$ (resp. to $H_*(M; \mathbb{Q}))$. If $M$ is a $\mathbb{Q}$-sphere, if $(A'/A)$ is a rational LP-surgery, and if $(J, K)$ is a two-component link of $M \setminus A$, then the linking number of $J$ and $K$ in $M$ and the linking number of $J$ and $K$ in $M(A'/A)$ coincide.

**PROOF:** Exercise.

<figure><img src="image_140.png" /></figure>

Let $(A'/A)$ be a rational LP-surgery in a punctured rational homology sphere $\tilde{R}$. Let

$$ \partial_{MV}: H_2(A \cup_{\partial A} -A'; \mathbb{Q}) \to \mathcal{L}_A $$

be the morphism that maps the class of a closed surface in the closed 3-manifold $(A \cup_{\partial A} -A')$ to the boundary of its intersection with $A$. The Mayer-Vietoris long exact sequence (see Theorem A.11) shows that the above canonical morphism $\partial_{MV}$ is an isomorphism. This isomorphism carries the algebraic triple intersection of surfaces to a trilinear antisymmetric form $\mathcal{I}_{AA'}$ on $\mathcal{L}_A$. Explicitly, we have

$$ \mathcal{I}_{AA'}(a_i, a_j, a_k) = \langle \partial_{MV}^{-1}(a_i), \partial_{MV}^{-1}(a_j), \partial_{MV}^{-1}(a_k) \rangle_{A \cup -A'}. $$

Let $(a_1, a_2, \dots, a_g)$ be a basis of $\mathcal{L}_A$ and let $z_1, \dots, z_g$ be (curves representing) homology classes of $\partial A$ such that the system $(z_1, z_2, \dots, z_g)$ is dual to $(a_1, a_2, \dots, a_g)$ with respect to $\langle, \rangle_{\partial A}$:

$$ \langle a_i, z_j \rangle_{\partial A} = \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j. \end{cases} $$

Note that $(z_1, \dots, z_g)$ is a basis of $H_1(A; \mathbb{Q})$.

Represent $\mathcal{I}_{AA'}$ by the following combination $T(\mathcal{I}_{AA'})$ of tripods whose three univalent vertices form an ordered set

$$ T(\mathcal{I}_{AA'}) = \sum_{(i,j,k) \in \mathbb{g}^3 : i < j < k} \mathcal{I}_{AA'}(a_i, a_j, a_k) \stackrel{z_k}{\leftarrow} \stackrel{z_j}{\leftarrow} \stackrel{z_i}{\leftarrow} $$

where the tripods are considered up to the relations

$$ \stackrel{z}{\leftarrow}_x y = \stackrel{x}{\leftarrow}_y z = - \stackrel{z}{\leftarrow}_x y = - \stackrel{z}{\leftarrow}_y x. $$
**Notation 18.5.** Let $G$ be a graph with $2k$ oriented trivalent vertices and with univalent vertices. Assume that the univalent vertices of $G$ are decorated with disjoint curves of a punctured $\mathbb{Q}$-sphere $\tilde{R}$. Let $P(G)$ be the set of partitions of the set of univalent vertices of $G$ into disjoint pairs.

For $p \in P(G)$, identifying the two vertices of each pair provides a vertex-oriented trivalent Jacobi diagram $\Gamma_p$. Let $\ell(p)$ be the product, over the disjoint pairs of $p$, of the linking numbers of the curves corresponding to the two vertices in a pair. We get an element $[\ell(p)\Gamma_p]$ of $\mathcal{A}_k(\emptyset)$. Define

$$ \langle\langle G \rangle\rangle = \sum_{p \in P(G)} [\ell(p) \Gamma_p]. $$

The contraction $\langle\langle\cdot\rangle\rangle$ is linearly extended to linear combinations of graphs. The disjoint union of combinations of graphs is bilinear.

The universality theorem with respect to Lagrangian-preserving surgeries is the following one. It was proved in [Les04b] for the invariant $\mathcal{Z}$ of rational homology spheres. The statement below is more general since it applies to the invariant $\mathcal{Z}$ of Theorem 12.7, which satisfies the properties stated in Theorem 13.12. Nevertheless, its proof reproduced in this book, is identical to the proof of the preprint [Les04b], except for some editorial improvements. ([Les04b] has never been submitted for publication.)

**Theorem 18.6.** Let $L$ be a $q$-tangle representative in a rational homology cylinder $C$. Let $x$ be a positive integer. Let $\bigsqcup_{i=1}^x A^{(i)}$ be a disjoint union of rational homology handlebodies embedded in $C \setminus L$. Let $(A^{(i)\prime}/A^{(i)})$ be rational $LP$-surgeries in $C$. For a subset $I$ of $\underline{x}$, let $C_I = C((A^{(i)\prime}/A^{(i)})_{i \in I})$ be the rational homology cylinder obtained from $C$ by performing the $LP$-surgeries that replace $A^{(i)}$ by $A^{(i)\prime}$ for $i \in I$. Set $X = [C, L; (A^{(i)\prime}/A^{(i)})_{i \in \underline{x}}]$ and

$$ \mathcal{Z}_n(X) = \sum_{I \subseteq \underline{x}} (-1)^{x+|I|} \mathcal{Z}_n(C_I, L). $$

If $2n < x$, then $\mathcal{Z}_n(X)$ vanishes. If $2n = x$, then we have

$$ \mathcal{Z}_n(X) = \left[ \left\langle \left\langle \bigsqcup_{i \in \underline{x}} T(\mathcal{I}_{A^{(i)} A^{(i)\prime}}) \right\rangle \right\rangle \right]. $$

Before proving Theorem 18.6, we discuss some of its consequences and variants. Section 18.3 shows that Theorem 18.6 yields a direct proof of a surgery formula for the Theta invariant, as in [Les09, Section 9]. The article [Les09] presents many other surgery formulae implied by Theorem 18.6, which are not reproduced in this book.
Section 18.4 shows how Theorem 18.6 implies that $\mathcal{Z}$ restricts to a uni-
versal finite type invariant of integer homology 3-spheres. In Section 18.5, we
review the Moussard classification of finite type invariants of rational homol-
ogy 3-spheres [Mou12], and we show how $\mathcal{Z}$ can be augmented to provide a
universal finite type invariant of rational homology 3-spheres, too, following
an idea of Gwénaël Massuyeau. Section 18.6 shows how Theorem 18.6 also
implies that the invariant $\frac{1}{6}\Theta$ is the Casson–Walker invariant. Assuming this
identification, the surgery formula of Section 18.3 is nothing but a conse-
quence of the Casson–Walker surgery formula of [Wal92]. So a reader who
does not need examples can skip Section 18.3, at first.

We sketch the proof of Theorem 18.6 in Section 18.7. We complete the
details of the proof in the following two chapters.

Theorem 18.6 and the universality theorems 6.9 and 17.30 for knots or
tangles are put together in Theorem 18.35, which generalizes all of them.

Section 18.8 shows how the main ingredients of the proof of Theorem 18.6
also lead to Theorem 18.39, which allows us to compute the degree 2 part
of $\tilde{Z}$, for any null-homologous knot, in Theorem 18.43, with the help of the
contents of Section 18.3.

Section 18.2 below gives some background about Dehn surgeries. We use
this background in Sections 18.3 and 18.4.

18.2 On Dehn surgeries

In this section, we define the manifold $R_{(K;p/q)}$ obtained by $p/q$-surgery on a
$\mathbb{Q}$-sphere $R$ along a knot $K$. We also introduce the lens spaces $L(p, q)$ and
give some examples of surgeries on links used in Section 18.4.

Let $K$ be a knot in a 3-manifold $M$, and let $N(K)$ be a tubular neighborhood of $K$. The *exterior* $E(K)$ of $K$ is the closure of the complement of $N(K)$ in $M$. Let $\mu$ be a nonseparating simple closed curve of the boundary $\partial N(K)$ of $N(K)$. The manifold obtained from $M$ by *Dehn surgery* on $K$ with respect to $\mu$ is the union $E(K) \cup_{\partial N(K)} T$ of $E(K)$ and a solid torus $T$, where $E(K)$ and $T$ are glued along $\partial N(K)$ by an orientation-reversing homeomorphism from $\partial T$ to $\partial N(K)$, which maps a meridian of $T$ to $\mu$. The result is then smoothed in a standard way. (Since the gluing of $T$ can be achieved by gluing a meridian disk of $T$ along $\mu$, thickening it, and gluing a 3-dimensional ball to the resulting boundary, this surgery operation is well-defined.<fnref n="1" />)

<footnote n="1">¹The operation of gluing a 3-ball $B^3$ along $S^2$ is well-defined, because any homeomor-
phism $f$ from $S^2$ to $S^2$ extends to $B^3$ as the homeomorphism that maps $tx$ to $tf(x)$ for
$t \in [0, 1]$ and $x \in S^2$.</footnote>
**Example 18.7.** As the reader can check, the manifold obtained by Dehn surgery on the unknot U of $S^3$ with respect to its meridian $m(U)$ is $S^2 \times S^1$.

Let $K$ be a knot in a rational homology sphere. If $K$ is null-homologous, then $K$ has a unique parallel $\ell(K)$ such that $lk(K, \ell(K)) = 0$. This parallel is called the preferred longitude of $K$. (A parallel of $K$ is also called a longitude of $K$.) Let $\mu$ be a simple closed curve in the boundary $\partial N(K)$ of a tubular neighborhood of $K$ such that $\mu$ does not separate $\partial N(K)$. The class of the curve $\mu$ in $H_1(\partial N(K))$ may be expressed as $pm(K) + q\ell(K)$, where $m(K)$ is the meridian of $K$. The coefficient of the Dehn surgery along $K$ with respect to $\mu$ is $p/q$. We refer to this Dehn surgery as the $p/q$-surgery on $K$. This coefficient $p/q$ may be expressed as $lk(K, \mu)/\langle m(K), \mu \rangle_{\partial N(K)}$. The $p/q$-surgery along a non-necessarily null-homologous knot $K$ in a rational homology 3-sphere $R$ is the Dehn surgery with respect to a nonseparating simple closed curve $\mu$ of $\partial N(K)$ such that

$$ \frac{lk(K, \mu)}{\langle m(K), \mu \rangle_{\partial N(K)}} = \frac{p}{q}. $$

Let $R_{(K;p/q)}$ denote the result of a $p/q$-Dehn surgery on $R$ along $K$. As shown in Example 18.7, we have $S^3_{(U;0)} = S^2 \times S^1$.

According to a theorem independently proved by Raymond Lickorish [Lic62] and Andrew Wallace [Wal60] in 1960, every closed oriented 3-manifold can be obtained from $S^3$ by surgery along a link of $S^3$ whose components are equipped with integers. (Surgeries are performed simultaneously along all the components of the link.) In [Rou85], Colin Rourke gave a quick and elementary proof of this result.

**Examples 18.8.** As the reader can check, the manifold obtained by Dehn surgery on the trivial link of $S^3$ with $g$ components, all equipped with the coefficient 0, is the connected sum of $g$ copies of $S^2 \times S^1$. Furthermore, this connected sum is homeomorphic to the manifold $H_g \cup_{1_{\partial H_g}} (-H_g)$.

<figure><img src="image_141.png" /><figcaption>Figure 18.1: Borromean link</figcaption></figure>

As a more challenging exercise, the reader can prove the following fact. The manifold obtained by Dehn surgery on the *Borromean link* of $S^3$, represented in Figure 18.1, whose components are equipped with the coefficient 0, is diffeomorphic to $(S^1)^3$. A hint can be found in [Thu78, Example 13.1.5].
Let $p$ and $q$ be two coprime integers, $p > 0$. View $S^3$ as the unit sphere of $\mathbb{C}^2$. The lens space $L(p, q)$ is the quotient of $S^3$ by the action of $\mathbb{Z}/p\mathbb{Z}$ on $S^3$, where the generator $[1]$ of $\mathbb{Z}/p\mathbb{Z}$ acts on a unit vector $(z_1, z_2)$ of $\mathbb{C}^2$ by mapping it to $(\exp(2i\pi/p)z_1, \exp(2i\pi q/p)z_2)$.

Let us study more surgeries along *unknots* or *trivial knots*, which are knots that bound an embedded disk, and prove the following well-known lemma.

**Lemma 18.9.** Let $k$ be an integer. Let $U$ be a trivial knot. Then $S^3_{(U;1/k)} \cong S^3$. More generally, for any pair $(a, b)$ of coprime integers such that $a > 0$, $S^3_{(U;a/(b+ka))}$ is diffeomorphic to $S^3_{(U;a/b)}$, and we have

$$S^3_{(U;a/b)} = L(a, -b).$$

If $U$ is a trivial knot of a 3-manifold $M$, then $M_{(U;a/b)}$ is the connected sum $M\#L(a, -b)$.

**PROOF:** The exterior $E$ of the unknot $U$ in $S^3$ is a solid torus whose meridian $m(E)$ is the preferred longitude $\ell(U)$ of $U$. The meridian $m(U)$ is a longitude $\ell(E)$ of $E$. Performing $(a/b)$-surgery along $U$ on $S^3$ amounts to gluing a solid torus with meridian

$$\mu = am(U) + b\ell(U) = bm(E) + a\ell(E)$$

to $E$, where $\langle m(E), \mu \rangle_{\partial E} = a$ and $\langle \mu, \ell(E) \rangle_{\partial E} = b$. The manifold $S^3_{(U;a/b)}$ is the union of two solid tori $E$ and $T$ glued by a homeomorphism from $(-\partial T)$ to $\partial E$ mapping the meridian of $T$ to a curve $\mu$ as above.

Let $k$ be an integer. Then $(\ell(E) - km(E))$ is another longitude of $E$. This shows that for any coprime integers $a$ and $b$, the manifold $S^3_{(U;a/(b+ka))}$ is diffeomorphic to $S^3_{(U;a/b)}$. In particular, we have $S^3_{(U;1/k)} \cong S^3$.

For a trivial knot $U$ in a 3-manifold $M$, the manifold $M_{(U;a/b)}$ is the connected sum of $S^3_{(U;a/b)}$ and $M$. (The connected sum replaces a ball in the interior of the above solid torus $E$ by the exterior of a ball containing $U$ in $M$.)

Below, $S^3$ is viewed as the sphere of $\mathbb{C}^2$ with radius $\sqrt{2}$. The action of $\mathbb{Z}/p\mathbb{Z}$ on $S^3$ that defines the lens space $L(p, q)$ preserves the solid torus $|z_1| \le |z_2|$ and the solid torus $|z_1| \ge |z_2|$. Let $E$ be the quotient of the second torus. We have

$$E = \frac{\left\{ \left( \exp\left(\frac{2i\pi t}{p}\right) \sqrt{2 - |z_2|^2}, z_2 \right) : t \in [0, 1], z_2 \in \mathbb{C}, |z_2| \le 1 \right\}}{\left( \sqrt{2 - |z_2|^2}, z_2 \right) \sim \left( \exp\left(\frac{2i\pi}{p}\right) \sqrt{2 - |z_2|^2}, z_2 \exp\left(\frac{2i\pi q}{p}\right) \right)}.$$
The meridian of the solid torus E is

$$m(E) = \left\{ (1, \exp(2i\pi u)) : u \in [0, 1] \right\}$$

and the possible (homology classes of) longitudes of E are all the $\ell(E) + km(E)$ for $k \in \mathbb{Z}$, where

$$\ell(E) = \left\{ \left( \exp\left(\frac{2i\pi t}{p}\right), \exp\left(\frac{2i\pi tq}{p}\right) \right) : t \in [0, 1] \right\}.$$

The boundary of E is oriented as $(-S^1) \times S^1$. The quotient of the solid torus $|z_1| \le |z_2|$ is also a solid torus whose meridian is

$$\begin{aligned} \mu &= \left\{ (\exp(2i\pi s), 1) : s \in [0, 1] \right\} \\ &= \cup_{j=0}^{p-1} \left\{ \left( \exp\left(\frac{2i\pi(t+j)}{p}\right), 1 \right) : t \in [0, 1] \right\} \\ &= \cup_{j=0}^{p-1} \left\{ \left( \exp\left(\frac{2i\pi t}{p}\right), \exp\left(-\frac{2i\pi qj}{p}\right) \right) : t \in [0, 1] \right\}. \end{aligned}$$

So we have $\langle m(E), \mu \rangle_{\partial E} = p$ and $\langle \mu, \ell(E) \rangle_{\partial E} = -q$. (There are $|q|$ pairs $(t, j)$ with $t \in [0, 1[$ and $j \in \{0, 1, \dots, p-1\}$ such that $(t+j) \in \frac{p}{q}\mathbb{Z}$.) $\square$

**Remark 18.10.** A homeomorphism from $M$ to $M_{(U;1/k)}$ can also be directly described as follows. Let $D$ be a disk bounded by $U$, and let $d$ be a smaller disk inside $D$. The disk $D$ is parametrized by the disk of radius 2 in $\mathbb{C}$, and the unit disk parametrizes $d$. The exterior $E$ of $U$ is homeomorphic to

$$(M \setminus (\mathring{D} \times [0, 2\pi])) \cup_{d \times \{0, 2\pi\}} (d \times [0, 2\pi])$$

by a homeomorphism mapping the meridian of $U$ to

$$(\{1\} \times [0, 2\pi]) \cup ([1, 2] \times \{2\pi\}) \cup (-\{2\} \times [0, 2\pi]) \cup (-[1, 2] \times \{0\}).$$

See Figure 18.2.

<figure><img src="image_142.png" /><figcaption>Figure 18.2: The gray image $m$ of $m(U)$ in $D \times [0, 2\pi]$</figcaption></figure>

The homeomorphism of $E$ restricting to $(M \setminus (\mathring{D} \times [0, 2\pi]))$ as the identity map and sending $(z, \theta) \in d \times [0, 2\pi]$ to $(z \exp(ik\theta), \theta)$ sends the above meridian $m(U)$ to a curve homologous to $m(U) + k\ell(U)$. So this homeomorphism extends to provide a homeomorphism from $M$ to $M_{(U;\frac{1}{k})}$.
a symplectic basis of $\Sigma$. For a curve $c$ of $\Sigma$, let $c^+$ denote its push-off in the direction of the positive normal to $\Sigma$. Set

$$a_2(\Sigma) = \sum_{(i,j) \in \underline{g}^2} (lk(x_i, x_j^+)lk(y_i, y_j^+) - lk(x_i, y_j^+)lk(y_i, x_j^+)).$$

Then we have

$$\Theta(R_{(K;p/q)}) = \Theta(R) - \Theta(L(p, q)) + 6 \frac{q}{p} a_2(\Sigma).$$

We will prove this proposition in Subsection 18.3.2 in these words. Thus, this proposition implies that $a_2(\Sigma)$ is an invariant of $K$. This invariant will be denoted by $a_2(K)$. It is equal to $\frac{1}{2}\Delta_K''(1)$, where $\Delta_K$ is the Alexander polynomial of $K$ normalized so that $\Delta_K(t) = \Delta_K(t^{-1})$ and $\Delta_K(1) = 1$.

**Definition 18.13.** Here is a possible quick definition of the Alexander polynomial $\Delta_K$ of the null-homologous knot $K$. Rewrite the symplectic basis $(x_i, y_i)_{i \in \underline{g}}$ as the basis $(z_j)_{j \in \underline{2g}}$ such that $z_{2i-1} = x_i$ and $z_{2i} = y_i$ for $i \in \underline{g}$. Let $V = [lk(z_j, z_k^+)]_{(j,k) \in \underline{2g}^2}$ denote the associated Seifert matrix, and let $t^V$ denote its transpose, then we have

$$\Delta_K(t) = \det (t^{1/2}V - t^{-1/2}t^V).$$

See [Ale28] or [Les96, Chapter 2] for other definitions of the Alexander polynomial, which will be mentioned later but not used in this book anymore.

**Remark 18.14.** Proposition 18.12 is also a consequence of the identification of $\Theta$ with $6\lambda_{CW}$ in Theorem 18.31, which is proved independently in Section 18.6, and of the Walker surgery formula proved in [Wal92, Theorem 5.1].

### 18.3.1 A Lagrangian-preserving surgery associated to a Seifert surface

**Definition 18.15.** Let $c(S^1)$ be a curve embedded in the interior of an oriented surface $F$, and let $c(S^1) \times [-1, 1]$ be a collar neighborhood of $c(S^1)$ in $F$. A right-handed (resp. left-handed) Dehn twist about the curve $c(S^1)$ is a homeomorphism of $F$ that coincides with the identity map of $F$ outside $c(S^1) \times [-1, 1]$ and that maps

$$(c(z), t) \in c(S^1) \times [-1, 1] \text{ to } \left(c\left(z \exp(i f(t))\right), t\right)$$

for $f(t) = \pi(t + 1)$ (resp. for $f(t) = -\pi(t + 1)$).
Let $\Sigma$ be a Seifert surface of a knot $K$ in a manifold $M$. Consider an annular neighborhood $[-3, 0] \times K$ of $(\{0\} \times K =)K = \partial\Sigma$ in $\Sigma$, a small disk $D$ inside $]-2, -1[ \times K$, and a small open disk $d$ whose closure is in the interior of $D$. Let $F = \Sigma \setminus d$. Let $h_F$ be the composition of the two left-handed Dehn twists on $F$ about $c = \partial D$ and $K_2 = \{-2\} \times K$ with the right-handed one about $K_1 = \{-1\} \times K$. See Figure 18.4.

<figure><img src="image_143.png" /><figcaption>Figure 18.4: $K$, $\Sigma$, $F$, $c$, $K_1$, and $K_2$</figcaption></figure>

View $F$ as $F \times \{0\}$ in the boundary of a handlebody $A_F = F \times [-1, 0]$ of $M$. Extend $h_F$ to a homeomorphism $h_A$ of $\partial A_F$ that is the identity map outside $F \times \{0\}$. Let $A'_F$ be a copy of $A_F$. Identify $\partial A'_F$ with $\partial A_F$ with

$$h_A: \partial A'_F \to \partial A_F.$$

Define the surgery associated to $\Sigma$ to be the surgery $(A'_F/A_F)$ associated to $(A_F, A'_F; h_A)$. If $j$ denotes the embedding from $\partial A_F$ to $M$. This surgery replaces

$$M = A_F \cup_j (M \setminus \hat{A}_F)$$

by

$$M_F = A'_F \cup_{j \circ h_A} (M \setminus \hat{A}_F).$$

**Proposition 18.16.** With the above notation, the surgery $(A'_F/A_F)$ associated to $\Sigma$ is a Lagrangian-preserving surgery with the following properties. There is a homeomorphism from $M_F$ to $M$,

* which extends the identity map of

$$M \setminus ([-3, 0] \times K \times [-1, 0]),$$

* which transforms a curve passing through $d \times [-1, 0]$ by a band sum with $K$,
* which transforms a 0-framed meridian m of K passing through d × [-1, 0], viewed as a curve of M \ Â_F (which may be expressed as ĥ_A⁻¹(m) in A'_F), to a 0-framed copy of K isotopic to the framed curve ĥ_A⁻¹(m) of Figure 18.5 (with the framing induced by ∂A_F).

<figure><img src="image_144.png" /><figcaption>Figure 18.5: m and h_A^{-1}(m)</figcaption></figure>

PROOF: Observe that $h_A|_{(F \times \{-1,0\}) \cup (\partial\Sigma \times [-1,0])}$ extends to $\Sigma \times [-1,0]$ as

$$h: \begin{array}{ccc} \Sigma \times [-1,0] & \to & \Sigma \times [-1,0] \\ (\sigma, t) & \mapsto & h(\sigma, t) = (h_t(\sigma), t), \end{array}$$

where $h_{-1}$ is the identity map of $\Sigma$, $h_0$ is the extension of $h_F$ by the identity map on $d$,<fnref n="3" /> $h_t$ coincides with the identity map outside [-5/2, -1/2] × K(S¹), and $h_t$ is defined as follows on [-5/2, -1/2] × K(S¹).

* When $t \le -1/2$, $h_t$ coincides with the identity map $h_{-1}$ outside the disk D, whose elements are written as $D(z \in \mathbb{C})$, with $|z| \le 1$. The elements of d are the $D(z)$ for $|z| < 1/2$. On D, $h_t$ describes the isotopy between the identity map and the left-handed Dehn twist about $\partial D$ located on $\{D(z) : 1/2 \le |z| \le 1\}$. We have

$$h_t(z \in D) = z \exp(i\pi(2t + 2)4(|z| - 1)) \quad \text{if } |z| \ge 1/2$$

$$h_t(z \in D) = z \exp(-2i\pi(2t + 2)) \quad \text{if } |z| \le 1/2.$$

<footnote n="3">³This extension is isotopic to the identity map of $\Sigma$.</footnote>
• When $t \ge -1/2$, $h_t$ describes the following isotopy between $h_{-1/2}$ and the composition $h_0$ of $h_{-1/2}$ with the left-handed Dehn twist about $K_2$ and the right-handed Dehn twist about $K_1$, where the first twist is supported on $[-5/2, -2] \times K(S^1)$ and the second one is supported on $[-1, -1/2] \times K(S^1)$:

$$h_t(u, K(z)) = \left( u, K \left( z \exp \left( i(2t+1)(4\pi(u+5/2)) \right) \right) \right) \quad \text{if } -5/2 \le u \le -2,$$

$$h_t(u, K(z)) = h_{-1/2} \left( u, K \left( z \exp \left( i(2t+1)(2\pi) \right) \right) \right) \quad \text{if } -2 \le u \le -1,$$

$$h_t(u, K(z)) = \left( u, K \left( z \exp \left( i(2t+1)(4\pi(-u-1/2)) \right) \right) \right) \quad \text{if } -1 \le u \le -1/2.$$

Now, $M_F$ is naturally homeomorphic to

$$\left( A'_{F} \cup_{h|_{\partial A'_{F} \setminus (\partial d \times [-1, 0])}} (M \setminus \mathrm{Int}(\Sigma \times [-1, 0])) \right) \cup_{\partial(d \times [-1, 0])} (d \times [-1, 0]),$$

and hence to

$$(\Sigma \times [-1, 0]) \cup_{h|_{\partial(\Sigma \times [-1, 0])}} (M \setminus \mathrm{Int}(\Sigma \times [-1, 0))),$$

which is mapped homeomorphically to $M$ by the identity map outside $\Sigma \times [-1, 0]$, and by $h$ on $\Sigma \times [-1, 0]$. Therefore, we indeed have a homeomorphism from $M_F$ to $M$. This homeomorphism is the identity map outside $[-3, 0] \times K \times [-1, 0]$. It maps $d \times [-1, 0]$ to a cylinder running along $K$ after being twisted negatively.

View the meridian $m$ as a curve of $M \setminus \hat{A}_F$ with its framing induced by the boundary of $A_F$. Thicken $m$ as a band $[0, 1] \times m$ in $\partial A_F$. Assume that a part of this band lies in the vertical boundary $\partial d \times [-1, 0]$ of $d \times [-1, 0]$ and can be written as a rectangle $[\theta, \theta'] \times [-1, 0] \subset \partial d \times [-1, 0]$. Then $h$ sends $[\theta, \theta'] \times [-1, -1/2]$ to some other rectangle in $\partial d \times [-1, -1/2]$. The image under $h$ of $[\theta, \theta'] \times [-1/2, 0]$ together with a small additional piece of the thickened meridian can be isotoped in a tubular neighborhood of $K$ to a band on $\partial A_F$, which is first vertical in $\partial d \times [-1/2, 0]$, and which then runs along $K$. So $h$ sends the framed meridian $m$ to a curve isotopic to $h_A^{-1}(m)$ in a tubular neighborhood of $K$ with the framing induced by the boundary of $A_F$. See Figure 18.5.

Now, $H_1(\partial A_F)$ is generated by the generators of $H_1(\Sigma) \times \{0\}$, the generators of $H_1(\Sigma) \times \{-1\}$, and the homology classes of $c = \partial D$ and $m$. Among
them, the only generator that could be affected by $h_A$ is the class of $m$, which is not. Thus $h_A$ acts trivially on $H_1(\partial A_F)$, and the defined surgery is an LP-surgery. $\square$

Let $\Sigma \times [-1, 2]$ be an extension of the previous neighborhood of $\Sigma$. Set $B_F = F \times [1, 2]$. Define the homeomorphism $h_B$ of $\partial B_F$ to be the identity map anywhere except on $F \times \{1\}$, where it coincides with the homeomorphism $h_F$ of $F$, with the obvious identification.

Let $B'_F$ be a copy of $B_F$. Identify $\partial B'_F$ with $\partial B_F$ with

$$h_B: \partial B'_F \to \partial B_F.$$

Define the inverse surgery associated to $\Sigma$ to be the surgery associated to $(B_F, B'_F)$ (or $(B_F, B'_F; h_B)$). We can apply the previous study to this surgery by using the central symmetry of $[-1, 2]$.

The following obvious lemma, which we will not prove, justifies the terminology.

**Lemma 18.17.** With the above notation, performing both surgeries $(B'_F/B_F)$ and $(A'_F/A_F)$ affects neither $M$ nor the curves in the complement of $F \times [-1, 2]$ (up to isotopy). Performing $(A'_F/A_F)$ (resp. $(B'_F/B_F)$) changes a 0-framed meridian of $K$ passing through $d \times [-1, 2]$ into a 0-framed copy of $K$ (resp. $(-K)$).

**Lemma 18.18.** Let $(x_i, y_i)_{i=1,\dots,g}$ be a symplectic basis of $\Sigma$. Then the tripod combination $T(\mathcal{I}_{A_F A'_F})$ associated to the surgery $(A'_F/A_F)$ is

$$T(\mathcal{I}_{A_F A'_F}) = \sum_{i=1}^{g} \left\langle \begin{array}{c} x_i \\ y_i \\ c \end{array} \middle| \begin{array}{c} \cdot \\ \partial D \end{array} \right\rangle.$$

For a curve $\gamma$ of $F$, let $\gamma^+$ denote $\gamma \times \{1\}$. The tripod combination $T(\mathcal{I}_{B_F B'_F})$ associated to the surgery $(B'_F/B_F)$ is

$$T(\mathcal{I}_{B_F B'_F}) = - \sum_{i=1}^{g} \left\langle \begin{array}{c} x_i^+ \\ y_i^+ \\ c^+ \end{array} \middle| \begin{array}{c} \cdot \\ \partial D \end{array} \right\rangle.$$

**PROOF:** For a curve $\gamma$ of $F$, $\gamma^-$ denotes $\gamma \times \{-1\}$. In order to compute the intersection form of $(A_F \cup -A'_F)$, use the basis $(m, (x_i - x_i^-)_{i \in \underline{g}}, (y_i - y_i^-)_{i \in \underline{g}})$ of the Lagrangian of $A_F$. The system $(c, (y_i)_{i \in \underline{g}}, (-x_i)_{i \in \underline{g}})$ is dual to this basis. The only curve of the Lagrangian basis modified by $h_A$ is $m$, and $h_A^{-1}(m)$
may be expressed as a path composition $mc^{-1}K_2$. Let $D_m$ be a disk of $A_F$ bounded by $m$. The isomorphism $\partial_{MV}^{-1}$ from $\mathcal{L}_{AF}$ to $H_2(A_F \cup -A'_F)$ satisfies:

$$ \partial_{MV}^{-1}(x_i - x_i^-) = S(x_i) = -(x_i \times [-1, 0]) \cup (x_i \times [-1, 0] \subset A'_F), $$

$$ \partial_{MV}^{-1}(y_i - y_i^-) = S(y_i) = -(y_i \times [-1, 0]) \cup (y_i \times [-1, 0] \subset A'_F), $$

$$ \partial_{MV}^{-1}(m) = S_A(m) = D_m - (\Sigma \setminus ([-2, 0] \times K)) \cup (D_m \subset A'_F), $$

where the given expression of $\partial_{MV}^{-1}(m)$ must be completed in

$$ \partial A_F \cap ([-2, 0] \times K \times [-1, 0]) $$

so that the boundary of $\partial_{MV}^{-1}(m)$ actually vanishes, as it does algebraically.

Since $x_i$ intersects only $y_i$ among the curves $x_j$ and $y_j$ for $j \in g$, the surface $S(x_i)$ intersects only $S(y_i)$ and $S_A(m)$ in our basis of $H_2(A_F \cup -A'_F)$. The algebraic intersection of $S(x_i)$, $S(y_i)$, and $S_A(m)$ is $-1$.

For the surgery $(B'_F/B_F)$, use the reflection $T: [-1, 2] \to [-1, 2]$ such that $T(x) = 1-x$. The induced reflection $T_F = 1_F \times T$ of $F \times [-1, 2]$ maps $A_F$ onto $-B_F$. Use the image by $T_F$ of the above basis of $\mathcal{L}_{AF}$ for $\mathcal{L}_{BF}$. The system

$$ (-c^+, (-y_i)_{i \in g}, (x_i^+)_{i \in g}) $$

is dual to the obtained basis of $\mathcal{L}_{BF}$. (Since $T_F$ reverses the orientation, the intersection numbers on $\partial A_F$ are multiplied by $-1$.) Use the images under $T_F$ of the former surfaces. Their triple intersection numbers are the same since their positive normals and the ambient orientation are reversed. $\square$

### 18.3.2 A direct proof of the Casson surgery formula

In this subsection, we prove Proposition 18.12, assuming Theorem 18.6, which will be proved independently.

Note the following easy, well-known lemma.

**Lemma 18.19.** The variation of the linking number of two knots J and K in a rational homology 3-sphere R after a p/q-surgery on a knot V disjoint from J $\sqcup$ K in R is given by the following formula.

$$ lk_{R(V;p/q)}(J, K) = lk_R(J, K) - \frac{q}{p}lk_R(V, J)lk_R(V, K). $$

**Proof:** The p/q-surgery on V is the surgery with respect to a curve $\mu_V \subset \partial N(V)$. Set $q_V = \langle m(V), \mu_V \rangle_{\partial N(V)}$ and $p_V = lk(V, \mu_V)$. We have $\frac{p}{q} = \frac{p_V}{q_V}$. In $H_1(R \setminus (V \cup K); \mathbb{Q})$, we have

$$ J = lk_R(J, K)m(K) + lk_R(V, J)m(V) \text{ and } \mu_V = p_Vm(V) + q_Vlk_R(V, K)m(K). $$
## 18.4 Finite type invariants of $\mathbb{Z}$-spheres

In this section, we state the fundamental theorem of finite type invariants for integer homology 3-spheres due to Thang Lê [L97], and we show how we may use Theorem 18.6 in its proof. This shows in what sense Theorem 18.6 implies that $\mathcal{Z}$ restricts to a *universal* finite type invariant of integer homology 3-spheres. In order to do this, we first follow Mikhail Goussarov [GGP01] and Kazuo Habiro [Hab00] and construct surjective maps from $A_n(\emptyset)$ to $\mathcal{F}_{2n}(M)/\mathcal{F}_{2n+1}(M)$.

**Mapping $A_n(\emptyset)$ to $\mathcal{F}_{2n}(M)/\mathcal{F}_{2n+1}(M)$**. Let $\Gamma$ be a degree $n$ trivalent Jacobi diagram whose vertices are numbered in $2n$. Let $\Sigma(\Gamma)$ be an oriented surface containing $\Gamma$ in its interior such that $\Sigma(\Gamma)$ is a regular neighborhood of $\Gamma$ in $\Sigma(\Gamma)$. Equip $\Gamma$ with its vertex-orientation induced by the orientation of $\Sigma(\Gamma)$. Embed $\Sigma(\Gamma)$ in a ball inside $\mathbb{R}^3$. Replace neighborhoods

<figure><img src="image_145.png" /></figure>

Thus, $\Sigma(\Gamma)$ is transformed into a collection of disjoint oriented surfaces

$$ \Sigma(Y) = \bigcirc_{j=1}^{m_1} \bigcirc_{j=2}^{m_2} \dots \bigcirc_{j=m_3}^{m_3}, $$

one for each trivalent vertex. The graph $\bigcirc_{j=1}^{m_1} \bigcirc_{j=2}^{m_2} \dots \bigcirc_{j=m_3}^{m_3}$ equipped with its framing induced by $\Sigma(Y)$ is called a *Y-graph*. Its looped edges are called *leaves*. Thickening the $\Sigma(Y)$ transforms each of them into a standard genus 3 handlebody $H$. The handlebody $H$ has three handles with meridians $m_j$ and longitudes $\ell_j$, such that $\langle m_i, \ell_j \rangle_{\partial H} = \delta_{ij}$. Its longitudes $\ell_j$ are on $\Sigma(Y)$ as in Figure 18.6.

<figure><img src="image_146.png" /><figcaption>Figure 18.6: Meridians and longitudes of the genus 3 handlebody $H$</figcaption></figure>
The Matveev *Borromean surgery* on $H$ is the Dehn surgery on the six-component link $L_6$ inside $H$ with respect to the parallels of its components that are parallel in Figure 18.7. Sergei Matveev studied it in [Mat87].

<figure><img src="image_147.png" /><figcaption>Figure 18.7: Y-graph and associated LP-surgery</figcaption></figure>

**Lemma 18.20.** *The Matveev Borromean surgery changes the handlebody $H$ to an integer homology handlebody $H'$ with the same boundary and Lagrangian as $H$. The manifold $H \cup_{\partial} (-H')$ is diffeomorphic to $(S^1)^3$, and there exists $\varepsilon_S = \pm 1$ such that*

$$T(\mathcal{I}_{HH'}) = \varepsilon_S < \ell_2^{\ell_3} .$$

*(The sign $\varepsilon_S$ is well-determined by the data. We do not need its explicit value.)*

**PROOF:** First observe that the Lagrangian of $H'$ is the same as the Lagrangian of $H$. As in Example 18.8, $H \cup_{\partial} (-H)$ is obtained from $S^3$ by surgery on three 0-framed meridians of three handles of $H$, where $H$ is embedded in $S^3$ in a standard way. So $H' \cup_{\partial} (-H)$ is obtained by surgery on the zero-framed nine-component link obtained from the six-component link $L_6$ of Figure 18.7 by adding a meridian for each outermost component of $L_6$. Lemma 18.11 implies that $H' \cup_{\partial} (-H)$ is obtained by surgery on the zero-framed Borromean link. Therefore, according to Example 18.8, $H' \cup_{\partial} (-H)$ is diffeomorphic to $(S^1)^3$. So is $H \cup_{\partial} (-H')$. Easy homological computations imply that $H'$ is an integer homology handlebody. $\square$

Denote the Borromean surgeries associated to the Y-graphs corresponding to the vertices of $\Gamma$ by $(A^{(i)\prime}/A^{(i)})$. With the notation of Section 6.1, define $\psi_n(\Gamma)$ to be the class of

$$[S^3; (A^{(i)\prime}/A^{(i)})_{i \in \underline{2n}}] = \sum_{I \subseteq \underline{2n}} (-1)^{|I|} S^3((A^{(i)\prime}/A^{(i)})_{i \in I})$$
in $\mathcal{F}_{2n}(\mathcal{M})/\mathcal{F}_{2n+1}(\mathcal{M})$. The coefficient field $\mathbb{K}$ of Section 6.1 for $\mathcal{F}_{2n}(\mathcal{M})$ is $\mathbb{R}$, from now on.<fnref n="4" />

In [GGP01, Theorem 4.13, Section 4], Stavros Garoufalidis, Mikhail Gous-sarov, and Michael Polyak proved the following theorem.

**Theorem 18.21** (Garoufalidis, Goussarov, Polyak). Let $n \in \mathbb{N}$. For a degree $n$ trivalent Jacobi diagram $\Gamma$, the element $\psi_n(\Gamma)$ of $\mathcal{F}_{2n}(\mathcal{M})/\mathcal{F}_{2n+1}(\mathcal{M})$ constructed above depends only on the class of $\Gamma$ in $\mathcal{A}_n(\emptyset)$, and the map

$$ \psi_n: \mathcal{A}_n(\emptyset) \to \frac{\mathcal{F}_{2n}(\mathcal{M})}{\mathcal{F}_{2n+1}(\mathcal{M})} $$

is surjective. Furthermore, we have $\frac{\mathcal{F}_{2n+1}(\mathcal{M})}{\mathcal{F}_{2n+2}(\mathcal{M})} = \{0\}$.

Assuming the above theorem, the following Lê fundamental theorem on finite type invariants of $\mathbb{Z}$-spheres becomes a corollary of Theorem 18.6.

**Theorem 18.22** (Lê). There exists a family $(Y_n: \mathcal{F}_0(\mathcal{M}) \to \mathcal{A}_n(\emptyset))_{n \in \mathbb{N}}$ of linear maps such that

* $Y_n(\mathcal{F}_{2n+1}(\mathcal{M})) = 0$,

* the restriction $\bar{Y}_n$ to $\mathcal{F}_{2n}(\mathcal{M})/\mathcal{F}_{2n+1}(\mathcal{M})$ of the morphism induced by $Y_n$ on $\mathcal{F}_0(\mathcal{M})/\mathcal{F}_{2n+1}(\mathcal{M})$ to $\mathcal{A}_n(\emptyset)$ is a left inverse of $\psi_n$.

In particular, for any $n \in \mathbb{N}$, we have

$$ \frac{\mathcal{F}_{2n}(\mathcal{M})}{\mathcal{F}_{2n+1}(\mathcal{M})} \cong \mathcal{A}_n(\emptyset) \text{ and } \frac{\mathcal{I}_{2n}(\mathcal{M})}{\mathcal{I}_{2n-1}(\mathcal{M})} \cong \mathcal{A}_n^*(\emptyset). $$

An invariant $Y$ satisfying the properties in the statement of Theorem 18.22 above is called a *universal finite type invariant of $\mathbb{Z}$-spheres*. In order to prove Theorem 18.22, Thang Lê proved that the Lê–Murakami–Ohtsuki invariant $Z^{LMO} = (Z_n^{LMO})_{n \in \mathbb{N}}$ of [LMO98] is a universal finite type invariant of $\mathbb{Z}$-spheres in [L97].

As a corollary of Theorem 18.6, we get the following Kuperberg–Thurston theorem [KT99].

**Theorem 18.23** (Kuperberg, Thurston). The restriction of $\mathcal{Z}$ to $\mathbb{Z}$-spheres is a universal finite type invariant of $\mathbb{Z}$-spheres.

<footnote n="4">⁴For the statements involving only invariants $\mathcal{Z}$ valued in spaces of Jacobi diagrams with rational coefficients (when no interval components are involved), we can fix the coefficient field to be $\mathbb{Q}$, provided that we also restrict the coefficient field of our related spaces of Jacobi diagrams to be $\mathbb{Q}$.</footnote>
**PROOF:** Theorem 18.6 ensures $\mathcal{Z}_n(\mathcal{F}_{2n+1}(\mathcal{M})) = 0$. This reduces the proof of Theorem 18.23 to the proof of the following lemma. $\square$

**Lemma 18.24.** For any trivalent Jacobi diagram $\Gamma$, we have $\mathcal{Z}_n \circ \psi_n([\Gamma]) = [\Gamma]$.

**PROOF:** Let us show how this lemma follows from Theorem 18.6. Number the vertices of $\Gamma$ in $2n$. Call $(A^{(i)}/A^{(i)})$ the Borromean surgery associated to the vertex $i$. The associated tripod is

$$T(\mathcal{I}_{A^{(i)}A^{(i)'}}) = \varepsilon_S \left\langle \begin{array}{c} \ell_3^{(i)} \\ \ell_2^{(i)} \\ \ell_1^{(i)} \end{array} \right\rangle$$

for the fixed $\varepsilon_S = \pm 1$ of Lemma 18.20. Embed the tripods $T(\mathcal{I}_{A^{(i)}A^{(i)'}})$ into the graph $\Gamma$, naturally, so that the half-edge of $\ell_k^{(i)}$ is on the half-edge that gave rise to the leaf of $\ell_k^{(i)}$ in the $Y$-graph associated to $i$. In order to contribute to

$$\langle\langle\bigsqcup_{i \in 2n} T(\mathcal{I}_{A^{(i)}A^{(i)'}})\rangle\rangle_n,$$

a partition must pair a half-edge associated to a leaf of some $\ell_k^{(i)}$ with the half-edge of the only leaf that links $\ell_k^{(i)}$, which is the other half-edge of the same edge. We get

$$[\langle\langle\bigsqcup_{i \in 2n} T(\mathcal{I}_{A^{(i)}A^{(i)'}})\rangle\rangle_n] = [\Gamma].$$

$\square$

**Remark 18.25.** In the original work of Thang Lê [L97] and in the article [GGP01], the primary filtration used for the space of $\mathbb{Z}$-spheres is defined from Borromean surgeries rather than from integral LP-surgeries. In [AL05], Emmanuel Auclair and I proved that the two filtrations coincide. We also proved that a universal finite type invariant of $\mathbb{Z}$-spheres automatically satisfies the more general formula of Theorem 18.6 for any $X = [\tilde{R}; (A^{(i)}/A^{(i)})_{i \in \underline{X}}]$ such that $R$ is a $\mathbb{Z}$-sphere and the $(A^{(i)}/A^{(i)})$ are integral LP-surgeries in $\tilde{R}$. In [GGP01], Stavros Garoufalidis, Mikhail Goussarov, and Michael Polyak compare other filtrations of the space of $\mathbb{Z}$-spheres, including the original filtration of Tomotada Ohtsuki using surgeries on algebraically split links. This original Ohtsuki filtration defined in his introduction of finite invariants of $\mathbb{Z}$-spheres [Oht96] gives rise to the same notion of real-valued finite-type invariants.
## 18.5 Finite type invariants of $\mathbb{Q}$-spheres

For a $\mathbb{Q}$-sphere $R$, the cardinality of $H_1(R; \mathbb{Z})$ is the product over the prime numbers $p$ of $p^{\nu_p(R)}$, where $\nu_p(R)$ is called the $p$-valuation of the order of $H_1(R; \mathbb{Z})$. In [Mou12, Proposition 1.9], Delphine Moussard proved that $\nu_p$ is a degree 1 invariant of $\mathbb{Q}$-spheres with respect to $\mathcal{O}_{\mathcal{L}}^{\mathbb{Q}}$, which is defined in Section 6.1. She also proved [Mou12, Corollary 1.10] that the degree 1 invariants of $\mathbb{Q}$-spheres with respect to $\mathcal{O}_{\mathcal{L}}^{\mathbb{Q}}$ are (possibly infinite) linear combinations of the invariants $\nu_p$ and of a constant map.

Define an augmented trivalent Jacobi diagram to be the disjoint union of a trivalent Jacobi diagram and a finite number of isolated 0-valent vertices equipped with prime numbers. The degree of such a diagram is half the number of its vertices. It is a half-integer. For a half-integer $h$, let $\mathcal{A}_h^{\text{aug}}$ denote the quotient of the $\mathbb{Q}$-vector space generated by degree $h$ augmented trivalent Jacobi diagrams, by the Jacobi relation and the antisymmetry relation. The product induced by the disjoint union turns $\mathcal{A}^{\text{aug}} = \prod_{h \in \frac{1}{2}\mathbb{N}} \mathcal{A}_h^{\text{aug}}$ to a graded algebra. In [Mou12], Delphine Moussard proved that

$$ \frac{\mathcal{F}_n(\mathcal{M}_{\mathbb{Q}})}{\mathcal{F}_{n+1}(\mathcal{M}_{\mathbb{Q}})} \cong \mathcal{A}_{n/2}^{\text{aug}} $$

for any integer $n$. Her proof used the configuration space integral $Z_{KKT}$, described in [KT99] and [Les04a], and the splitting formulae of [Les04b] stated in Theorem 18.6. See [Mou12, Theorem 1.7]. The invariant $Z_{KKT}$ is the restriction to $\mathbb{Q}$-spheres of the invariant $\mathcal{Z}$ described in this book.

The maps $\psi_n$ of Section 18.4 can be generalized to canonical maps

$$ \psi_h : \mathcal{A}_h^{\text{aug}} \to \frac{\mathcal{F}_{2h}(\mathcal{M}_{\mathbb{Q}})}{\mathcal{F}_{2h+1}(\mathcal{M}_{\mathbb{Q}})} $$

as follows. For any prime number $p$, let $B_p$ be a rational homology ball such that $|H_1(B_p; \mathbb{Z})| = p$. Let $\Gamma^a$ be the disjoint union of a degree $k$ trivalent Jacobi diagram $\Gamma$ and $r$ isolated 0-valent vertices $v_j$ equipped with prime numbers $p_j$ for $j \in \underline{r}$. Embed $\Gamma^a$ in $\mathbb{R}^3$. Thicken it, replace $\Gamma$ by $2k$ genus 3 handlebodies $A^{(i)}$ associated to the vertices of $\Gamma$ as in Section 18.4, and replace each vertex $v_j$ by a small ball $B(v_j)$ around it so that the $B(v_j)$ and the $A^{(i)}$ form a family of $2k + r$ disjoint rational homology handlebodies. Define $\psi_{k+r/2}(\Gamma^a)$ to be the class of

$$ [S^3; (A^{(i)}/A^{(i)})_{i \in \underline{2k}}, (B_{p_j}/B(v_j))_{j \in \underline{r}}] \text{ in } \frac{\mathcal{F}_{2k+r}(\mathcal{M}_{\mathbb{Q}})}{\mathcal{F}_{2k+r+1}(\mathcal{M}_{\mathbb{Q}})}, $$

where the $(A^{(i)}/A^{(i)})$ are the Borromean surgeries associated to the $Y$-graphs corresponding to the vertices of $\Gamma$ as in Section 18.4.
**Lemma 18.26.** *The map $\psi_h$ is well-defined.*

**PROOF:** According to [Mou12, Lemma 6.11], if $B'_{pj}$ is a $\mathbb{Q}$-ball whose $H_1(\cdot; \mathbb{Z})$ has the same cardinality as $H_1(B_{pj}; \mathbb{Z})$, then

$$ (S^3(B'_{pj}/B^3) - S^3(B_{pj}/B^3)) $$

belongs to $\mathcal{F}_2(\mathcal{M}_{\mathbb{Q}})$. This guarantees that $\psi_{r/2+k}(\Gamma^a)$ does not depend on the chosen balls $B_{pj}$. Thus, Theorem 18.21 implies that $\psi_h$ is well-defined.

□

This map $\psi_h$ is canonical. The generators of $\mathcal{F}_{2h}(\mathcal{M}_{\mathbb{Q}})/\mathcal{F}_{2h+1}(\mathcal{M}_{\mathbb{Q}})$ exhibited in [Mou12, Section 6.2 and Proposition 6.9] are in the image of $\psi_h$. So $\psi_h$ is surjective.

Let $\mathcal{A}_h^{\text{aug},c}$ denote the subspace of $\mathcal{A}_h^{\text{aug}}$ generated by connected degree $h$ diagrams. So, if $\mathcal{A}_h^{\text{aug},c} \neq 0$, then $h \in \mathbb{N}$ or $h = 1/2$. Set $\mathcal{A}^{\text{aug},c} = \prod_{h \in \frac{1}{2}\mathbb{N}} \mathcal{A}_h^{\text{aug},c}$. Let $z^{\text{aug}}$ denote the $\mathcal{A}^{\text{aug},c}$-valued invariant $z^{\text{aug}}$ of $\mathbb{Q}$-spheres such that, for any $\mathbb{Q}$-sphere $R$, we have

* $z_0^{\text{aug}}(R) = 0$,

* $z_{1/2}^{\text{aug}}(R) = \sum_{p \text{ prime}} \nu_p(R)\bullet_p$, and

* $z_n^{\text{aug}}(R)$ is the natural projection $z_n^{\text{aug}}(R, \emptyset) = p^c(\mathcal{Z}_n(R, \emptyset))$ of $\mathcal{Z}_n(R, \emptyset)$ to the subspace $\mathcal{A}_n^c(\emptyset)$ of $\mathcal{A}_n(\emptyset)$ generated by connected diagrams. (Recall from Notation 7.16 that the projection $p^c$ maps disconnected diagrams to 0.)

Define an $\mathcal{A}^{\text{aug}}$-valued invariant $Z^{\text{aug}} = (Z_n^{\text{aug}})_{n \in \frac{1}{2}\mathbb{N}}$ to be $Z^{\text{aug}} = \exp(z^{\text{aug}})$ for the $\mathcal{A}^{\text{aug},c}$-valued invariant $z^{\text{aug}}$ (meaning $Z^{\text{aug}}(R) = \exp(z^{\text{aug}}(R))$ for any $\mathbb{Q}$-sphere $R$).

As noticed by Gwénaël Massuyeau, the Moussard fundamental theorem for finite type invariants of $\mathbb{Q}$-spheres can be stated as follows.

**Theorem 18.27 (Moussard).** The family $(Z_h^{\text{aug}} : \mathcal{F}_0(\mathcal{M}_{\mathbb{Q}}) \to \mathcal{A}_h^{\text{aug}})_{h \in \frac{1}{2}\mathbb{N}}$ of linear maps is such that, for any $h \in \frac{1}{2}\mathbb{N}$,

* we have $Z_h^{\text{aug}}(\mathcal{F}_{2h+1}(\mathcal{M}_{\mathbb{Q}})) = 0$, and

* $Z_h^{\text{aug}}$ induces a left inverse to $\psi_h$ from $\frac{\mathcal{F}_{2h}(\mathcal{M}_{\mathbb{Q}})}{\mathcal{F}_{2h+1}(\mathcal{M}_{\mathbb{Q}})}$ to $\mathcal{A}_h^{\text{aug}}$.

In particular, we have $\frac{\mathcal{F}_n(\mathcal{M}_{\mathbb{Q}})}{\mathcal{F}_{n+1}(\mathcal{M}_{\mathbb{Q}})} \cong \mathcal{A}_{n/2}^{\text{aug}}$ and $\frac{\mathcal{I}_n(\mathcal{M}_{\mathbb{Q}})}{\mathcal{I}_{n-1}(\mathcal{M}_{\mathbb{Q}})} \cong (\mathcal{A}_{n/2}^{\text{aug}})^*$ for any $n \in \mathbb{N}$.
PROOF: The first assertion follows from Theorem 18.6 and Lemma 6.3. Let $\Gamma^a$ be the disjoint union of a degree $k$ trivalent Jacobi diagram $\Gamma$ and $r$ isolated 0-valent vertices $v_j$ equipped with prime numbers $p_j$ for $j \in \mathbb{Z}$. Let $\psi_\Gamma$ be a representative of $\psi_{k+r/2}(\Gamma^a)$ in $\mathcal{F}_{2k+r}(M_\mathbb{Q})$. Let us prove

$$Z_{\le k+r/2}^{\text{aug}}(\psi_{\Gamma}) = [\Gamma^a]$$

Write $\psi_\Gamma$ as $[S^3; (A^{(i)\prime}/A^{(i)})_{i \in 2k}, (B_{p_j}/B(v_j))_{j \in \mathbb{Z}}]$. If $r=0$, then $\psi_\Gamma$ belongs to $\mathcal{F}_{2k+r}(M)$ and $Z^{\text{aug}}(\psi_\Gamma)$ is equal to $\mathcal{Z}(\psi_\Gamma)$. So we have $Z_k^{\text{aug}}(\psi_\Gamma) = [\Gamma]$, thanks to Lemma 18.24. (All the involved manifolds are $\mathbb{Z}$-spheres.) The general case follows by induction. If $r > 0$, let $\Gamma'$ be obtained from $\Gamma^a$ by forgetting the vertex $v_r$. Let $\psi_{\Gamma'}$ be obtained from $\psi_\Gamma$ by forgetting the surgery $(B_{p_r}/B(v_r))$. This surgery is nothing but a connected sum with $S_{p_r} = B_{p_r} \cup_{S^2} B^3$. Since $Z^{\text{aug}}$ is multiplicative under connected sum according to Theorem 10.26, we have

$$Z^{\text{aug}}(\psi_{\Gamma}) = Z^{\text{aug}}(\psi_{\Gamma'}) (Z^{\text{aug}}(S_{p_r}) - 1).$$

Then, identifying the nonvanishing terms with minimal degree yields

$$Z_{\le k+r/2}^{\text{aug}}(\psi_{\Gamma}) = Z_{k+(r-1)/2}^{\text{aug}}(\psi_{\Gamma'})[\bullet_{p_r}],$$

which allows us to conclude the proof. Indeed, any $\lambda \in (\mathcal{F}_n(M_\mathbb{Q})/\mathcal{F}_{n+1}(M_\mathbb{Q}))^*$ extends to the linear form $\lambda \circ \psi_{n/2} \circ Z_{n/2}^{\text{aug}}$ of $\mathcal{I}_n(M_\mathbb{Q})$. So the natural injection

$$\frac{\mathcal{I}_n(M_\mathbb{Q})}{\mathcal{I}_{n-1}(M_\mathbb{Q})} \hookrightarrow \left( \frac{\mathcal{F}_n(M_\mathbb{Q})}{\mathcal{F}_{n+1}(M_\mathbb{Q})} \right)^*$$

is surjective.

<figure><img src="image_148.png" /></figure>

According to Corollary 10.11, we have

$$\mathcal{Z}_1(R, \emptyset) = \frac{1}{12} \Theta(R) [\ominus].$$

In particular, the invariant $\Theta$ is of degree at most 2 with respect to $\mathcal{O}_\mathcal{L}^\mathbb{Q}$ according to Theorem 18.6. Furthermore, Lemma 18.24 implies

$$\Theta(\psi_1(\ominus)) = 12.$$

The following easy corollary of Theorem 18.27 can be proved as Corollary 6.10.
**Corollary 18.28.** For any real-valued invariant $\nu$ of $\mathbb{Q}$-spheres of degree at most 2 with respect to $\mathcal{O}_{\mathcal{L}}^{\mathbb{Q}}$, there exist real numbers $a_{\theta}$, $a_0$, $a_p$ for any prime number $p$, and $a_{p,q}$ for any pair $(p, q)$ of prime numbers such that $p \le q$, such that

$$ \nu(R) = a_0 + \sum_{p \text{ prime}} a_p \nu_p(R) + \sum_{p,q \text{ prime: } p \le q} a_{p,q} \nu_p(R) \nu_q(R) + a_{\theta} \Theta(R) $$

for any $\mathbb{Q}$-sphere $R$.

Note that the above infinite sums of the statement do not cause problems since they are finite when applied to a $\mathbb{Q}$-sphere $R$.

According to Proposition 5.15 (or to Theorem 10.29), we have $\Theta(-R) = -\Theta(R)$ for any $\mathbb{Q}$-sphere $R$.

**Theorem 18.29.** Let $\nu$ be a real-valued invariant of $\mathbb{Q}$-spheres such that

* the invariant $\nu$ is of degree at most 2 with respect to $\mathcal{O}_{\mathcal{L}}^{\mathbb{Q}}$ and

* we have $\nu(-R) = -\nu(R)$ for any $\mathbb{Q}$-sphere $R$.

Then there exists a real number $a_{\theta}$ such that $\nu = a_{\theta}\Theta$.

**PROOF:** Apply Corollary 18.28. We have

$$ (\nu - a_{\theta}\Theta)(-R) = (\nu - a_{\theta}\Theta)(R) = -(\nu - a_{\theta}\Theta)(R) $$

for any $\mathbb{Q}$-sphere $R$. We get $\nu = a_{\theta}\Theta$. $\square$

<figure><img src="image_149.png" /></figure>

**Remark 18.30.** A similar result was proved in [Les04b, Proposition 6.2] without using the Moussard theorem.

## 18.6 Identifying $\Theta$ with the Casson–Walker invariant

In 1984, Andrew Casson introduced an invariant of $\mathbb{Z}$-spheres, which counts the conjugacy classes of irreducible $SU(2)$-representations of their fundamental groups using Heegaard splittings. See [AM90, GM92, Mar88]. This invariant lifts the Rohlin $\mu$-invariant of Definition 5.29 from $\mathbb{Z}/2\mathbb{Z}$ to $\mathbb{Z}$. In 1988, Kevin Walker generalized the Casson invariant to $\mathbb{Q}$-spheres in [Wal92]. Here, the Casson–Walker invariant $\lambda_{CW}$ is normalized as in [AM90, GM92, Mar88] for integer homology 3-spheres, and as $\frac{1}{2}\lambda_W$ for rational homology 3-spheres,
where $\lambda_W$ is the Walker normalisation in [Wal92]. [Wal92, Lemma 3.1] implies

$$ \lambda_{CW}(-R) = -\lambda_{CW}(R) $$

for any $\mathbb{Q}$-sphere $R$. [Les98] shows that the Casson–Walker generalization satisfies the same splitting formulae as $\frac{1}{6}\Theta$. So $\lambda_{CW}$ is of degree at most 2 with respect to $O_L^\mathbb{Q}$, and we have

$$ \lambda_{CW}(\psi_1(\Theta)) = 2. $$

(This is a consequence of [Les98, Theorem 1.3].)

As a direct corollary of Theorem 18.29, we obtain the following theorem first proved by Greg Kuperberg and Dylan Thurston in [KT99] for $\mathbb{Z}$-spheres, and generalized to $\mathbb{Q}$-spheres in [Les04b, Section 6]. See [Les04b, Theorem 2.6].

**Theorem 18.31.** We have $\Theta = 6\lambda_{CW}$.

**PROOF:** Recall that Lemma 18.24 and Corollary 10.11 imply $\Theta(\psi_1(\Theta)) = 12$. $\square$

## 18.7 Sketch of the proof of Theorem 18.6

Fix $\cup_{i=1}^x A^{(i)}$ and a representative of $L$ in $\mathcal{C}$ as in the statement of Theorem 18.6. We have $\cup_{i=1}^x A^{(i)} \subset \mathcal{C} \setminus L$.

For $I \subseteq \underline{x}$, recall $\mathcal{C}_I = \mathcal{C}((A^{(i)\prime}/A^{(i)})_{i \in I})$. Set $\tilde{R}_I = \tilde{R}(\mathcal{C}_I)$, $R_I = R(\mathcal{C}_I)$, $\tilde{R} = \tilde{R}_\emptyset = \tilde{R}(\mathcal{C})$, and $R = R_\emptyset = R(\mathcal{C})$.

For any part $X$ of $R_I$, $C_2(X)$ denotes the preimage of $X^2$ under the blowdown map from $C_2(R_I)$ to $R_I^2$.

In order to prove Theorem 18.6, we will compute the $\mathbb{Z}_n(\mathcal{C}_I, L)$ with antisymmetric homogeneous propagating forms $\omega_I$ on the $C_2(R_I)$ such that the $\omega_I$ coincide with each other as much as possible (with respect to Definition 3.14 of antisymmetric propagating forms). More precisely, for any subsets $I$ and $J$ of $\underline{x}$, our forms will satisfy

$$ \omega_I = \omega_J \text{ on } C_2\left(\left(R \setminus \cup_{i \in I \cup J} \text{Int}(A^{(i)})\right) \cup_{i \in I \cap J} A^{(i)\prime}\right). $$

When dealing with integral LP-surgeries, such forms will be associated with parallelizations $\tau_I$ of the $\mathcal{C}_I$ (as in Definition 12.4), which coincide as much as possible with each other, i.e., such that

$$ \tau_I = \tau_J \text{ on } \left(\left(R \setminus \cup_{i \in I \cup J} \text{Int}(A^{(i)})\right) \cup_{i \in I \cap J} A^{(i)\prime}\right) \times \mathbb{R}^3. $$
Unfortunately, such a consistent choice of parallelizations is not always possible for rational LP-surgeries. See Section 19.1 and Example 19.8. To remedy this problem, we will make the definition of $\mathcal{Z}$ more flexible by allowing more general propagating forms associated with generalizations of parallelizations, called pseudo-parallelizations.

We define these pseudo-parallelizations in Chapter 19, and we show that they satisfy the following properties.

* They generalize parallelizations. They are genuine parallelizations outside a link tubular neighborhood, inside which they can be thought of as an average of genuine parallelizations.

* A parallelization defined near the boundary of a rational homology handlebody always extends to this rational homology handlebody as a pseudo-parallelization. (See Lemma 19.10.)

* When $\tilde{R}$ is an asymptotic rational homology $\mathbb{R}^3$, a pseudo-parallelization $\tilde{\tau}$ of $\tilde{R}$ induces a homotopy class of special complex trivializations $\tilde{\tau}_C$ of $T\tilde{R} \otimes_{\mathbb{R}} \mathbb{C}$, which has a Pontrjagin number $p_1(\tilde{\tau}_C)$. Outside the link tubular neighborhood considered above, $\tilde{\tau}_C$ is $\tilde{\tau} \otimes_{\mathbb{R}} 1_C$. We set $p_1(\tilde{\tau}) = p_1(\tilde{\tau}_C)$. (See Definitions 19.11 and 19.12.)

* The notion of a homogeneous propagating form of $(C_2(R), \tilde{\tau})$ is presented in Definition 19.15. This definition allows us to extend the definition of $\mathcal{Z}$ of Theorem 12.7 using pseudo-parallelizations instead of parallelizations as follows. For any long tangle representative

$$L: \mathcal{L} \hookrightarrow \tilde{R}(\mathcal{C})$$

in a rational homology cylinder equipped with a pseudo-parallelization $\tilde{\tau}$ restricting to a neighborhood of the image of $L$ as a genuine parallelization, for any $n \in \mathbb{N}$, and for any family $(\omega(i))_{i \in \underline{3n}}$ of homogeneous propagating forms of $(C_2(R(\mathcal{C})), \tilde{\tau})$, the sum

$$Z_n(C, L, (\omega(i))) = \sum_{\Gamma \in \mathcal{D}_n^c(\mathcal{L})} \zeta_{\Gamma} I(C, L, \Gamma, (\omega(i))_{i \in \underline{3n}}) [\Gamma] \in \mathcal{A}_n(\mathcal{L})$$

depends only on $(C, L, p_1(\tilde{\tau}_C))$ and on the $I_\theta(K_j, \tilde{\tau})$, which are defined as in Lemma 7.15 and Definition 12.6, for the components $K_j, j \in \underline{k}$, of $L$. It is denoted by $Z_n(C, L, \tilde{\tau})$. Set

$$Z(C, L, \tilde{\tau}) = (Z_n(C, L, \tilde{\tau}))_{n \in \mathbb{N}} \in \mathcal{A}(\mathcal{L}).$$
Then Theorem 19.17 ensures that we have

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

In the general case, we begin by choosing pseudo-parallelizations $\tau_I$ of the $C_I$ such that $\tau_I$ and $\tau_J$ coincide on

$$ \left( (R \setminus \cup_{i \in I \cup J} \mathrm{Int}(A^{(i)})) \cup_{i \in I \cap J} A^{(i)'} \right) \times \mathbb{R}^3. $$

A reader only interested in the cases for which pseudo-parallelizations are not necessary, as in the applications of Sections 18.3 and 18.4, can skip Chapter 19, and substitute the word pseudo-parallelization with parallelization in the rest of the proof below and in Chapter 20.

Set $\tau_\emptyset = \tau$. The following lemma relates the $p_1(\tau_I)$.

**Lemma 18.32.** Set $p(i) = p_1(\tau_{\{i\}}) - p_1(\tau)$. For any subset $I$ of $\underline{x}$, we have

$$ p_1(\tau_I) = p_1(\tau) + \sum_{i \in I} p(i). $$

**PROOF:** Proceed by induction on the cardinality of $I$. The lemma is obviously true if $|I|$ is zero or one. Assume that $|I| \ge 2$. Let $j \in I$. It suffices to prove $p_1(\tau_I) - p_1(\tau_{I \setminus \{j\}}) = p_1(\tau_{\{j\}}) - p_1(\tau)$. This follows by applying twice the second part of Proposition 5.26, where $M_0 = A^{(j)}$, $M_1 = A^{(j)'}$, and $D = C$ or $D = C_{I \setminus \{j\}}$. The first application identifies $(p_1(\tau_{\{j\}}) - p_1(\tau))$ with $p_1(\tau|_{A^{(j)}}, \tau_{\{j\}|A^{(j)'}})$. The second one yields the conclusion. $\square$

For any $i \in \underline{x}$, fix pairwise disjoint simple closed curves $(a_j^i)_{j=1,...,g_i}$ and pairwise disjoint simple closed curves $(z_j^i)_{j=1,...,g_i}$ on $\partial A^{(i)}$ such that

$$ \mathcal{L}_{A^{(i)}} = \bigoplus_{j=1}^{g_i} [a_j^i] $$

and

$$ \langle a_j^i, z_k^i \rangle_{\partial A^{(i)}} = \delta_{jk} = \begin{cases} 0 & \text{if } j \neq k \\ 1 & \text{if } j = k. \end{cases} $$

Let $[-4, 4] \times (\sqcup_{i \in N} \partial A^{(i)})$ be a tubular neighborhood of $(\sqcup_{i \in N} \partial A^{(i)})$ in $C$. This neighborhood intersects $A^{(i)}$ as $[-4, 0] \times \partial A^{(i)}$. Let $[-4, 0] \times \partial A^{(i)}$ be a neighborhood of $\partial A^{(i)'} = \partial A^{(i)}$ in $A^{(i)'}$. The manifold $C_{\{i\}} = C_i$ is obtained from $C$ by removing $(A^{(i)} \setminus ([-4, 0] \times \partial A^{(i)}))$ and by gluing back $A^{(i)'}$ along $[-4, 0] \times \partial A^{(i)}$.