\draw[line width=1.2,-] (-0.4,-0.52) -- (.4,.53);
\draw[line width=1.2,-] (0.4,-0.52) -- (0.1,-0.12);
\draw[line width=1.2,-] (-0.1,0.12) -- (-.4,.53);
\end{tikzpicture}
= q^{1/2}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2] (-0.4,-0.52) ..controls +(.3,.5).. (-.4,.53);
\draw[line width=1.2] (0.4,-0.52) ..controls +(-.3,.5).. (.4,.53);
\end{tikzpicture}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90]
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2] (-0.4,-0.52) ..controls +(.3,.5).. (-.4,.53);
\draw[line width=1.2] (0.4,-0.52) ..controls +(-.3,.5).. (.4,.53);
\end{tikzpicture}
, \quad
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90]
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2,black] (0,0)  circle (.4)   ;
\end{tikzpicture}
=  -(q+q^{-1})
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90]
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\end{tikzpicture}

The composition is obtained by gluing the tangles together.

* Define a right module $F_M: \TL\to \Mod_k$ by letting  $F_M(n)$ be the quotient of the $k$-module freely generated by isotopy classes of tangles $T$ in $\mathbf{M}$ such that $\iota_M^{-1}(T)=[n]$, by the ideal generated by the Kauffman-bracket relations. When $[T'] \in \TL(n,m)$ is the class of   a $[n]-[m]$ tangle we define $F_M([T]) : F_M(n) \to F_M(m)$ to be the linear map sending the class $[T]\in F_M(n)$  of a tangle in $\mathbf{M}$ to $[T'\cup T]$ obtained by gluing the bigon $\mathbb{B}$ to $\mathbf{M}$ while gluing $\mathbb{D}_M$ to $a_R$. The functor $F_M$ was called the internal skein module in [26] and its restriction to marked surfaces  appeared earlier in [13, 18].

* In addition to the filtration of Definition <ref>, the skein quantum plane $\mathcal{S}_q[\mathbb{A}]$ has a graduation $\mathcal{S}_q[\mathbb{A}]= \oplus_{n\geq 0} \mathcal{S}_q[\mathbb{A}]^{<n>}$, where

$$\mathcal{S}_q[\mathbb{A}]^{<n>}:= \Span\left( [T,s^L], |\partial T \cap a_R| = n \right).$$

So whereas the filtration of Definition <ref> is based on the number of intersection points of $T$ with $a_L$, this graduation counts the number of intersection points of $T$ with $a_R$.
Clearly $\mathcal{S}_q[\mathbb{A}]^{<n>} \cdot \mathcal{S}_q[\mathbb{A}]^{<m>}= \mathcal{S}_q[\mathbb{A}]^{<n+m>}$, so we have an algebra graduation. The comodule structure of $\mathcal{S}_q[\mathbb{A}]$ preserves each graded component so induces comodule maps
$\Delta^L_{<n>} : \mathcal{S}_q[\mathbb{A}]^{<n>} \to \mathcal{S}_q(\mathbb{B}) \otimes \mathcal{S}_q[\mathbb{A}]^{<n>}$.

* Define a left module $F_{\mathbb{A}} : \TL^{op}\to \Mod_k$  by sending $n$ to $\mathcal{S}_q[\mathbb{A}]^{<n>}$. The action on morphisms is defined as follows. For  $T' \in \TL(m,n)$ then $F_{\mathbb{A}} (T')$ sends $[T, s^L] \in \mathcal{S}_q[\mathbb{A}]^{<n>}$ to $[T\cup T', s^L]$ where $T\cup T'$ is obtained by gluing the bigon $\mathbb{B}$ where lives $T$ to the bigon $\mathbb{B}'$ where lives $T'$ by gluing $a_R$ to $a'_L$. The comodules maps $\Delta^L_{<n>}$ define a natural morphism $\Delta^L : F_{\mathbb{A}} \to \mathcal{S}_q(\mathbb{B})\otimes F_{\mathbb{A}}$.

The functor $\bullet \otimes_{\TL} F_M : \widehat{\TL} \to \Mod_k$, sending a functor $G: \TL^{op} \to \Mod_k$ to the $k$-module $G\otimes_{\TL}F_M$, is right exact while working over the ring $k_{\SL_2}$ and becomes exact when working over the field $K_{\SL_2}$.

Let $\iota: \TL\to \mathcal{C}_q^{\SL_2}$ be the ribbon functor sending $[n]$ to $V^{\otimes n}$ (see [66] for details), which identifies $\mathcal{C}_q^{\SL_2}$ with the Cauchy closure of $\TL$. Then the functor $\iota^*: \widehat{\mathcal{C}_q^{\SL_2}} \to \widehat{\TL}$, defined by $\iota^*(G):= G\circ \iota$, is an equivalence of categories. Still denote by $F_M\in \widehat{\mathcal{C}_q^{\SL_2}} $ a fixed lift of $F_M$ by $\iota^*$ (unique up to unique isomorphism). It is sufficient to prove that the functor $\bullet \otimes_{\TL} F_M : \widehat{\mathcal{C}_q^{\SL_2}} \to \Mod_k$ is exact.
Recall from Lemma <ref> the $\dot{U}_q\SL_2-\mathcal{C}_q^{\SL_2}$ bimodule $E'$ such that $E'\otimes_{\mathcal{C}_q^{\SL_2}} \bullet$ is an equivalence of categories. Write $G_M:= E'\otimes_{\mathcal{C}_q^G} F_M$ so that the following diagram commutes:

\begin{equation}\label{diagram_KiTu}
\begin{tikzcd}
\widehat{\mathcal{C}_q^{\SL_2}} \ar[rrd, "\bullet \otimes_{\TL}F_M"] \ar[dd, "E'\otimes_{\mathcal{C}_q^{\SL_2}} \bullet", "\cong"'] &{}&{}\\
{}&{}& \Mod_k \\
\overline{\mathcal{C}_q^{\SL_2}} \ar[rru, "\bullet {\otimes}_{\dot{U}_q\SL_2} G_M"'] &{}&{}
\end{tikzcd}
\end{equation}

Since $\bullet {\otimes}_{\dot{U}_q[\SL_2]} G_M$ is right exact,  the commutativity of Diagram <ref> implies that $\bullet \otimes_{\TL} F_M$ is right exact as well.
By Lemma <ref>, $\overline{\mathcal{C}_q^{\SL_2, rat}}$ is semi-simple, so all its modules are projective and thus flat. In particular $G_M$ is flat when we work over $K_{\SL_2}$ so the functor $\bullet {\otimes}_{\dot{U}_q[\SL_2]} G_M$ is exact and the commutativity of Diagram <ref> implies that $\bullet \otimes_{\TL} F_M$ is exact as well.

It was conjectured in <cit.> and proved in <cit.> that $G_M$ is isomorphic to the stated skein module $\mathcal{S}^{rat}_q(\mathbf{M})$.

By Lemma <ref>, the submodule of coinvariant vectors of $F_{\mathbb{A}}(n)= \mathcal{S}_q[\mathbb{A}]^{<n>}$ is the submodule $F^{(0)}_{\mathbb{A}}(n) \subset F_{\mathbb{A}}(n)$ spanned by $[T,s^L]$ where $T\cap a_L = \emptyset$. Let $F^{(0)}_{\mathbb{A}}: \TL^{op} \to \Mod_k$ the associated functor and $i: F^{(0)}_{\mathbb{A}}\hookrightarrow F_{\mathbb{A}}$ the inclusion morphism.
We have a left exact sequence in $\widehat{\TL}$:

$$0\to  F_{\mathbb{A}}^{(0)} \xrightarrow{i} F_{\mathbb{A}} \xrightarrow{ \Delta^L - \eta\otimes \id} \mathcal{S}_q(\mathbb{B})\otimes F_{\mathbb{A}}.$$

By tensoring with $F_{\mathbf{M}}$ we get a sequence:

$$0\to  F_{\mathbb{A}}^{(0)}\otimes_{\TL} F_{\mathbf{M}}  \xrightarrow{i\otimes \id} F_{\mathbb{A}}\otimes_{\TL} F_{\mathbf{M}}  \xrightarrow{ (\Delta^L - \eta\otimes \id)\otimes \id} \mathcal{S}_q(\mathbb{B})\otimes F_{\mathbb{A}}\otimes_{\TL} F_{\mathbf{M}}.$$

which is exact while working over $K_{\SL_2}$ and only right exact while working over $k_{\SL_2}$ by  Lemma <ref>.
Define an  isomorphism $f: F_{\mathbb{A}}\otimes_{\TL} F_{\mathbf{M}} \xrightarrow{\cong} \mathcal{S}_q(\mathbf{M}) $ as follows. For $[T, s^L] \in F_{\mathbb{A}}(n)$ and $T' \in F_{\mathbf{M}}(n)$, set $f\left( [ [T,s^L] \otimes [T']]\right):= [T\cup T', s^L]$ where $T\cup T'$ is the tangle obtained by gluing $T$ to $T'$ while gluing $a_R$ to $\mathbb{D}_M$. The inverse morphism $f^{-1}$ is defined by splitting stated tangles in the same manner than in the definition of $\theta_{a_R\# a_L}$. Define an isomorphism $f_0: F_{\mathbb{A}}^{(0)}\otimes_{\TL} F_{\mathbf{M}} \xrightarrow{\cong} \mathcal{S}_q(M)$ in the same manner. The commutativity of the following diagram is a straightforward consequence of the definitions:

\begin{tikzcd}
0 \ar[r] & F_{\mathbb{A}}^{(0)}\otimes_{\TL} F_{\mathbf{M}}
\ar[d, "f_0", "\cong"'] \ar[r, "i \otimes \id"] &
F_{\mathbb{A}}\otimes_{\TL} F_{\mathbf{M}}
\ar[d, "f", "\cong"'] \ar[rr, " (\Delta^L - \eta\otimes \id)\otimes \id" ]&{}&
\mathcal{S}_q(\mathbb{B})\otimes F_{\mathbb{A}}\otimes_{\TL} F_{\mathbf{M}}
\ar[d, "\id \otimes f", "\cong"']     \\
0 \ar[r] &
\mathcal{S}^{(rat)}_q(M) \ar[r, "\iota_*"] &
\mathcal{S}^{(rat)}_q(\mathbf{M}) \ar[rr, "\Delta^L - \eta \otimes \id"] &{}&
\mathcal{S}^{(rat)}_q(\mathbb{B})\otimes \mathcal{S}_q(\mathbf{M})
\end{tikzcd}

So the exactness of the first line when working over $K_{\SL_2}$ (resp. the right exactness of the first line when working over $k_{\SL_2}$)  implies the exactness (resp. right exactness) of the second which concludes the proof.

Identifying locally finite $U_q\mathfrak{sl}_2$ modules with $\mathcal{O}_q[\SL_2]$-comodules, the $n+1$ dimensional irreducible representation $V_n$ of $U_q\mathfrak{sl}_2$ (so $V_1=V$ in our notations) corresponds to the comodule $\mathcal{O}_q[\mathbb{A}]^{(n)}$. It is easy to see that the quotient $\widetilde{\mathcal{S}_q[\mathbb{A}]}$ of
${\mathcal{S}_q[\mathbb{A}]}$ by the ideal generated by the $\adjustbox{valign=c}{\includegraphics[width=1cm]{TangleXi.eps}}$ is isomorphic to the limit $\lim_{n\geq 0} (\mathcal{O}_q[\mathbb{A}]^{(1)})^{\otimes n}$ and the equality $\Gr( \widetilde{\mathcal{S}_q[\mathbb{A}]})= \oplus_{n\geq 0}\mathcal{O}_q[\mathbb{A}]^{(n)}$ in Lemma <ref> can be reinterpreted dually in terms of $U_q\mathfrak{sl}_2$-modules,   by the equality in $K^0(\mathcal{C}_q^{\SL_2})$:

$$ [(V_{1}) ^{\otimes n}] = [V_{n}] + \mbox{lower terms}, $$

where "lower terms" is a linear combinations of $[V_i]$ with $i< n$. The latter equality comes from the fact that $[V_n]= S_n([V_1])$ in $K^0(\mathcal{C}_q^{\SL_2})$ (by the quantum Clebsch-Gordan formula) and that the $n$-th Chebyshev polynomial of second species satisfies $S_n(X) = X^n + \mbox{lower terms}$.

§.§ Spherical boundary component

Recall that $k_G= \mathbb{Z}[q^{\pm 1/n}]$ for $n=n_G$ and consider the field of fractions $K_G:=\mathbb{Q}(q^{1/n})$ and the $K_G$ vector spaces $\mathcal{S}_q^{rat}(\mathbf{M}):= \mathcal{S}_q(\mathbf{M})\otimes_{k_{\SL_2}}K_{\SL_2}$ and $\Rep_q^{G,rat}(\mathbf{M}):=\Rep_q^G(\mathbf{M})\otimes_{k_G}K_G$. They both admit a comodule structure over $\mathcal{O}_qG^{rat}:=\mathcal{O}_qG\otimes_{k_G}K_G$.
The goal of this subsection is to prove the

Let $\mathbf{M} \in \mathcal{M}_{\con}^{(1)}$ be such that the connected component of $\partial M$ containing the boundary disc $\mathbb{D}_M$ is a sphere. Then every element of $\Rep_q^{G, rat}(\mathbf{M})$ are coinvariant, i.e. $\Rep_q^{G, rat}(\mathbf{M})=\Char_q^{G, rat}(\mathbf{M})$. Similarly, every element of $\mathcal{S}^{rat}_q(\mathbf{M})$ are coinvariant, i.e. $\mathcal{S}^{rat}_q(\mathbf{M})=\mathcal{S}^{rat}_q(\mathbf{M})^{coinv}$.

Together with Theorem <ref> this implies the

Let $\mathbf{M} \in \mathcal{M}_{\con}^{(1)}$ be such that the connected component of $\partial M$ containing the boundary disc $\mathbb{D}_M$ is a sphere. Then the usual (non stated) skein module of $M$ over $K_{\SL_2}$ is equal to the stated skein module of $\mathbf{M}$ over $K_{\SL_2}$, i.e. the inclusion $i_*: \mathcal{S}^{rat}_q(M)\to \mathcal{S}^{rat}_q(\mathbf{M})$ is an isomorphism.

Theorem <ref> and its proof are very similar and largely inspired (though different) from the work of Gunningham-Jordan-Safronov in <cit.> however our argument, illustrated in Figure <ref>, is more topological and arguably more enlightening (in [26] no marked $3$-manifold with spherical boundary component is mentioned).
As explained in Remark <ref>, Theorem <ref> would not hold if we were working over the ring $k_G$ instead of $K_G$. Let $\mathcal{C}_q^{G,rat}=\mathcal{C}_q^G\otimes_{k_G}K_G$ the category of finite dimensional $\mathcal{O}_qG^{rat}$ comodules and $\overline{\mathcal{C}_q^{G,rat}} := \overline{\mathcal{C}_q^G}\otimes_{k_G}K_G$ the category of possibly infinite dimensional comodules. The heart of the proof and the reason why we need to work over the field $K_G$ is the fact that the Müger center of $\overline{\mathcal{C}_q^{G,rat}}$ is trivial whereas the Müger center of $\overline{\mathcal{C}_q^{G}}$ is not.
Throughout this subsection, we fix $\mathbf{M}$ satisfying the hypothesis of Theorem <ref>. Recall that in the braided quantum group $B_qG:= \int^{W\in \mathcal{C}_q^G} W^* \otimes W \in  \overline{\mathcal{C}_q^G}$, the counit $\epsilon: B_qG \to k$ is defined by $\epsilon([w^*\otimes w]):= \overrightarrow{ev}_W(w^*\otimes w)$. For $V, W \in  \overline{\mathcal{C}_q^G}$,
consider the map

$$ \tau_{V, W} := V\otimes W^* \otimes W \to W^* \otimes W \otimes V, \quad \tau_{V,W}:= (\id_{W^*}\otimes c_{V, W})(c_{W^*, V}^{-1} \otimes \id_W) =   \adjustbox{valign=c}{\includegraphics[width=1.5cm]{field_goal.eps}}.$$

Then taking the colimit over $W\in \mathcal{C}_q^G$, one gets a morphism $\tau_V : V\otimes B_qG \to B_qG \otimes V$ named the field goal transformed by Lyubashenko (see [53]). The following lemma is essentially a reformulation of <cit.>. We add the (very simple) proof for the reader convenience.

Suppose that $V\in  \overline{\mathcal{C}_q^{G, rat}}$ satisfies

$$ \epsilon(x) v = (\epsilon \otimes \id_V)\circ \tau_V (v\otimes x) , \quad \mbox{ for all }x\in B_qG, v\in V, $$

then $V$ is a direct sum of some copies of the trivial comodule $\mathds{1}\in \overline{\mathcal{C}_q^{G, rat}}$.

Let us show that $V$ belongs to the  Müger center $\mathcal{Z}^{M\ddot{u}ger} $ of $\overline{\mathcal{C}_q^{G, rat}}$:

\begin{align*}
V\in \mathcal{Z}^{M\ddot{u}ger} & \Leftrightarrow c_{W, V} \circ c_{V,W} = \id_{V \otimes W} , \quad \mbox{ for all } W\in  \mathcal{C}_q^G, \\
{} & \Leftrightarrow (\id_V \otimes \overrightarrow{ev}_W) \circ (c_{W, V} c_{V,W} \otimes \id_{W^*}) =  (\id_V \otimes \overrightarrow{ev}_W) , \quad \mbox{ for all } W\in  \mathcal{C}_q^G, \\
{} & \Leftrightarrow (\overrightarrow{ev}_W \otimes \id_V) \tau_{V,W} = \id_V \otimes \overrightarrow{ev}_W , \quad \mbox{ for all } W\in  \mathcal{C}_q^G, \\
{} & \Leftrightarrow (\id_V \otimes \epsilon) = (\epsilon \otimes \id_V) \circ \tau_V \\
{} & \Leftrightarrow \epsilon(x) v = (\epsilon \otimes \id_V)\circ \tau_V (v\otimes x) , \quad \mbox{ for all }x\in B_qG, v\in V.
\end{align*}

We conclude using the  fact that any element in $\mathcal{Z}^{M\ddot{u}ger}$ is a trivial comodule.

Write $V= \Rep_q^{G, rat}(\mathbf{M})$ and let us prove that it satisfies the hypothesis of Lemma <ref>. The proof is illustrated in Figure <ref>. Let $x\in B_qG$ and  $v\in \Rep_q^{G,rat}(\mathbf{M})$ be an element of the form $v:= [y\otimes \alpha] $ with $\alpha\in P_n(M)$ a $n$-bottom tangle and $y\in (B_qG)^{\otimes n}$. Let us prove that $ \epsilon(x) v = (\epsilon \otimes \id_V)\circ \tau_V (v\otimes x) $. Let $\beta_0 \in P_1(M)$ be the trivial $1$-bottom tangle made of an arc lying in a small neighborhood of $\mathbb{D}_M$. More precisely, if $\boldsymbol{\eta}\in BT(1,0)$ is the counit of Figure <ref> and $\eta_M$ is the only element of $P_0(M)$ then $\beta_0:= \eta_M\circ \boldsymbol{\eta}$. Let $\beta_0 \cup \alpha \in P_{n+1}(M)$ be the $n+1$ bottom tangle obtained by stacking $\beta_0$ on top (in the height direction) of $\alpha$. Similarly, let $\alpha\cup \beta_0 \in P_{n+1}(M)$ be the bottom tangle obtained by stacking $\beta_0$ on the bottom of $\alpha$. In more precise terms, we set

$$ \beta_0 \cup \alpha:= \alpha \circ (\boldsymbol{\eta} \wedge \mathds{1} \wedge \ldots \wedge \mathds{1}), \quad \alpha\cup \beta_0 := \alpha \circ ( \mathds{1} \wedge \ldots \wedge \mathds{1} \wedge \boldsymbol{\eta}).$$

So by definition of the counit $\epsilon$, one has

$$ \epsilon(x) v = [(x\otimes y) \otimes (\beta_0\cup \alpha)] = [(y\otimes x) \otimes (\alpha\cup \beta_0)] \in \Rep_q^G(\mathbf{M}).$$

$$T:=     \adjustbox{valign=c}{\includegraphics[width=2cm]{BT_field_goal.eps}}   \in \mathfrak{bt}(n+1, n+1).$$

Then by definition of the field goal transform, one has

$$ (\epsilon \otimes \id_V) \circ \tau_V(v\otimes x) = [ (x\otimes y) \otimes \left((\alpha\cup \beta_0)\cdot T\right)] \in \Rep_q^{G, rat}(\mathbf{M}).$$

The equality $ \epsilon(x) v = (\epsilon \otimes \id_V)\circ \tau_V (v\otimes x) $ follows from the fact, illustrated in Figure <ref>, that the $n+1$ bottom tangles $\beta_0\cup \alpha$ and $(\alpha \cup \beta_0)\cdot T$ are isotopic in $\mathbf{M}$ because the boundary component containing $\mathbb{D}_M$ is a sphere. We conclude using Lemma <ref>.

An illustration of the isotopy $\beta_0\cup \alpha \sim (\alpha \cup \beta_0)\cdot T$.

Note that in Theorem <ref>, it is important to work over the field $K_G$ instead of the ring $k_G$. Indeed, as we shall prove in Section <ref>, the module $\mathcal{S}_q(\mathbf{M}) \otimes_{q^{1/4}=1}\mathbb{Z}$, obtained by setting $q^{1/4}=1$, is isomorphic as an $\mathcal{O}[\SL_2]$-comodule to the ring of regular functions of the representation scheme with coaction given by conjugacy. Since the action of $\SL_2$ on the space $\Hom(\pi_1(M), \SL_2)$ by conjugacy is not trivial, this implies that $\mathcal{S}_q(\mathbf{M})$ does not have a trivial coaction, i.e. that the analogue of Theorem <ref> over $k_{\SL_2}$ does not hold. Let us illustrate this phenomenon on a concrete example. Let $\alpha$ be a non contractible $1$-bottom tangle in $\mathbf{M}$ and for $i,j=\pm$ let $\alpha_{ij}\in \mathcal{S}_q(\mathbf{M})$ denote the class of the arc $\alpha$ with state $v_i$ at its higher endpoint and $v_j$ at its lower endpoint. Let $\alpha^0\in \mathcal{S}_q(\mathbf{M})$ be the class of the closed arc obtained from $\alpha$ by gluing its two endpoints together in the interior of $M$.
Write $\begin{pmatrix} D_{++} & D_{+-} \\ D_{-+} & D_{--} \end{pmatrix} := \begin{pmatrix} 0& -A^{5/2} \\ A^{1/2} & 0 \end{pmatrix}$ the matrix coefficients of the half-twist.
A simple skein computation illustrated in Figure <ref> shows that in $\mathcal{S}_q(\mathbf{M})$ we have the equality in $\mathcal{S}_q(\mathbf{M})$:

\begin{equation}\label{eq_spherical_cool}
(q^4-1) \alpha_{ij} = q(1-q^2) D_{ij} \alpha^0.
\end{equation}

In particular $\alpha_{++}\in \mathcal{S}_q(\mathbf{M})$ is not a coinvariant vector, but since $(q^4-1) \alpha_{++} = 0$, its class in $\mathcal{S}_q^{rat}(\mathbf{M})$ vanishes. In general, Equation (<ref>) shows that the class of any stated arc $\alpha_{ij}$ in $\mathcal{S}_q^{rat}(\mathbf{M})$ is proportional to the coinvariant vector $\alpha^0$, so is coinvariant. The proof of Theorem <ref> shows that $\mathcal{S}_q(\mathbf{M})$ belongs to the Müger center of $\overline{\mathcal{C}_q^G}$ even though it is not a trivial comodule because of the existence of torsion elements which are not coinvariant.

An illustration of Equation (<ref>). The isotopy $\sim$ is similar to the isotopy of Figure <ref>. The second equality is obtained by resolving the crossings using the Kauffman-bracket skein relations.

§ CLASSICAL LIMIT

Let $\mathcal{S}_{+1}(\mathbf{M}):= \mathcal{S}_q(\mathbf{M}) \otimes_{q^{1/4}=1}\mathbb{Z}$. The goal of this section is to endow $\mathcal{S}_{+1}(\mathbf{M})$ with a ring structure and to prove that it is isomorphic to the ring of regular functions of the representation scheme $\mathcal{R}_{\SL_2}(\mathbf{M})$.

§.§ Ring structure on $\mathcal{S}_{+1}(\mathbf{M})$

Let $\mathcal{O}[\SL_2]:= \mathcal{O}_q[\SL_2]\otimes_{q^{1/4}=1}\mathbb{Z}$: it is the classical integral form of $\SL_2$. Let $\mathcal{S}_{+1}: \mathcal{M}_{\con}^{(1)} \to \RComod_{\mathcal{O}[\SL_2]}$ be the composition of $\mathcal{S}_q$ with the change of scalars $\bullet\otimes_{q^{1/4}=1}\mathbb{Z}: \Mod_k \to \Mod_{\mathbb{Z}}$.

$\mathcal{S}_{+1}$ is the left Kan extension of $\restriction{\mathcal{S}_{+1}}{\BT}$ along $\iota: \BT\to \mathcal{M}_{\con}^{(1)}$.

The lemma follows from Lemma <ref> together with the fact that the functor $\bullet \otimes_{q^{1/4}=1}  \mathbb{Z}$ has right adjoint $\Hom_{k}(\cdot, \mathbb{Z})$.

Note that, thanks to the skein relation
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2,-] (-0.4,-0.52) -- (.4,.53);
\draw[line width=1.2,-] (0.4,-0.52) -- (0.1,-0.12);
\draw[line width=1.2,-] (-0.1,0.12) -- (-.4,.53);
\end{tikzpicture}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2] (-0.4,-0.52) ..controls +(.3,.5).. (-.4,.53);
\draw[line width=1.2] (0.4,-0.52) ..controls +(-.3,.5).. (.4,.53);
\end{tikzpicture}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90]
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2] (-0.4,-0.52) ..controls +(.3,.5).. (-.4,.53);
\draw[line width=1.2] (0.4,-0.52) ..controls +(-.3,.5).. (.4,.53);
\end{tikzpicture}
$, the elements of $\mathcal{S}_{+1}(\mathbf{M})$ are transparent, i.e. the class of a stated tangle in $\mathcal{S}_{+1}(\mathbf{M})$ does not change when we change a crossing $\Crosspos$ to $\Crossneg$.

Equip $\mathcal{S}_{+1}(\mathbf{M})$ with the product defined on two classes of stated tangles  $[T_1,s_1], [T_2, s_2] \in \mathcal{S}_{+1}(\mathbf{M})$ by first isotoping $T_1$ and $T_2$ such that they do not intersect and such that the heights of $\partial T_1$ are bigger than the heights of $\partial T_2$ and then defining
$[T_1,s_1] \cdot [T_2, s_2] := [T_1 \cup T_2, (s_1, s_2)]$.

The transparence skein relation $\Crosspos=\Crossneg$ implies that the class $[T_1 \cup T_2, (s_1, s_2)]$ does not depend on the choice of the representatives for $T_1\cup T_2$. Moreover, thanks to the height exchange relations
$\heightexch{->}{i}{j}=  \heightexch{<-}{i}{j}$ (proved in <cit.>), the class $[T,s]$ of a stated tangle does not change if we change the height order of the elements of $\partial T$. In particular $\mathcal{S}_{+1}(\mathbf{M})$  is a commutative ring.

If we had chosen $q^{1/4}=-1$, we still would have obtained a commutative ring; it is not difficult to prove that this ring is (non canonically) isomorphic to $\mathcal{S}_{+1}(\mathbf{M})$. However for the more standard choice $q^{1/4}=\sqrt{-1}$, for which $A=-1$, the product is still defined but the stated skein module is no longer commutative (unlike the usual skein module) because the height exchange relation $\heightexch{->}{i}{j}=  \heightexch{<-}{i}{j}$  does not hold anymore. The inconvenient of having chosen $A=+1$ is that, unlike for $A=-1$, the class of a stated tangle depends on the framing (up to a sign) because of the skein relation
$\adjustbox{valign=c}{\includegraphics[width=2cm]{TwistRel.eps}}$. This is the main reason why we will need to consider spin functions in the next subsection.

§.§ Spin functions

Let us define a functor $\tau \in \widehat{\BT}_{\mathbb{Z}}=[\BT^{op}, \Mod_{\mathbb{Z}}]$. For $n\geq 0$  set $\tau(\mathbf{H}_n):=\mathrm{H}^1(\mathbf{H}_n; \mathbb{Z}/2\mathbb{Z})$ and for $\mu: \mathrm{H}_a \to \mathrm{H}_b$ set $\tau(\mu):=\mu^* : \mathrm{H}^1(\mathbf{H}_b; \mathbb{Z}/2\mathbb{Z}) \to \mathrm{H}^1(\mathbf{H}_a; \mathbb{Z}/2\mathbb{Z})$. Let us identify $\mathrm{H}^1(\mathbf{H}_1; \mathbb{Z}/2\mathbb{Z})$ with $\mathbb{Z}/2\mathbb{Z}$.

For $\mathbf{M} \in \mathcal{M}_{\con}^{(1)}$, a spin function is a natural morphism $w: P_M \to \tau$ such that   $w_1\left(\adjustbox{valign=c}{\includegraphics[width=1cm]{BT_Theta.eps}}  \right) = 1 \in \mathbb{Z}/2\mathbb{Z} \cong \mathrm{H}^1(\mathbf{H}_1; \mathbb{Z}/2\mathbb{Z})$.

So a spin function is a collection of maps $w_n: P_n(\mathbf{M}) \to \mathrm{H}^1(\mathbf{H}_n; \mathbb{Z}/2\mathbb{Z})$ such that $w_1\left(\adjustbox{valign=c}{\includegraphics[width=1cm]{BT_Theta.eps}}  \right)  =1$ and such that for every morphism $\mu : \mathbf{H}_a \to \mathbf{H}_b$, the following diagram commutes:

\begin{tikzcd}
P_b(\mathbf{M}) \ar[r, "\mu^*"] \ar[d, "w_b"] & P_a(M) \ar[d, "w_a"] \\
\mathrm{H}^1(\mathbf{H}_b; \mathbb{Z}/2\mathbb{Z})\ar[r, "\mu^*"] & \mathrm{H}^1(\mathbf{H}_a; \mathbb{Z}/2\mathbb{Z})
\end{tikzcd}

* A spin function is completely determined by the map $w_1 : P_1(\mathbf{M}) \to \mathbb{Z}/2\mathbb{Z}$. Indeed, let $[\beta_i] \in \mathrm{H}_1(\mathbf{H}_n; \mathbb{Z}/2\mathbb{Z})$ be the class of a simple closed curve encircling the $i$-th hole of $\mathbb{D}_n\times \{1/2\} \subset \mathbf{H}_n$ once, and consider the isomorphism $\mathrm{H}^1(\mathbf{H}_n; \mathbb{Z}/2\mathbb{Z}) \cong (\mathbb{Z}/2\mathbb{Z})^n$ sending $\varphi$ to $(\varphi([\beta_1]), \ldots, \varphi([\beta_n]))$.
If $\alpha= \alpha^{(1)}\cup \ldots \cup \alpha^{(n)}$ is a $n$-bottom tangle with connected components $\alpha^{(i)}$, by naturality of $w$ and under the above isomorphism one has $w_n(\alpha) = (w_1(\alpha^{(1)}), \ldots, w_1(\alpha^{(n)}))$ so the maps $w_n$ are determined by $w_1$.

* For $\alpha$ a $1$-bottom tangle and $s$ a state on $\alpha$, the class $(-1)^{w_1(\alpha)}[\alpha, s] \in \mathcal{S}_{+1}(\mathbf{M})$ does not depend on the framing of $\alpha$ thanks to the condition $w_1\left(\adjustbox{valign=c}{\includegraphics[width=1cm]{BT_Theta.eps}}  \right) = 1$ and to the skein relation $\adjustbox{valign=c}{\includegraphics[width=2cm]{TwistRel.eps}}$.

* When $\mathbf{M}= \mathbf{\Sigma}\times I$ is a thickened marked surface, then the set of spin functions is in one-to-one correspondence with the set of isomorphism classes of spin structures on $\Sigma$ (hence the name "spin functions"). Indeed, fix a Riemannian metric on $\Sigma$ and let $\pi: UM\to M$ be the unitary tangent bundle. Let $[\theta] \in \mathrm{H}_1(UM; \mathbb{Z}/2\mathbb{Z})$ be the homology class of a loop in $UM$ whose projection in $M$ is the constant point $p_M$  and such that $\theta$ makes a full positive twist in the fiber $\pi^{-1}(p_M)$. For $w$ a spin function, one associates $\mathbf{w} :  \mathrm{H}_1(UM; \mathbb{Z}/2\mathbb{Z}) \to \mathbb{Z}/2\mathbb{Z}$ a group morphism such that $\mathbf{w}([\theta])=1$ as follows. Every class $[\alpha]\in  \mathrm{H}_1(UM; \mathbb{Z}/2\mathbb{Z})$ can be represented by a $1$-bottom tangle $\alpha$ (where we isotope both endpoints to $p_M$) and we set $\mathbf{w}([\alpha]):= w_1(\alpha)$. That $\mathbf{w}$ is a group morphism follows from the naturality with respect to the morphism $\mu: \mathbf{H}_2\to \mathbf{H}_1$ of Figure <ref>  and that $\mathbf{w}([\theta])=1$ follows from the fact that  $w_1\left(\adjustbox{valign=c}{\includegraphics[width=1cm]{BT_Theta.eps}}  \right) =1$. The morphism $\mathbf{w}$ defines a (regular) double covering $p:\widetilde{U\Sigma}\to U\Sigma$ and the condition $w([\theta])\neq 0$ ensures that each fiber of $\pi$ lifts to a non trivial double covering by $p$. Since $\Spin(2)$ is the only non trivial double covering of $\SO(2)$, the composition $\pi\circ p : \widetilde{U\Sigma} \to \Sigma$ is a principal $\Spin(2)$ bundle over $U\Sigma$, i.e. a spin structure (see Milnor [57] for details).

§.§ Representation and character schemes

Let $\Gamma$ be a finitely presented group and consider the commutative ring

$$\mathcal{O}[\mathcal{R}_{\SL_2}(\Gamma)]:= \quotient{\mathbb{Z}[X^{\gamma}_{ij}, i,j\in \{+, -\}, \gamma \in \Gamma]}{(M_{\alpha}M_{\beta}=M_{\alpha\beta}, \det(M_{\alpha})=1, \alpha, \beta\in \Gamma)}, $$

where $M_{\gamma}=\begin{pmatrix} X^{\gamma}_{++} & X^{\gamma}_{+-} \\ X^{\gamma}_{-+} & X^{\gamma}_{--} \end{pmatrix}$ is a $2\times 2$ matrix with coefficients in the polynomial ring $\mathbb{Z}[X^{\gamma}_{ij}]$ (so the relation $M_{\alpha}M_{\beta}=M_{\alpha\beta}$ represents in fact four relations). Set $\mathcal{O}[\SL_2]:= \quotient{\mathbb{Z}[x_{ij}, i,j\in \{-, +\}]}{(x_{++}x_{--}-x_{+-}x_{-+}-1)}$.
The set of characters $\Hom_{\Alg}\left(\mathcal{O}[\mathcal{R}_{\SL_2}(\Gamma)], \mathbb{C}\right)$ is in bijection with the set of representations $\rho: \Gamma \to \SL_2(\mathbb{C})$. The group $\SL_2(\mathbb{C})$ acts (on the left) by conjugacy on the set of representations by the formula:

$$ (g\cdot \rho) (\gamma):= g \rho(\gamma)g^{-1} , \quad \mbox{ for all }g\in \SL_2, \gamma \in \Gamma.$$

The above action is algebraic, i.e. induced by a comodule map $\Delta^R: \mathcal{O}[\mathcal{R}_{\SL_2}(\Gamma)] \to \mathcal{O}[\mathcal{R}_{\SL_2}(\Gamma)] \otimes_{\mathbb{Z}} \mathcal{O}[\SL_2]$ defined by the formula:

$$ \Delta^R (X_{ij}^{\gamma}):= \sum_{a,b= \pm}  X_{ab}^{\gamma} \otimes x_{i a}S(x_{bj}) .$$

For instance

$$\Delta^R (X_{++}^{\gamma}) =  X_{++}^{\gamma} \otimes x_{++}x_{--} +  X_{-+}^{\gamma}\otimes x_{+-}x_{--} -  X_{+-}^{\gamma}\otimes x_{-+}x_{++} -  X_{--}^{\gamma}\otimes x_{-+}x_{+-}.$$

Let $\mathcal{O}[\mathcal{X}_{\SL_2}(\mathbf{M})] \subset \mathcal{O}[\mathcal{R}_{\SL_2}(\Gamma)]$ be the subalgebra of coinvariant vectors.
Both $\mathcal{O}[\mathcal{R}_{\SL_2}(\Gamma)]$ and $\mathcal{O}[\mathcal{X}_{\SL_2}(\Gamma)]$ are finitely generated, however they might be non reduced, i.e. their nilradical $\sqrt{0}$ might be non trivial.

The $\SL_2$-representation scheme is $\mathcal{R}_{\SL_2}(\Gamma):= \Spec(\mathcal{O}[\mathcal{R}_{\SL_2}(\Gamma)])$. The $\SL_2$-character scheme is $\mathcal{X}_{\SL_2}(\Gamma):=\Spec(\mathcal{O}[\mathcal{X}_{\SL_2}(\Gamma)] )$.
The $\SL_2$-representation variety is $\mathcal{R}_{\SL_2}^{red}(\Gamma):= \Spec(\quotient{\mathcal{O}[\mathcal{R}_{\SL_2}(\Gamma)])\otimes_{\mathbb{Z}}\mathbb{C}}{\sqrt{0}})$. The $\SL_2$-character variety is $\mathcal{X}_{\SL_2}^{red}(\Gamma):= \Spec(\quotient{\mathcal{O}[\mathcal{X}_{\SL_2}(\Gamma)])\otimes_{\mathbb{Z}}\mathbb{C}}{\sqrt{0}})$.

Recall that to  $\mathbf{M}\in \mathcal{M}^{(1)}_{\con}$ one associates a canonical based point $p_M:=\iota_M(0)\in \mathbb{D}_M$. Since every compact $3$-manifold can be triangulated (see e.g. [32]), we easily deduced that the fundamental group $\pi_1(\mathbf{M}):=\pi_1(M, p_M)$ is finitely presented.
Write $\mathcal{R}_{\SL_2}(\mathbf{M}):= \mathcal{R}_{\SL_2}(\pi_1(\mathbf{M}))$ and $\mathcal{X}_{\SL_2}(\mathbf{M}):= \mathcal{X}_{\SL_2}(\pi_1(\mathbf{M}))$. In <cit.>, Kapovitch-Millson proved the existence of large families of $3$-manifolds for which $\mathcal{X}_{\SL_2}(\mathbf{M})$ (and thus $\mathcal{R}_{\SL_2}(\mathbf{M})$) is non reduced, so it is important to distinguish between representation schemes and varieties. However, when $M$ is a thickened surface, $\mathcal{R}_{\SL_2}(\mathbf{\Sigma}\times I)$ is reduced (see [61, 17]). An embedding $f: \mathbf{M} \to \mathbf{M}'$ between two marked $3$-manifolds induces a group morphism $f_{*}: \pi_1(\mathbf{M})\to \pi_1(\mathbf{M}')$ and thus a  morphism $f_* : \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{M})] \to \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{M}')]$ of $\mathcal{O}[\SL_2]$ comodules. We denote by $\mathcal{O}[\mathcal{R}] : \mathcal{M}_{\con}^{(1)} \to {\mathcal{O}[\SL_2]}-\RComod$ the functor sending $\mathbf{M}$ to $\mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{M})]$.

The following diagram is a left Kan extension:

\begin{tikzcd}
{} & \mathcal{M}^{(1)}_{\con}
\ar[rd, "{\mathcal{O}[\mathcal{R}]}"]
& {} \\
\BT  \ar[ru, "i"]
\ar[rr, "\restriction{\mathcal{O}[\mathcal{R}]}{\BT}"]
&{}&  {\mathcal{O}[\SL_2]}-\RComod
\end{tikzcd}

In particular, one has:

$$ \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{M})] \cong \restriction{\mathcal{O}[\mathcal{R}]}{\BT}\otimes_{\BT} \mathbb{Z}[P_M].$$

For each $n\geq 0$, we fix a basis of the free group $\pi_1(\mathbf{H}_n)$ inducing an isomorphism $ \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{H}_n)]\cong \mathcal{O}[\SL_2]^{\overline{\otimes}n}$.
Let $L:= \Lan_i \restriction{\mathcal{O}[\mathcal{R}]}{\BT}$ be the left Kan extension of $\restriction{\mathcal{O}[\mathcal{R}]}{\BT}$ along $i$ given by the formula:

$$ L(\mathbf{M}):= \restriction{\mathcal{O}[\mathcal{R}]}{\BT}\otimes_{\BT} \mathbb{Z}[P_M] =\int^{n\geq 0} \mathcal{O}[\SL_2]^{\overline{\otimes}n} \otimes \mathbb{Z}[P_n(\mathbf{M})].$$

By definition of the coend, one has

$$ L(\mathbf{M})= \quotient{ \left( \oplus_{n\geq 0}\mathcal{O}[\SL_2]^{\overline{\otimes}n} \otimes \mathbb{Z}[P_n(\mathbf{M})]\right)}{\mathcal{J}}, $$

where $\mathcal{J}$ is the ideal generated by elements $(\mu_* X \otimes \gamma) - (X\otimes \mu^* \gamma)$ with $\mu: \mathbf{H}_n \to \mathbf{H}_m$, $\gamma\in P_m(\mathbf{M})$ and $X\in \mathcal{O}[\SL_2]^{\overline{\otimes}n}$.
Let $\pi_{M} \in \widehat{\BT}$ be the functor sending $\mathbf{H}_n$ to $\Hom_{\Gp}(\pi_1(\mathbf{H}_n), \pi_1(\mathbf{M}))\cong \pi_1(\mathbf{M})^n$ (the latter isomorphism is given by the fixed basis of $\pi_1(\mathbf{H}_n)$). A $1$-bottom tangle $\alpha\subset M$ defines an element $[\alpha] \in \pi_1(\mathbf{M})$ by forgetting the framing and isotoping locally both endpoints of $\alpha$ to $p_M$. For a $n$-bottom tangle $\alpha \in P_n(\mathbf{M})$ with components $\alpha= \alpha^{(1)}\cup \ldots \cup \alpha^{(n)}$,  ordered such that the heights of the endpoints $\partial \alpha^{(i)}$ is bigger than the heights of the endpoints $\partial \alpha^{(i+1)}$, we set $[\alpha]:= ([\alpha^{(1)}], \ldots, [\alpha^{(n)}]) \in \pi_1(\mathbf{M})^n$.

Define a natural morphism $\eta^M: P_M \Rightarrow \pi_M$ by sending $\alpha\in P_n(\mathbf{M})\cong P_M(\mathbf{H}_n)$ to $\eta^M_n(\alpha):= [\alpha]$. The map $\eta^M_n: P_n(\mathbf{M}) \to \pi_1(\mathbf{M})^n$ is clearly surjective and two bottom tangles $\alpha$, $\alpha'$ have the same image if and only if one can pass from $\alpha$ to $\alpha'$ by a finite sequence of crossings changes $\Crosspos \leftrightarrow \Crossneg$ and twist moves $\adjustbox{valign=c}{\includegraphics[width=2cm]{TwistRel+1.eps}}$. Write $\alpha\sim \alpha'$ if $\alpha$ and $\alpha'$ are related by such a finite sequence, so that $\eta^M_n$ induces a bijection $\quotient{P_n(\mathbf{M})}{\sim} \xrightarrow{\cong} \pi_1(\mathbf{M})^n$. We claim that

$$ L(\mathbf{M}):= \restriction{\mathcal{O}[\mathcal{R}]}{\BT}\otimes_{\BT} \mathbb{Z}[P_M] \xrightarrow[\id \otimes \eta^M]{\cong}  \restriction{\mathcal{O}[\mathcal{R}]}{\BT}\otimes_{\BT} \mathbb{Z}[\pi_M],$$

i.e. that for $X\in \mathcal{O}[\SL_2]^{\overline{\otimes}n}$ and $\alpha, \alpha' \in P_n(\mathbf{M})$ which differ by some change of crossings $\Crosspos \leftrightarrow \Crossneg$, then $[X\otimes \alpha]=[X \otimes \alpha'] \in L(\mathbf{M})$. Indeed, in such case, one has an endomorphism $\mu : \mathbf{H}_n \to \mathbf{H}_n$ such that $\mu(\alpha)= \alpha'$ and such that $\eta^M(\mu)=\id$. For such an endomorphism $\mu$ whose action on $\pi_1(\mathbf{M})$ is trivial, the induced morphism $\mu_*\in \End(\mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{H}_n)])$ is the identity, therefore $\alpha \otimes X - \alpha' \otimes X \in \mathcal{J}$, which proves the claim.

We can thus write

$$L(\mathbf{M})= \quotient{ \left( \oplus_{n\geq 0}\mathcal{O}[\SL_2]^{\overline{\otimes}n} \otimes \mathbb{Z}[\pi_1(M)^n]\right)}{\mathcal{J}'}, $$

with $\mathcal{J}'$ the image of $\mathcal{J}$ though the quotient map.
On the other hand, write

$$ \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{M})] = \quotient{A}{\mathcal{I}}, $$

$$ A:= \quotient{\mathbb{Z}[X^{\gamma}_{ij}, i,j\in \{+, -\}, \gamma \in \pi_1(\mathbf{M})]}{(\det(M_{\alpha})=1, \alpha\in \pi_1(\mathbf{M}))},$$

and $\mathcal{I}$ is the ideal generated by the relations given by the matrix coefficients of the equalities $M_{\alpha}M_{\beta}=M_{\alpha \beta}$, $\alpha, \beta \in \pi_1(\mathbf{M})$. Let

$$\widetilde{\kappa}_{\mathbf{M}}:  A \cong \oplus_{n\geq 0}\mathcal{O}[\SL_2]^{\overline{\otimes}n} \otimes \mathbb{Z}[\pi_1(M)^n]$$

be the isomorphism sending a monomial $X_{a_1 b_1}^{\gamma_1} \ldots X_{a_n b_n}^{\gamma_n}$ to $[(x_{a_1 b_1}\otimes \ldots \otimes x_{a_n b_n}) \otimes (\gamma_1, \ldots, \gamma_n)]$. By definition of $\mathcal{J}'$, we have $\widetilde{\kappa}_{\mathbf{M}}(\mathcal{I}) = \mathcal{J}'$, so $\widetilde{\kappa}$ induces an isomorphism

$$ \kappa_{\mathbf{M}} :  \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{M})]  \cong L(\mathbf{M}).$$

That $\kappa_{\mathbf{M}}$ is an isomorphism of $\mathcal{O}[\SL_2]$ comodules and that it is natural in $\mathbf{M}$  are immediate consequences of the definitions. We thus have defined a natural isomorphism $\kappa: \mathcal{O}[\mathcal{R}]\cong L$. This concludes the proof.

§.§ Classical limit of stated skein modules

Let $\mathbf{M}\in \mathcal{M}^{(1)}_{\con}$ and $\gamma \in \pi_1(\mathbf{M})$. Let $T(\gamma)$ be a tangle isotopic to $\gamma$ with arbitrary framing with distinct endpoints $\partial{T}=\{s,t\} \in \mathbb{D}_M$ such that $\gamma$ is oriented from $s$ (starting point) to $t$ (target point). For $i,j \in \{+, -\}$, let $\gamma_{ij}:= [T(\gamma), s_{ij}] \in \mathcal{S}_{+1}(\mathbf{M})$ be the class of the tangle $T(\gamma)$ with state defined by $s_{ij}(s):= v_i$ and $s_{ij}(t):= v_j$. The main result of this section is the following:

Let $w$ be a spin function.
We have a ring isomorphism $\eta_{w}: \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{M})] \xrightarrow{\cong} \mathcal{S}_{+1}(\mathbf{M})$ whose image on the generators $X_{ij}^{\gamma}$ is given by

$$ \eta_{w} \begin{pmatrix} X^{\gamma}_{++} & X^{\gamma}_{+-} \\ X^{\gamma}_{-+} & X^{\gamma}_{--} \end{pmatrix} := (-1)^{w(\gamma)}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \gamma_{++} & \gamma_{+-} \\ \gamma_{-+} & \gamma_{--} \end{pmatrix}.$$

Moreover $\eta_w$ is a morphism of $\mathcal{O}[\SL_2]$ comodules.

In particular, we obtain a variant of a  classical theorem of Bullock:
One has an isomorphism $\mathcal{O}[\mathcal{X}_{\SL_2}({M})] \cong \mathcal{S}_{+1}(M)$ sending a trace function $\tau_{\gamma}$ to $(-1)^{w(\gamma)}[\gamma]$.

* In the particular case where $\mathbf{M}=\mathbf{\Sigma}\times I$ is a thickened marked surface, Theorem <ref> was proved independently in [43] and [15] using triangulations. An alternative proof using explicit finite presentations of stated skein algebras was also presented in [40].

* Corollary <ref> is closely related to a theorem of Bullock in [11] who constructed an isomorphism between $\mathcal{O}[\mathcal{X}_{\SL_2}(\mathbf{M})] $ and the skein algebra  $\mathcal{S}_{A=-1}(M)$ evaluated in $A=-1$ by sending a trace function $\tau_{\gamma}$ to $-[\gamma]$. Putting these two results together, one obtains an isomorphism (depending on $w$) between the skein algebra in $A=-1$ and the skein algebra in $A=+1$ sending $[\gamma]$ to $(-1)^{w(\gamma)+1}[\gamma]$. In the particular case where  $\mathbf{M}=\mathbf{\Sigma}\times I$ is a thickened surface (so $w$ is given by a spin structure on $\Sigma$  as explained in Remark <ref>), the existence of such an isomorphism $\mathcal{S}_{+1}(\Sigma)\cong \mathcal{S}_{-1}(\Sigma)$ was proved by Barett in [4].

Let us define two quotients $\BT_0$ and $\BT_1$ of $\BT$. The objects of $\BT_0$ and of $\BT_1$ are the same than the objects of $\BT$, so are the handlebodies $\mathbf{H}_n$ for $n\geq 0$. Identify $\BT(a,b)=\BT(\mathbf{H}_a, \mathbf{H}_b)$ with the set $P_a(\mathbf{H}_b)$ of isotopy classes of $a$-bottom tangles in $\mathbf{H}_b$.
The set  $\BT_0(a,b)$ is defined as the quotient of $P_a(\mathbf{H}_b)$ by the skein relations $\Crosspos \leftrightarrow \Crossneg$ and $\adjustbox{valign=c}{\includegraphics[width=2cm]{TwistRel+1.eps}}$.
The set $\BT_1(a,b)$ is defined as the quotient of $P_a(\mathbf{H}_b)$ by the skein relations $\Crosspos \leftrightarrow \Crossneg$ and $\adjustbox{valign=c}{\includegraphics[width=2cm]{TwistReldouble.eps}}$.
The compositions in $\BT_0$ and $\BT_1$ are induced by the compositions in $\BT$ after passing to the quotient. We thus have projection functors

$$ \BT \xrightarrow{p'} \BT_1 \xrightarrow{p} \BT_0$$

which are the identity on objects. For $1\leq i \leq q$, let $\theta_i^{a} \in \BT_1(a,a)$ be the bottom tangle

$$\theta_i^a:= \adjustbox{valign=c}{\includegraphics[width=3cm]{BT_theta_i.eps}}. $$
Then $(\theta_i^a)^2=\id$ and the $\theta_i^a$ pairwise commute so they generate a group $G_a \cong (\mathbb{Z}/2\mathbb{Z})^a \subset \BT_1(a,a)$. This group naturally acts freely on the right of $\BT_1(a,b)$ and the quotient map $p: \BT_1(a,b) \to \BT_0(a,b)$ induces an isomorphism

$$ \widetilde{p} : \quotient{\BT_1(a,b)}{G_a} \xrightarrow{\cong} \BT_0(a,b).$$

Define also a section $s: \BT_0(a,b) \hookrightarrow \BT_1(a,b)$ such that $s\circ p = \id$ by lifting a class $[\alpha] \in \BT_0(a,b)$ to the class $[\alpha_0] \in \BT_1(a,b)$ where the bottom tangle $\alpha_0$ is chosen such that each of its component has self-linking number $0$. The section $s$ defines an isomorphism $\BT_1(a,b)\cong \BT_0(a,b)\times G_a$. We will now denote a morphism $\mu \in \BT_1(a,b)$ by $\mu=(\mu_0, g)$ where $\mu_0=s(\mu)$ and $g\in G_a$ is the unique element such that $\mu_0 \cdot g = \mu$. Note that $s: \BT_0 \to \BT_1$ is a functor.

Note that the right $\BT$-module $\restriction{\mathcal{O}[\mathcal{R}]}{\BT}$ passes to the quotient to a right $\BT_0$ module:  this follows from the facts that for $\mu : \mathbf{H}_a \to \mathbf{H}_b$ an embedding, the morphism $\mu_* : \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{H}_a)] \to \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{H}_b)]$ only depends on the morphism $\mu_* : \pi_1(\mathbf{H}_a)\to \pi_1(\mathbf{H}_b)$ i.e. only depends on the class of $\mu$ in $\BT_0(a,b)$.
We defined during the proof of Lemma <ref> an explicit isomorphism $\mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{M})] \cong \restriction{\mathcal{O}[\mathcal{R}]}{\BT} \otimes_{\BT} \mathbb{Z}[\pi_M]$ where $\pi_M$ is a right $\BT_0$ module obtained as a quotient of $P_M$. Therefore one has  an isomorphism

$$ \kappa_{\mathbf{M}}: \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{M})] \cong \restriction{\mathcal{O}[\mathcal{R}]}{\BT_0} \otimes_{\BT_0} \mathbb{Z}[\pi_M].$$

On the other hand, the right $\BT$-module $\restriction{\mathcal{S}_{+1}}{\BT}$ passes to the quotient to a right $\BT_1$ module: for $\mu : \mathbf{H}_a \to \mathbf{H}_b$ an embedding, the fact that the morphism $\mu_*: \mathcal{S}_{+1}(\mathbf{H}_a) \to \mathcal{S}_{+1}(\mathbf{H}_b)$ is invariant under the skein relations $\Crosspos \leftrightarrow \Crossneg$ and $\adjustbox{valign=c}{\includegraphics[width=2cm]{TwistReldouble.eps}}$ follows from the fact that the same skein relations hold in $\mathcal{S}_{+1}(\mathbf{M})$. By Lemma <ref>, one has an isomorphism

$$ \mathcal{S}_{+1}(\mathbf{M}) \cong \restriction{\mathcal{S}_{+1}}{\BT} \otimes_{\BT}  \mathbb{Z}[P_M] $$

which is explicited  in the proof of Theorem <ref> where it is denoted by $G^{-1}$. Let $\pi_M^{fr}$ be the left $\BT_1$ module sending $\mathbf{H}_n$ to the quotient of $P_n(\mathbf{M})$ by the relations $\Crosspos \leftrightarrow \Crossneg$ and $\adjustbox{valign=c}{\includegraphics[width=2cm]{TwistReldouble.eps}}$: so $\pi_M^{fr}$ is a quotient of $P_M$ and we have quotient maps (of left $\BT$-modules) $P_M \to \pi_M^{fr} \to \pi_M$. The isomorphism of Lemma <ref> can rewritten as

$$ g_M : \mathcal{S}_{+1}(\mathbf{M}) \cong \restriction{\mathcal{S}_{+1}}{\BT_1} \otimes_{\BT_1}  \mathbb{Z}[\pi^{fr}_M].$$

In order to prove Theorem <ref>, we are thus reduced to construct an isomorphism
$$\restriction{\mathcal{S}_{+1}}{\BT_1} \otimes_{\BT_1}  \mathbb{Z}[\pi^{fr}_M] \cong \restriction{\mathcal{O}[\mathcal{R}]}{\BT_0} \otimes_{\BT_0} \mathbb{Z}[\pi_M].$$

By precomposing the functor $\restriction{\mathcal{S}_{+1}}{\BT_1} : \BT_1 \to \Mod_{\mathbb{Z}}$ with $s:\BT_0 \to \BT_1$, we can see $\restriction{\mathcal{S}_{+1}}{\BT_1} $ as a right $\BT_0$ module. We now define an explicit isomorphism between this right $\BT_0$-module and $ \restriction{\mathcal{O}[\mathcal{R}]}{\BT_0}$.

Let $\beta_i \in P_1(\mathbf{H}_n)$ be the oriented bottom tangle depicted by $\beta_i:= \adjustbox{valign=c}{\includegraphics[width=2cm]{BT_Ti.eps}}$.

For each $n\geq 0$, one has  isomorphisms $\omega_n: \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{H}_n)] \xrightarrow{\cong} \mathcal{S}_{+1}(\mathbf{H}_n)$ of $\mathcal{O}[\SL_2]$ comodules algebras  characterized by the formula

$$ \omega_n \begin{pmatrix} X^{\beta_i}_{++} & X^{\beta_i}_{+-} \\ X^{\beta_i}_{-+} & X^{\beta_i}_{--} \end{pmatrix} := \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} {\beta_i}_{++} & {\beta_i}_{+-} \\ {\beta_i}_{-+} & {\beta_i}_{--} \end{pmatrix}.$$

Moreover, for $\mu_0 \in \BT_0(a,b)$ then the following diagram commutes

$$ \begin{tikzcd}
\mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{H}_a)]
\ar[r, "(\mu_0)_*"] \ar[d, "\omega_a"] &
\mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{H}_b)]
\ar[d, "\omega_b"] \\
\mathcal{S}_{+1}(\mathbf{H}_a)
\ar[r, "s(\mu_0)_*"] &
\mathcal{S}_{+1}(\mathbf{H}_b)
\end{tikzcd}

The fact that the $\omega_n$ are isomorphisms of rings and of $\mathcal{O}[\SL_2]$-comodules is a particular (easy) case of <cit.> (see also <cit.>). By Theorem <ref>,  every morphism in $\BT_0$ is obtained by composition and tensoring the generating morphisms $(\mu, \eta, \Delta, \epsilon, S^{\pm 1},  \theta^{\pm 1}) $ of Figure <ref>. So to prove the naturality of $\omega$, we need to prove the commutativity of the diagram in the particular cases where $s(\mu_0)$ is one of these morphisms. For $(\mu, \eta)$ this follows from the fact that $\omega_n$ are algebras morphisms. For $\Delta, \epsilon, S^{\pm 1}$ this follows from the computations made in the proof of Theorem <ref> to prove that $f$ is a morphism of braided Hopf algebra object. For $\theta^{\pm 1}$ this is obvious. This completes the proof.

Recall from Notations <ref> the group $G_a\subset \BT_1(a,a)$ generated by the involutive elements $\theta_i^a$. Let us describe the right action of $G_a$ on $\mathcal{S}_{+1}(\mathbf{H}_a)$. First define $\Theta \in \Aut( \mathcal{O}[\SL_2])$ the automorphism sending the generator $x_{ij}$ for $i,j =\pm $ to $-x_{ij}$. Recall that as a $\mathbb{Z}$-module one has $\mathcal{S}_{+1}(\mathbf{H}_a) \cong \mathcal{S}_{+1}(\mathbf{H}_1)^{\otimes a} \cong (\mathcal{O}[\SL_2])^{\otimes a}$ and define $\Theta_i^a \in \Aut (\mathcal{S}_{+1}(\mathbf{H}_a))$ by $\Theta_i^a = \id^{\otimes i-1}\otimes \Theta \otimes \id^{\otimes a-i}$. Said differently, the automorphism $\Theta_i^a$ sends each generator $(\beta_i)_{\varepsilon, \varepsilon'}$ to $- (\beta_i)_{\varepsilon, \varepsilon'}$ and each $(\beta_j)_{\varepsilon, \varepsilon'}$ with $j\neq i$ to $+(\beta_j)_{\varepsilon, \varepsilon'}$. In general, for $g=\theta_1^{n_1}\ldots \theta_a^{n_a} \in G_a$, we write $\Theta_g:= (\Theta_1^a)^{n_1}\ldots (\Theta_a^a)^{n_a}$.

For $g\in G_a$ and $x\in \mathcal{S}_{+1}(\mathbf{H}_a)$, one has $x\cdot g = \Theta_g(x)$, where $x \cdot g$ denotes the right action coming from the functoriality of $\mathcal{S}_{+1}$.

We need to prove that for each generator $(\beta_i)_{\varepsilon, \varepsilon'}$ of $\mathcal{S}_{+1}(\mathbf{H}_a)$ and for each generator $\theta_j^a$ of $G_a$, one has $(\beta_i)_{\varepsilon, \varepsilon'} \cdot \theta_i^a = (-1)^{\delta_{i,j}} (\beta_i)_{\varepsilon, \varepsilon'}$. This follows from the skein relation $\adjustbox{valign=c}{\includegraphics[width=2cm]{TwistRel.eps}}$ in $\mathcal{S}_{+1}(\mathbf{H}_a)$.

Note that the left $\BT$ module $\eta$ of Definition <ref> passes to the quotient to a left $\BT_1$ module and that
a spin function is, by definition, a morphism $w: \pi_M^{fr} \to \eta$ of left $\BT_1$ modules.
Let $[\beta_i] \in \mathrm{H}_1(\mathbf{H}_a; \mathbb{Z}/2\mathbb{Z})$ be the homology class of the closed curve obtained from $\beta_i$ by joining its endpoints, i.e. the simple closed curve encircling the $i$-th hole of the punctured disc $\mathbb{D}_a$. Let $\varphi_i \in  \mathrm{H}^1(\mathbf{H}_a; \mathbb{Z}/2\mathbb{Z})$ be the dual element sending $[\beta_j]$ to $\delta_{ij}$.
Consider the isomorphism $G_a \cong \mathrm{H}^1(\mathbf{H}_a; \mathbb{Z}/2\mathbb{Z})$ sending $\theta_i^a$ to $\varphi_i$. Using this isomorphism, for $\alpha \in \pi_M^{fr}(\mathbf{H}_a)$, we can consider the automorphism $\Theta_{w_1(\alpha)} \in \Aut(\mathcal{S}_{+1}(\mathbf{H}_a))$ and the group element $\theta_{w_1(\alpha)}\in G_a$.

Let us define an isomorphism

$$ f : \restriction{\mathcal{S}_{+1}}{\BT_1} \otimes_{\BT_1}  \mathbb{Z}[\pi^{fr}_M] \cong \restriction{\mathcal{O}[\mathcal{R}]}{\BT_0} \otimes_{\BT_0} \mathbb{Z}[\pi_M].$$

Identify the tensor products as

\begin{align*}
& \restriction{\mathcal{S}_{+1}}{\BT_1} \otimes_{\BT_1}  \mathbb{Z}[\pi^{fr}_M] = \quotient{\left( \oplus_{n \geq 0} \mathcal{S}_{+1}(\mathbf{H}_n) \otimes_{\mathbb{Z}} \mathbb{Z}[\pi^{fr}_M(\mathbf{H}_n)] \right)}{\mathcal{I}_1}, \\
& \restriction{\mathcal{O}[\mathcal{R}]}{\BT_0} \otimes_{\BT_0} \mathbb{Z}[\pi_M] = \quotient{\left( \oplus_{n \geq 0} \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{H}_n)] \otimes_{\mathbb{Z}} \mathbb{Z}[\pi_M(\mathbf{H}_n)] \right)}{\mathcal{I}_0},
\end{align*}

where $\mathcal{I}_i$ is spanned by elements of the form $u \cdot \mu \otimes v - u\otimes \mu\cdot v$ for $\mu$ a morphism in $\BT_i$. Let us slightly re-write the first quotient as

$$ \restriction{\mathcal{S}_{+1}}{\BT_1} \otimes_{\BT_1}  \mathbb{Z}[\pi^{fr}_M] = \quotient{\left( \oplus_{n \geq 0} \mathcal{S}_{+1}(\mathbf{H}_n) \otimes_{\mathbb{Z}[G_n]} \mathbb{Z}[\pi^{fr}_M(\mathbf{H}_n)] \right)}{\mathcal{I}'_1}$$

where $\mathcal{I}_1'$ is spanned by elements $u \cdot \mu \otimes v - u\otimes \mu\cdot v$ where $\mu$ is a morphism in $\BT_1$ such that $s(\mu)=\mu$.
For $n\geq 0$, we define an isomorphism

$$f_n : \mathcal{S}_{+1}(\mathbf{H}_n) \otimes_{\mathbb{Z}[G_n]} \mathbb{Z}[\pi^{fr}_M(\mathbf{H}_n)] \xrightarrow{\cong}  \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{H}_n)] \otimes_{\mathbb{Z}} \mathbb{Z}[\pi_M(\mathbf{H}_n)] $$

$$f_n ( x\otimes y):=  \omega_n^{-1}(x\cdot w_n(y)) \otimes p(y). $$
Here $p: \pi^{fr}_M(\mathbf{H}_n) \to \pi_M(\mathbf{H}_n)$ is the quotient map and $w_n : \pi_M^{fr}(\mathbf{H}_n) \to \mathrm{H}^1(\mathbf{H}_a; \mathbb{Z}/2\mathbb{Z})\cong G_n$ is the spin function.
The fact that $f_n$ is well-defined, i.e. that $f_n(x\cdot g \otimes y)=f_n(x\otimes g\cdot y)$ for all $g\in G_n$, comes from the naturality of $w_n$ with respect to the morphisms in $G_n$ together with the fact that $w_1\left(\adjustbox{valign=c}{\includegraphics[width=1cm]{BT_Theta.eps}}  \right)  =1$.
The fact that $f_n$ is an isomorphism comes from the fact that $\omega_n$ is an isomorphism together with the fact that $\pi_M(\mathbf{H}_n)=\quotient{\pi_M^{fr}(\mathbf{H}_n)}{G_n}$. The fact that $f_n$ is $\mathcal{O}[\SL_2]$ equivariant follows from the fact that $\omega_n$ is equivariant.
Let $\widetilde{f}:= \oplus_{n\geq 0} f_n$ and let us prove that $\widetilde{f}(\mathcal{I}'_1)= \mathcal{I}_0$. Consider an element $X= u \cdot \mu \otimes v - u\otimes \mu\cdot v \in \mathcal{I}'_1$ with $\mu\in  \BT_1(a,b)$ and $u\in \mathcal{S}_{+1}(\mathbf{H}_b)$, $v\in \pi_M^{fr}(\mathbf{H}_a)$. Then

\begin{align*}
f_a (u \cdot \mu \otimes v) &:= \omega_a^{-1}( u \cdot \mu \cdot \theta_{w_a(v)}) \otimes p(v) & {} \\
{} & = \omega_a^{-1}(u \cdot \theta_{w_b(\mu\cdot v)} \cdot \mu) \otimes p(v) & \mbox{ by naturality of }w \\
{} &= \omega_a^{-1}(u \cdot \theta_{w_b(\mu \cdot v)})\cdot \mu \otimes p(v) & \mbox{ by  Lemma \ref{lemma_classical1} and the fact that }s(\mu)=\mu.
\end{align*}

Thus,  writing $x:= u\cdot \theta_{w_b(\mu \cdot v)}$ and $y:=p(v)$ one has

$$ f_a( u\cdot \mu \otimes v) - f_b(u\otimes \mu \cdot v) = x \cdot \mu \otimes y - x \otimes \mu \cdot y \in \mathcal{I}_0.$$

Therefore, we have proved the inclusion  $\widetilde{f}(\mathcal{I}'_1)\subset \mathcal{I}_0$. To prove the reverse inclusion $\mathcal{I}_0 \subset \widetilde{f}(\mathcal{I}'_1)$, consider $x \in \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{H}_b)]$, $y\in \pi_M(\mathbf{H}_a)$ and $\mu_0 \in \BT_0(a,b)$ and consider the generator $X:= x\cdot \mu_0 \otimes y - x\otimes \mu_0\cdot y \in \mathcal{I}_0$. Let $\mu\in \BT_1(a,b)$ be a lift of $\mu_0$ such that $s(\mu)=\mu$ and choose $v \in \pi_M^{fr}(\mathbf{H}_a)$ a lift of $y$ such that $w_a(v)=0$. Set $u:= \omega_b(x)$ and $Y:= u \cdot \mu \otimes v - u \otimes \mu \cdot v \in \mathcal{I}_1'$. Then by definition one has $\widetilde{f}(Y)=X$ so $\mathcal{I}_0 \subset \widetilde{f}(\mathcal{I}'_1)$. We thus have proved that $\widetilde{f}(\mathcal{I}'_1)= \mathcal{I}_0$ so the isomorphism $\widetilde{f}$ induces an isomorphism

$$ f : \restriction{\mathcal{S}_{+1}}{\BT_1} \otimes_{\BT_1}  \mathbb{Z}[\pi^{fr}_M] \cong \restriction{\mathcal{O}[\mathcal{R}]}{\BT_0} \otimes_{\BT_0} \mathbb{Z}[\pi_M]$$

which is $\mathcal{O}[\SL_2]$-equivariant since $\widetilde{f}$ is equivariant as well.
Define the isomorphism $\eta_w^{-1} :  \mathcal{S}_{+1}(\mathbf{M}) \xrightarrow{\cong} \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{M})]$ as the composition

$$ \eta_w^{-1} :  \mathcal{S}_{+1}(\mathbf{M}) \xrightarrow{g_M} \restriction{\mathcal{S}_{+1}}{\BT_1} \otimes_{\BT_1}  \mathbb{Z}[\pi^{fr}_M]  \xrightarrow{f} \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{M})] \xrightarrow{\kappa_M^{-1}}  \mathcal{O}[\mathcal{R}_{\SL_2}(\mathbf{M})].$$

That $\eta_w$ is equivariant follows from the fact that each above map is equivariant. It remains to prove the  explicit formula for a stated arc given in Theorem <ref>. Write $C_{++}=C_{--}:=0$, $C_{-+}:=-C_{+-}=1$ so that the formula we need to prove writes $\eta_w(X_{ij}^{\gamma})= (-1)^{w_1(\gamma)} \sum_{k=\pm} C_{ik} \gamma_{kj}$. The arc $\gamma$ defines a $1$-bottom tangles $\gamma\in P_1(M)$ and we denote by $\gamma_0, \gamma_1$ its images in the quotients $\pi_M(\mathbf{H}_1)$ and $\pi_M^{fr}(\mathbf{H}_1)$ respectively. Then $\kappa_M$ (defined in the proof of  Lemma <ref>) sends $X_{ij}^{\gamma}$ to the class $[X^{\beta_1}_{ij}\otimes \gamma_0]$. The isomorphism $g_M$ (defined from the isomorphism $G^{-1}$ in the proof of Theorem <ref> by tensoring by $\mathbb{Z}$) sends a stated arc $\gamma_{ab}$ to the class $[(\beta_1)_{ab} \otimes \gamma_1]$. Now the isomorphism $\omega_1$ in Lemma <ref> reads $\omega_1(X^{\beta_1}_{ij})=\sum_k C_{ik} (\beta_1)_{kj}$ so the desired formula follows from the equalities

$$ \eta_M(X_{ij}^{\gamma})= g_M^{-1}\circ f \circ \kappa_M(X_{ij}^{\gamma})= g_M^{-1}\circ f ([X^{\beta_1}_{ij} \otimes \gamma_0]) = (-1)^{w_1(\gamma)} g_M^{-1}(\sum_{k= \pm} C_{ik} [(\beta_1)_{kj} \otimes \gamma_1]) = (-1)^{w_1(\gamma)} \sum_{k=\pm} C_{ik} \gamma_{kj}, $$

where we used that $(\beta_1)_{kj} \cdot \theta_{w_1(\gamma)} = \Theta_{w_1(\gamma)}((\beta_1)_{kj})= (-1)^{w_1(\gamma)}(\beta_1)_{kj}$ as proved in Lemma <ref>. This concludes the proof.

§ QUANTUM VAN KAMPEN THEOREMS

§.§ Quantum Van Kampen for quantum fundamental groups

Recall that $\mathbb{D}^2$ is the unit disc of $\mathbb{R}^2$, $h: \mathbb{D}^2 \to [-1,1]$ is the projection on the $y$ axis  and let $\partial_{+}\mathbb{D}, \partial_{-}\mathbb{D} \subset \partial \mathbb{D}^2$ and $\mathbb{D}^{+}, \mathbb{D}^- \subset \mathbb{D}^2$ be the subsets of points $p$ for which $h(p)\geq 0$ and $h(p)\leq 0$ respectively. For $\mathbf{M}=(M, \iota_M) \in \mathcal{M}_{\con}^{(1)}$, we write $\partial_{\pm}\mathbb{D}_M:= \iota_M(\partial_{\pm} \mathbb{D})$ and $\mathbb{D}_M^{\pm}:= \iota_M(\mathbb{D}^{\pm})$.
Let $\mathbf{M}_1, \mathbf{M}_2 \in \mathcal{M}_{\con}^{(1)}$, $\mathbf{\Sigma}=(\Sigma, a) \in \MS$ a connected marked surface with a single boundary arc $a$. Consider oriented embeddings $\phi_1: \overline{\Sigma} \hookrightarrow \partial M_1$ and $\phi_2: \Sigma \hookrightarrow \partial M_2$ sending $a$ to $\partial_- \mathbb{D}_{M_1}$ and $\partial_+\mathbb{D}_{M_2}$ respectively. Here $\overline{\Sigma}$ is $\Sigma$ with the opposite orientation.

Let $\mathbf{M}_1\cup_{\Sigma} \mathbf{M}_2 \in \mathcal{M}_{\con}^{(1)}$ be the marked $3$-manifold where

$$M_1\cup_{\Sigma} M_2= \quotient{ M_1 \bigsqcup M_2}{(\phi_1(p)\sim \phi_2(p), p\in \Sigma)}$$
and the boundary disc $\mathbb{D}_{M_1\cup_{\Sigma} M_2}$ is obtained by gluing $\mathbb{D}_{M_1}$ with $\mathbb{D}_{M_2}$ by identifying $\phi_1(a)=\partial_-\mathbb{D}_{M_1}$ with $\phi_2(a)=\partial_+ \mathbb{D}_{M_2}$.

By construction, there is a natural projection map $\pi^0 : \mathbf{M}_1\bigsqcup \mathbf{M}_2 \to \mathbf{M}_1\cup_{\Sigma} \mathbf{M}_2$. Since we prefer to work in the monoidal category $(\mathcal{M}^{(1)}_{\con}, \wedge)$, let us define a morphism $\pi : \mathbf{M}_1 \wedge \mathbf{M}_2 \to  \mathbf{M}_1\cup_{\Sigma} \mathbf{M}_2$ such that the diagram

$$ \begin{tikzcd}
M_1 \bigsqcup M_2 \ar[rd, "\pi^0"] \ar[dd, "\iota"] & {} \\
{} & M_1\cup_{\Sigma} M_2 \\
M_1\wedge M_2 \ar[ru, "\pi"] & {}
\end{tikzcd}$$

commutes up to isotopy (i.e. such that $\pi^0$ is isotopic to $\pi\circ \iota$), where $\iota$ is the natural inclusion. Recall that $\mathbb{T}$ is a ball $\mathbb{B}^3$ with three boundary discs and that  $M_1\wedge M_2$ is obtained from $M_1\bigsqcup M_2\bigsqcup \mathbb{T}$ by gluing $\mathbb{D}_{M_1}$ to one disc, say $e_1$ of $\mathbb{T}$ and by gluing $\mathbb{D}_2$ to another disc, say $e_2$ of $\mathbb{T}$. The closed intervals $\partial_- \mathbb{D}_{M_1}$ and $\partial_+ \mathbb{D}_{M_2}$ are glued along closed intervals $\partial_- e_1$ and $\partial_+e_2$ respectively. Isotope $e_2$ to a disc $e_2'$  such that $\partial_- e_1$ coincides with  $\partial_+e_2'$ and let $M'$ be the $3$-manifold obtained by gluing $M_1$ to $e_1$ and $M_2$ to $e_2'$ as in Figure <ref>, so that $M'$ is isotopic to $M_1\wedge M_2$. The manifold $M:= \quotient{M'}{(\phi_1(p)\sim \phi_2(p), p\in \Sigma)}$ is isotopic to $M_1\cup_{\Sigma} M_2$ so the quotient map
$M'\to M$ defines a surjective morphism $\pi : \mathbf{M}_1 \wedge \mathbf{M}_2 \to  \mathbf{M}_1\cup_{\Sigma} \mathbf{M}_2$ and clearly  $\pi^0$ is isotopic to $\pi\circ \iota$.

The morphism
$\pi : \mathbf{M}_1 \wedge \mathbf{M}_2 \to  \mathbf{M}_1\cup_{\Sigma} \mathbf{M}_2$ is called the gluing morphism.

An illustration of the gluing morphism $\pi : \mathbf{M}_1 \wedge \mathbf{M}_2 \to  \mathbf{M}_1\cup_{\Sigma} \mathbf{M}_2$ .

The gluing operation is associative up to isotopy, i.e. one has $(\mathbf{M}_1\cup_{\Sigma}\mathbf{M}_2)\cup_{\Sigma'} \mathbf{M}_3 = \mathbf{M}_1\cup_{\Sigma}(\mathbf{M}_2\cup_{\Sigma'} \mathbf{M}_3)$ and if $f_{(12)}:\mathbf{M}_1\wedge \mathbf{M}_2 \to \mathbf{M}_1 \cup_{\Sigma}\mathbf{M}_2$, $f_{(23)}: \mathbf{M}_2\wedge \mathbf{M}_3 \to \mathbf{M}_2\cup_{\Sigma'} \mathbf{M}_3$,
$f_{(12,3)}: (\mathbf{M}_1\cup_{\Sigma}\mathbf{M}_2)\wedge \mathbf{M}_3 \to \mathbf{M}_1\cup_{\Sigma}\mathbf{M}_2\cup_{\Sigma'} \mathbf{M}_3$
and $f_{(1,23)}: \mathbf{M}_1\wedge (\mathbf{M}_2\cup_{\Sigma'} \mathbf{M}_3) \to  \mathbf{M}_1\cup_{\Sigma}\mathbf{M}_2\cup_{\Sigma'} \mathbf{M}_3$
are the four gluing morphisms, then $f_{(12,3)}\circ (f_{(12)}\wedge \id_{M_3})$ is isotopic to $f_{(1,23)}\circ (\id_{M_1}\wedge f_{(23)})$.

The lemma is a straightforward consequence of the definitions.

By replacing $\mathbf{M}_1$ and $\mathbf{M}_2$ by $\mathbf{\Sigma}\times I$ in the preceding discussion, while gluing $\Sigma \times \{-1\}$ of the first copy to $\Sigma \times \{+1\}$ in the second copy, we get a gluing morphism
$\mu : (\mathbf{\Sigma}\times I)\wedge (\mathbf{\Sigma}\times I) \to \mathbf{\Sigma}\times I$ which endows $\mathbf{\Sigma}\times I$ with a structure of algebra in $(\mathcal{M}^{(1)}_{\con}, \wedge)$. Similarly, by gluing $M_1$ to $\Sigma\times I$ while identifying $\phi_1(p)\sim (p, +1)$ for $p\in \Sigma$, we obtain a gluing morphism $\nabla_1: \mathbf{M}_1\wedge \mathbf{\Sigma}\times I \to \mathbf{M}_1$ turning $\mathbf{M}_1$ into a right $\mathbf{\Sigma}\times I$ module in
$(\mathcal{M}^{(1)}_{\con}, \wedge)$. Similarly, by gluing $\mathbf{\Sigma}\times I$ to $\mathbf{M}_2$ while identifying $\phi_2(p)\sim (p, -1)$, we get a gluing morphism $\nabla_2 : \mathbf{\Sigma}\times I \wedge \mathbf{M}_2 \to \mathbf{M}_2$ turning $\mathbf{M}_2$ into a left $\mathbf{\Sigma}\times I$ module. Note that the diagram

$$ \begin{tikzcd}
\mathbf{M}_1\wedge (\mathbf{\Sigma} \times I) \wedge \mathbf{M}_2
\ar[r,shift left=.75ex,"\nabla_1\wedge \id"]
\ar[r,shift right=.75ex,swap,"\id \wedge \nabla_2"]
&   \mathbf{M}_1\wedge \mathbf{M}_2 \ar[r, "\pi"] & \mathbf{M}_1\cup_{\Sigma} \mathbf{M}_2
\end{tikzcd}$$

commutes up to isotopy by Lemma <ref>. By abuse of notations, for $i=1,2$, still denote by $\nabla_i$ the image of $\nabla_i$ by the quantum fundamental group functor $P$ and let us identify the elements $P_{M_1\wedge M_2}\cong P_{M_1}\otimes_D P_{M_2}$ and $P_{M_1\wedge \Sigma \wedge M_2} \cong P_{M_1} \otimes_D P_{\Sigma}\otimes_D P_{M_2}$ in $\widehat{\BT}$ using the isomorphism of (the proof of) Lemma <ref>.

$P_{M_1} \otimes_{P_{\Sigma}} P_{M_2}\in \widehat{\BT}$ be the coequalizer

$$ \begin{tikzcd}
P_{M_1} \otimes_D P_{\Sigma} \otimes_D P_{M_2}
\ar[r,shift left=.75ex,"\nabla_1\otimes \id"]
\ar[r,shift right=.75ex,swap,"\id \otimes \nabla_2"]
&  P_{M_1} \otimes_D P_{M_2} \ar[r] & P_{M_1}\otimes_{P_{\Sigma}} P_{M_2}.
\end{tikzcd}$$

Since the diagram

$$ \begin{tikzcd}
P_{M_1} \otimes_D P_{\Sigma} \otimes_D P_{M_2}
\ar[r,shift left=.75ex,"\nabla_1\otimes \id"]
\ar[r,shift right=.75ex,swap,"\id \otimes \nabla_2"]
&  P_{M_1} \otimes_D P_{M_2} \ar[r, "P(\pi)"] & P_{M_1\cup_{\Sigma}M_2}
\end{tikzcd}$$

commutes, there exists a unique morphism $\kappa : P_{M_1} \otimes_{P_{\Sigma}} P_{M_2} \to P_{M_1\cup_{\Sigma}M_2}$ making commuting the obvious diagram.

The following was proposed without proof by Habiro in [29]:
(Quantum Van Kampen theorem for quantum fundamental groups)

The morphism $\kappa : P_{M_1} \otimes_{P_{\Sigma}} P_{M_2} \to P_{M_1\cup_{\Sigma}M_2}$ is an isomorphism.

The proof of Theorem <ref> is the most technical part of the paper and will be cut into several lemmas.
Let us first introduce some terminology. Write $M:= M_1\cup_{\Sigma}M_2$ for simplicity. Consider the embedding $\iota_{M_1\wedge M_2}: M_1\wedge M_2 \hookrightarrow M$ defined as the composition

$$ \iota_{M_1\wedge M_2}: M_1\wedge M_2 \subset M_1\wedge (\Sigma\times I) \wedge M_2 \xrightarrow{\pi \circ (\nabla_1\wedge \id)} M.$$

Here $M_1\wedge (\Sigma\times I) \wedge M_2 $ is, by definition, a ball $\mathbb{B}^3$ with three discs on the boundary to which we have attached $M_1, M_2, \Sigma \times I$ and we see $M_1\wedge M_2 \subset M_1\wedge (\Sigma\times I) \wedge M_2$ as the union of $\mathbb{B}^3$ with $M_1, M_2$ only. Define also an embedding $\iota_{\Sigma}$  as the composition:

$$ \iota_{\Sigma} : \Sigma\times I \subset M_1\wedge (\Sigma\times I) \wedge M_2 \xrightarrow{\pi \circ (\nabla_1\wedge \id)} M.$$

So $M$ is the union $M= \iota_{M_1\wedge M_2}(M_1\wedge M_2) \cup \iota_{\Sigma}(\Sigma\times I)$.
Write $\mathbb{B}_M := \pi \circ (\nabla_1\wedge \id)(\mathbb{B}^3)$, so $\mathbb{B}_M \subset M$ is a ball containing the boundary disc $\mathbb{D}_M$ and contained in the image of $\iota_{M_1\wedge M_2}$. We can further decompose $M$ as the union $M= \iota_{M_1\wedge M_2}(M_1) \cup \iota_{M_1\wedge M_2}(M_2) \cup  \iota_{\Sigma}(\Sigma\times I) \cup \mathbb{B}_M$ as illustrated in Figure <ref>.

An illustration of the decomposition $M= \iota_{M_1\wedge M_2}(M_1) \cup \iota_{M_1\wedge M_2}(M_2) \cup  \iota_{\Sigma}(\Sigma\times I) \cup \mathbb{B}_M$ .

Let $\overline{M}_1:= \iota_{M_1\wedge M_2}(M_1) \cup  \iota_{\Sigma}(\Sigma\times I) \cup \mathbb{B}_M \subset M$ and $\overline{M}_2:= \iota_{M_1\wedge M_2}(M_2) \cup  \iota_{\Sigma}(\Sigma\times I) \cup \mathbb{B}_M \subset M$. So $\overline{M}_1\cup \overline{M}_2 = M$ and $\overline{M}_1\cap \overline{M}_2= \mathbb{B}_M \cup  \iota_{\Sigma}(\Sigma\times I) $. For $n\geq 0$, by abuse of notations, we denote by $P_n (M_1\wedge M_2)$ the set of isotopy classes of $n$-bottom tangles in $\iota_{M_1\wedge M_2}(M_1\wedge M_2)$, i.e. we now see $M_1\wedge M_2$ as a submanifold of $M$ via $\iota_{M_1\wedge M_2}$.

For $n\geq 1$, and $\alpha, \beta \in P_n(M_1\wedge M_2)$, write:

* $\alpha \sim_{\phi} \beta$ if $\alpha$ and $\beta$ are related by a composition of isotopies, each having support either in $\overline{M}_1$ or $\overline{M}_2$;

* $\alpha \sim_{\psi} \beta$ if $\alpha$ and $\beta$ are isotopic in $M$.

Identify the spaces $P_M(\mathbf{H}_n)$ and $P_n(M)$ using Lemma <ref> so that $P_{M_1}\otimes_{P_{\Sigma}}P_{M_2} (\mathbf{H}_n)$ becomes identified with the coequalizer:

$$ \begin{tikzcd}
P_n(\mathbf{M}_1\wedge (\mathbf{\Sigma} \times I) \wedge \mathbf{M}_2)
\ar[r,shift left=.75ex,"\nabla_1\wedge \id"]
\ar[r,shift right=.75ex,swap,"\id \wedge \nabla_2"]
&   P_n(\mathbf{M}_1\wedge \mathbf{M}_2) \ar[r, "P(\pi)"] & P_{M_1}\otimes_{P_{\Sigma}}P_{M_2} (\mathbf{H}_n).
\end{tikzcd}$$

Here $\nabla_i$ has image $\overline{M}_i$, so we have an isomorphism

\begin{equation}\label{eq_id1}
\varphi: P_{M_1}\otimes_{P_{\Sigma}}P_{M_2} (\mathbf{H}_n) \cong \quotient{P_n(\mathbf{M}_1\wedge \mathbf{M}_2)}{\sim_{\phi}}.
\end{equation}

The embedding $\iota_{M_1\wedge M_2}$ induces a map $\Psi:= (\iota_{M_1\wedge M_2})^*: P_n(M_1\wedge M_2) \to P_n(M)$ such that $\Psi(\alpha) = \Psi(\beta) \Leftrightarrow \alpha \sim_{\psi} \beta$.
Let $\Sigma^0:= \iota_{\Sigma}(\Sigma\times \{0\}) \subset M$ and $\Sigma^0\times I:= \iota_{\Sigma}(\Sigma \times I)$.
To prove the surjectivity of $\Psi$, we introduce the:

Let $\alpha$ be an $n$ bottom tangle in $M_1\wedge M_2 \subset M$ which is transverse to $\Sigma^0$.  Let $v\in \alpha \cap \Sigma^0$, so $v=\iota_{\Sigma}(v_0)$ for $v_0 \in \Sigma$,  and suppose that the connected component $\alpha(v)$ of $\alpha \cap (\Sigma^0 \times I)$ is the straight line $\iota_{\Sigma}(v_0\times I)$ with a fixed framing of the form $\iota_{\Sigma}(\overrightarrow{v}_0\times I)$ for $\overrightarrow{v}_0$ a fixed vector in the unitary tangent bundle $U_{v_0}\Sigma$ of $v_0$. Let $w\in \mathbb{B}_M$ and $\gamma: v \to w$ a smooth path in $\Sigma_0\cup \mathbb{B}_M$ which only intersects $\alpha$ in $v$.
The push of $\alpha$ along $\gamma$ is the bottom tangle $\alpha'$ obtained from $\alpha$ by replacing the framed arc $\alpha(v)$ by an arc of the form $\alpha'(v)=\gamma^{-1}\gamma$, i.e. $\alpha'(v)$ goes from $\iota(v_0, -1)$ to $w$ along $\gamma$ and comes back to $\iota(v_0, +1)$ along $\gamma^{-1}$ so that $\alpha'(v)$ does not intersect $\Sigma_0$ in Figure <ref>. The framing is chosen such that $\alpha(v)$ and $\alpha'(v)$ are isotopic, so are $\alpha$ and $\alpha'$.

An illustration of the pushing operation.

The map  $\Psi: P_n(M_1\wedge M_2) \to P_n(M)$ is surjective. Therefore, $\Psi$ induces a bijection:

\begin{equation}\label{eq_id2}
P_M(\mathbf{H}_n) \cong \quotient{P_n(\mathbf{M}_1\wedge \mathbf{M}_2)}{\sim_{\psi}}.
\end{equation}

Starting with a $n$-bottom tangle $\alpha$ in $M$, we first isotope it such that it intersects $\Sigma^0$ transversally as in Definition <ref> and then push one-by-one each point of $\alpha \cap \Sigma^0$ inside $\mathbb{B}_M$ along arbitrary paths in order to get a bottom tangle $\alpha'$ such that: $(1)$ $\alpha'$ is isotopic to $\alpha$ and $(2)$ $\alpha'$ is in the image of $\iota_{M_1\wedge M_2}$. This proves the surjectivity.

Obviously, $\alpha \sim_{\phi} \beta$ implies $\alpha \sim_{\psi} \beta$ so we get a surjective map $\quotient{P_n(\mathbf{M}_1\wedge \mathbf{M}_2)}{\sim_{\phi}} \to \quotient{P_n(\mathbf{M}_1\wedge \mathbf{M}_2)}{\sim_{\psi}}$ which, using the isomorphisms of Equations (<ref>) and (<ref>), corresponds the morphism $\kappa : P_{M_1} \otimes_{P_{\Sigma}} P_{M_2} \to P_{M_1\cup_{\Sigma}M_2}$ of Theorem <ref>. To prove the injectivity, we need to prove the

For $n\geq 1$ and $\alpha, \beta \in P_n(M_1\wedge M_2)$, $\alpha \sim_{\psi} \beta$ implies $\alpha \sim_{\phi} \beta$.

If Proposition <ref> is true for $n=1$, then it is true for every $n\geq 1$.

Let $n\geq 2$ and $\alpha, \beta \in \boldsymbol{P}_n(\mathbf{M}_1\wedge \mathbf{M}_2)$ and suppose that $\alpha$ and $\beta$ are related by an isotopy $H$ in $M$.  Let $\alpha= \alpha^{(1)} \cup \ldots \cup \alpha^{(n)}$ be the $n$ components of $\alpha$. Up to changing $H$ by an isotopic isotopy, one can suppose that $H$ is a composition of isotopies each of which moves one strand while leaving the other $n-1$ strands fixed. Therefore, by induction on the number of such isotopies in the decomposition, we are reduced to prove the lemma in the case where $H$ leaves $\alpha^{(2)}, \ldots, \alpha^{(n)}$ fixed whereas it only isotopes the component $\alpha^{(1)}$. Choosing an open tubular neighborhood $N$ of $\alpha^{(2)}\cup \ldots \cup \alpha^{(n)}$ and replacing $M$ by $M'= M\setminus N$, the isotopy $H$ induces an isotopy $H'$ between $\alpha^{(1)}$ and $\beta^{(1)}$ in $\boldsymbol{P}_1(M')$. Since we assume that Proposition <ref> holds for $n=1$, we can decompose $H'$ as a composition of isotopies with support in $\overline{M_1} \setminus N$ and $\overline{M_2}\setminus N$. Therefore $\alpha \sim_{\phi} \beta$ and this concludes the proof.

We are reduced to prove Proposition <ref> in the case $n=1$. We will imitate the proof of the classical Van Kampen theorem as it appears in Hatcher's book [31].  Fix a Riemannian metric on $M$ and let $\pi: UM\to M$ be the unitary tangent bundle. A $1$-ribbon tangle $\alpha$ can be seen as a smooth map $c_{\alpha} : [0,1] \to UM$ such that $(1)$ $c_{\alpha}$ is an embedding, $(2)$ $c_{\alpha}(0), c_{\alpha}(1) \in \mathbb{D}_M$ have distinct heights and framings towards the height direction and $(3)$ the framing is normal to the direction, i.e. for all $t\in [0,1]$ if $v:=\pi \circ c_{\alpha}(t)$ then $c_{\alpha}(t)$ is orthogonal to $\restriction{\frac{d}{dt}\pi\circ c_{\alpha}}{t}$ in $U_vM$.
Conditions $(1)$, $(2)$ and $(3)$ are the reasons why the proof of the quantum Van Kampen theorem is slightly more involved than the proof of its classical version in [31].
An isotopy between $\alpha$ and $\beta$ is a smooth map $H: [0,1]^2 \to UM$ such that $H(0, \cdot) = c_{\alpha}(\cdot)$, $H(1, \cdot)=c_{\beta}(\cdot)$ and $H(s, \cdot)$ satisfies $(1)$, $(2)$ and $(3)$ for all $s\in [0,1]$.
Since $\overline{M}_1 \cup \overline{M}_2= M$, we can always find integers $a,b \geq 1$ and sequences $0=t_0 < \ldots < t_a=1$ and $0=s_0< \ldots < s_b=1$ such that $H([s_i, s_{i+1}] \times [t_j, t_{j+1}])$ is either included in $U\overline{M}_1$ or in $U\overline{M}_2$ for all $0\leq j \leq a-1$, $0\leq i \leq b-1$. The size of $H$ is the minimal such pair $(a,b)$. We will prove Lemma <ref> by decreasing induction on $a$ and $b$. More precisely, let $\mathcal{P}(a,b)$ be the property: if $\alpha, \beta\in P_1(\mathbf{M}_1\wedge \mathbf{M}_2)$ are related by an isotopy in $M$ of size $(n,m)$ with $n\leq a$, $m\leq b$ then $\alpha\sim_{\phi} \beta$.

If $\mathcal{P}(a,b)$ holds, so does $\mathcal{P}(a,b+1)$.

Suppose that $\alpha, \beta$ are related by an isotopy $H$ of size $(a,b+1)$ with parameters $0=t_0 < \ldots < t_a=1$ and $0=s_0< \ldots < s_b=1$. Let $\eta \in \boldsymbol{P}_1(\mathbf{M})$ be the bottom tangle parametrized by $c_{\eta}(t):= H(s_{a-1}, t)$. Let $H_1$ be the isotopy between $\alpha$ and $\eta$ defined by $H_1(s,t)=H(\frac{s}{s_{a-1}}, t)$ and $H_2$ the isotopy between $\eta$ and $\beta$ be $H_2(s,t)=H\left( (1-s_{a-1})s+s_{a-1}, t \right)$. One can suppose that  $\eta$ intersects $\Sigma_0$ transversally  in at most $a-1$ points. For each such point $v$, choose a path $\gamma_v$ between $v$ and a point in $\mathbb{B}_M$ not intersecting $\eta$ and let $\eta'$ be the bottom tangle obtained from $\eta$ by pushing each $v$ along $\gamma_v$. So $\eta' \subset M_1\wedge M_2$ and we have an isotopy $H_3$ between $\eta$ and $\eta'$ with support in $\overline{M}_1\cap \overline{M}_2$. By composing $H_1$ with $H_3$ we get an isotopy between $\alpha$ and $\eta'$ of size $(a,b)$ so $\alpha\sim_{\phi} \eta'$ by hypothesis. By composing the inverse of $H_3$ with $H_2$ we get an isotopy between $\eta'$ and $\beta$ of size $(a,1)$ so $\eta' \sim_{\phi} \beta$ by hypothesis. Therefore $\alpha\sim_{\phi} \beta$.

If $\mathcal{P}(a,1)$ holds, so does $\mathcal{P}(a+1,1)$.

Suppose that $\mathcal{P}(a,1)$ holds.
Let $\alpha$ and $\beta$ be related by an isotopy $H$ of size $(a+1,1)$ and parameters $0=t_0<\ldots< t_{a+1}$.
Consider the two squares $R_1=[0, t_1]\times [0,1]$ and $R_2=[t_1, t_2]\times [0,1]$. By minimality of $a$, $H(R_1)$ and $H(R_2)$ lies in distinct $\overline{M}_i$. Without loss of generality, one can suppose that $H(R_1)\subset \overline{M}_1$ and $H(R_2)\subset \overline{M}_2$. Let $e:= \{t_1\} \times [0,1]=R_1\cap R_2$ so that $H(e) \subset \overline{M}_1\cap \overline{M}_2 = \mathbb{B}_M \cup \iota_{\Sigma}(\Sigma\times I)$. Let $\alpha_s$ be the bottom tangle parametrized by $H(\cdot, s)$ so that $\alpha_0=\alpha$ and $\alpha_1=\beta$.

We want to "cut" the homotopy $H$ in two parts: the restriction to $R_1$, seen as an isotopy of size $(1,1)$  and the restriction to $[t_1,1]\times [0,1]$ seen an isotopy of size $(a,1)$. The equivalence $\alpha\sim_{\phi}\beta$ will then result from $\mathcal{P}(a,1)$.  We will proceed in three steps.
Step 1: Reduction to the case where $H(t_1, 1)\in \mathbb{B}_M$

Let $v_1:= \pi (H(t_1,1)) \in  \mathbb{B}_M \cup \iota_{\Sigma}(\Sigma\times I)$. By slightly isotoping $\beta$, one can suppose that $v_1\in \mathbb{B}_M\cup \Sigma^0$. If $v_1\in \Sigma^0$, choose $w_1\in \mathbb{B}_M$ and $\gamma$ path between $v_1$ and $w_1$ not intersecting $\beta$ and let $\beta'$ be the push of $\beta$ along $\gamma$. By post-composing $H$ with an isotopy between $\beta$ and $\beta'$ with support in $\mathbb{B}_M \cup \iota_{\Sigma}(\Sigma\times I)$, one gets an isotopy $H'$ between $\alpha$ and $\beta'$ of the same size $(a+1,1)$. Since $\beta\sim_{\phi} \beta'$, up to replacing $\beta$ and $H$ by $\beta'$ and $H'$, we can, and do,  suppose that $H(t_1, 1)\in \mathbb{B}_M$.

Step 2: Reduction to the case where $H( e)$ is a constant vector  in $U\mathbb{B}_M$

For $s\in [0,1]$, write $\overrightarrow{v}_s:= H(t_1,s) \in  U (\mathbb{B}_M\cup \iota_{\Sigma}(\Sigma \times I))$ and $v_s= \pi(\overrightarrow{v}_s)$. So $s\mapsto v_s$ parametrizes a smooth path $\delta$ in $\mathbb{B}_M\cup \iota_{\Sigma}(\Sigma \times I)$ between $v_0$ and $v_1 \in \mathbb{B}_M$. Up to slightly isotoping $\alpha$ and $H$, we can suppose that $\delta$ does not intersect $\alpha$. Let $\alpha'$ be the push of $\alpha$ along $\delta$ and let $H^0$ be an isotopy between $\alpha'$ and $\alpha$ with support in $\mathbb{B}_M \cup \iota_{\Sigma}(\Sigma\times I)$. Let $H'$ be the isotopy between $\alpha'$ and $\beta$ obtained by composing $H^0$ with $H$. By replacing $\alpha$ by $\alpha'$ and $H$ by $H'$, we are reduced to the case where the path $s\to \overrightarrow{v}_s$ is a contractible loop in $U\mathbb{B}_M$ (with base point $\overrightarrow{v}_1$). So, up to isotoping $H$, one can (and do) suppose that $H(e)$ is the constant vector

Step 3: Cutting the isotopy in two
Let $\eta$ be the $1$-bottom tangle parametrized by $c_{\eta}(t) = H(1, t)$ for $0\leq t \leq t_1$ and $c_{\eta}(t)=H(0,t)$ for $t_1\leq t \leq 1$. Define an isotopy $H^L$ between $\alpha$ and $\eta$ by $H^L(s,t)=H(s,t)$ for $0\leq t \leq t_1$ and $H^L(s,t)=H(0,t)$ for $t_1\leq t \leq 1$. Since $H^L$ has support in $H(R_1)\subset \overline{M}_1$, one has $\alpha \sim_{\phi} \eta$. Define an isotopy $H^R$ between $\eta$ and $\beta$ by $H^R(s,t)= H(1,t)$ for $0\leq t \leq t_1$ and $H^R(s,t)=H(s,t)$ for $t_1\leq t \leq 1$. Then $H^R$ has size $(1,a)$ so by hypothesis, we have $\eta\sim_{\phi} \beta$. Therefore $\alpha \sim_{\phi} \beta$.

Since the property $\mathcal{P}(1,1)$ is trivially satisfied, by induction on $a$ Lemma <ref> implies that $\mathcal{P}(a, 1)$ holds for every $a\geq 1$ and by induction on $b$, Lemma <ref> implies that $\mathcal{P}(a,b)$ holds for every $a,b\geq 1$. Therefore Proposition <ref> holds for $n=1$ and so, by Lemma <ref>, it holds for every $n\geq 1$. This completes the proof.

The theorem follows from the commutativity of the square:

$$ \begin{tikzcd}
P_{M_1}\otimes_{P_{\Sigma}}P_{M_2} (\mathbf{H}_n)
\ar[r, "\kappa"] \ar[d, "\cong", "\varphi"'] &
\ar[d, "\cong", "\Psi"'] \\
\quotient{P_n(\mathbf{M}_1\wedge \mathbf{M}_2)}{\sim_{\phi}}
\ar[r, "="] &
\quotient{P_n(\mathbf{M}_1\wedge \mathbf{M}_2)}{\sim_{\psi}}
\end{tikzcd}$$

where the equality on the bottom follows from Proposition <ref>.

§.§ Quantum Van Kampen for quantum representation spaces and stated skein modules

Let $\mathbf{M}_1$, $\mathbf{M}_2$ and $\mathbf{\Sigma}$ be as in the last subsection. Since $\mathbf{M}_1$ and $\mathbf{M}_2$ are right and left modules over $\mathbf{\Sigma}\times I$ in $(\mathcal{M}^{(1)}_{\con}, \wedge)$, then $\Rep^G_q(\mathbf{M}_1)$ and $\Rep^G_q(\mathbf{M}_2)$ are left and right modules over $\Rep^G_q(\mathbf{\Sigma})$ by monoidality of $\Rep^G_q$. The following was suggested by Habiro in [29].

One has an isomorphism $\Rep^G_q(\mathbf{M}_1\cup_{\Sigma} \mathbf{M}_2) \cong \Rep^G_q(\mathbf{M}_1) \otimes_{\Rep^G_q(\mathbf{\Sigma})}\Rep^G_q(\mathbf{M}_2)$.

By Theorem <ref>, the following is a coequalizer

$$ \begin{tikzcd}
P_{M_1} \otimes_D P_{\Sigma} \otimes_D P_{M_2}
\ar[r,shift left=.75ex,"\nabla_1\otimes \id"]
\ar[r,shift right=.75ex,swap,"\id \otimes \nabla_2"]
&  P_{M_1} \otimes_D P_{M_2} \ar[r, "P(\pi)"] & P_{M_1 \cup_{\Sigma} M_2}
\end{tikzcd}$$

thus the following sequence is exact:

$$ \begin{tikzcd}
k[P_{M_1}] \otimes_D k[P_{\Sigma}] \otimes_D k[P_{M_2}]
\ar[rr,"\nabla_1\otimes \id- \id \otimes \nabla_2"]
&{}&  k[P_{M_1}] \otimes_D k[P_{M_2}] \ar[r, "P(\pi)"] & k[P_{M_1 \cup_{\Sigma} M_2}]
\ar[r] & 0.
\end{tikzcd}$$

Since tensoring on the right preserves right exact sequences, the following sequence is exact as well:

\begin{multline*}
(k[P_{M_1}] \otimes_D k[P_{\Sigma}] \otimes_D k[P_{M_2}]) \otimes_{\BT} Q_{B_qG}
\xrightarrow{\nabla_1\otimes \id- \id \otimes \nabla_2}
(k[P_{M_1}] \otimes_D k[P_{M_2}])\otimes_{\BT} Q_{B_qG}
\\ \xrightarrow{P(\pi)} k[P_{M_1 \cup_{\Sigma} M_2}]\otimes_{\BT} Q_{B_qG}
\to 0.
\end{multline*}

Using Lemmas <ref> and  <ref>, we get the exact sequence

$$ \begin{tikzcd} \Rep_q^G(\mathbf{M}_1)\otimes_{\Rep_q^G(\mathbf{\Sigma})} \Rep_q^G(\mathbf{M}_2)  \ar[rr,"\nabla_1\otimes \id- \id \otimes \nabla_2"]
&{}&\Rep_q^G(\mathbf{M}_1)\otimes \Rep_q^G(\mathbf{M}_2) \ar[r, "P(\pi)"] & \Rep_q^G(\mathbf{M}_1\cup_{\mathbf{\Sigma}}\mathbf{M}_2)
\ar[r] & 0,
\end{tikzcd}$$

which says that $\Rep_q^G(\mathbf{M}_1\cup_{\mathbf{\Sigma}} \mathbf{M}_2)\cong \Rep_q^G(\mathbf{M}_1)\otimes_{\Rep_q^G(\mathbf{\Sigma})}\Rep_q^G(\mathbf{M}_2)$.

Putting Corollary <ref> and Theorem <ref> together, we obtain the following reformulation of Theorem <ref>.

One has an isomorphism $\mathcal{S}_q(\mathbf{M}_1\cup_{\Sigma} \mathbf{M}_2) \cong \mathcal{S}_q(\mathbf{M}_1) \otimes_{\mathcal{S}_q(\mathbf{\Sigma})}\mathcal{S}_q(\mathbf{M}_2)$.

One can derive from Corollary <ref> and Theorem <ref> an alternative proof of a theorem of Gunningham-Jordan-Safronov in [26] as follows. Let $M$ be a closed compact connected oriented $3$-manifold and suppose that $M=N_1\cup_{\Sigma}N_2$ is obtained by gluing two (compact, oriented, connected) $3$-manifolds along their connected boundary $\Sigma$. Choose a ball $\mathbb{B}^3 \subset M$ which intersects $\Sigma\subset M$ along a disc $\mathbb{D}= \mathbb{B}^3\cap \Sigma$ such that $\mathbb{D}$ cuts $\mathbb{B}^3$ in two hemispheres. Let $\mathbb{D}_M \subset \partial \mathbb{B}^3$ be a disc such that $\mathbb{D}_M$ intersects $\mathbb{D}$ along a closed interval $a$ which cuts $\mathbb{D}_M$ into two half-discs $\mathbb{D}_M^+\subset N_1$ and $\mathbb{D}_M^- \subset N_2$. Choose an oriented homeomorphism $\iota_M: \mathbb{D}^2 \cong \mathbb{D}_M$ sending $\mathbb{D}^{\pm}$ to $\mathbb{D}_M^{\pm}$ and set $\iota_{\pm}: \mathbb{D}_2\cong \mathbb{D}_{\pm} \xrightarrow{\iota_M} \mathbb{D}_M^{\pm}$.
Let $\mathbf{M}:= (M\setminus \mathring{\mathbb{B}}^3, \iota_M) \in \mathcal{M}_{\con}^{(1)}$ and consider $\mathbf{N}_1:= (N_1\setminus \mathring{\mathbb{B}}^3_+, \iota^+)$ and $\mathbf{N}_2:= (N_2 \setminus \mathring{\mathbb{B}}^3_-, \iota^-)$ and write $\mathbf{\Sigma}:= (\Sigma\setminus \mathring{\mathbb{D}}, a) \in \MS_{\con}^{(1)}$ such that $\mathbf{M}= \mathbf{N}_1\cup_{\mathbf{\Sigma}} \mathbf{N}_2$. By Theorem <ref>, the embedding $\iota: M \to \mathbf{M}$ induces an isomorphism $\iota_*: \mathcal{S}_q(M) \cong (\mathcal{S}_q(\mathbf{M}))^{coinv}$ and by Corollary <ref>, one has an isomorphism $\mathcal{S}_q(\mathbf{M})\cong \mathcal{S}_q(\mathbf{N}_1)\otimes_{\mathcal{S}_q(\mathbf{\Sigma})} \mathcal{S}_q(\mathbf{N}_2)$. Therefore, using Theorem <ref>, we have re-proved the

(Gunningham-Jordan-Safronov <cit.>) We have  isomorphisms:

\begin{align*}
{}& \mathcal{S}_q(M) \cong  \left(\mathcal{S}_q(\mathbf{N}_1)\otimes_{\mathcal{S}_q(\mathbf{\Sigma})} \mathcal{S}_q(\mathbf{N}_2)\right)^{coinv}, \quad \mbox{ and }\\
{}& \mathcal{S}^{rat}_q(M) \cong  \mathcal{S}^{rat}_q(\mathbf{N}_1)\otimes_{\mathcal{S}^{rat}_q(\mathbf{\Sigma})} \mathcal{S}^{rat}_q(\mathbf{N}_2).
\end{align*}

Applied to a Heegaard splitting $M=H_g\cup_{\Sigma_g}H_g$ of $M$, this theorem, together with an algebraic analogue of a theorem of Kashiwara-Schapira, was proved in <cit.> to imply that $\mathcal{S}^{rat}_q(M)$ has finite dimension. Note that, because of Corollary <ref>, $\mathcal{S}_q(M)$ is infinitely generated whenever the character variety of $M$ is not $0$ dimensional so the torsion part of $\mathcal{S}_q(M)$ is infinitely generated in this case.

§.§ Self-gluing and quantum HNN extensions

In the last subsections, we have considered the operation of gluing two $3$ manifolds along a surface and proved that $P_{M_1\cup_{\Sigma} M_2}$ is isomorphic to $P_{M_1}\otimes_{P_{\Sigma}}P_{M_2}$. We now consider the operation, giving a $3$ manifold $M$ and two attaching maps $\phi_1: \Sigma \hookrightarrow \partial M$, $\phi_2: \overline{\Sigma} \hookrightarrow \partial M$ of gluing the two copies of $\Sigma$ together to get a new manifold $M_{\phi_1 \# \phi_2}$ and will prove that $P_{M_{\phi_1 \# \phi_2}}$ is isomorphic to $\mathrm{HH}^0(P_{\Sigma}, P_M)$, where $\mathrm{HH}^0$ is defined in a braided sense. We will proceed by identifying $M_{\phi_1 \# \phi_2}$ with $\Sigma\times I \cup_{\overline{\Sigma} \wedge \Sigma} M$ and use the quantum Van Kampen theorem. The proof will then be a braided version of the classical proof of the fact that for $M$ a bimodule over some algebra $A$, then $\mathrm{HH}^0(A, M)$ is isomorphic to $A \otimes_{A^{op}\otimes A} M$.

Let $\mathbf{M}\in \mathcal{M}_{\con}^{(1)}$, $\mathbf{\Sigma}\in \MS_{\con}^{(1)}$ and $\phi_1: \Sigma \hookrightarrow \partial M$, $\phi_2: \overline{\Sigma} \hookrightarrow \partial M$ two embeddings such that: $(1)$ the boundary arc $a$ of $\mathbf{\Sigma}$ is sent to $\partial_+ \mathbb{D}_M$ by both $\phi_1$ and $\phi_2$ and $(2)$ the
images of $\phi_1$ and $\phi_2$  intersect in a contractible neighborhood of  $\partial_+ \mathbb{D}_M$. We define an embedding $\phi: \overline{\Sigma}\wedge \Sigma \hookrightarrow \partial M$, illustrated in Figure <ref>, as follows.
Divide the arc $\partial_+ \mathbb{D}_M$ into two segments $\partial_+ \mathbb{D}_M= \partial_{+,L} \mathbb{D}_M\cup \partial_{+, R} \mathbb{D}_M$ where $\partial_{+,L} \mathbb{D}_M= \iota_M \left( \partial \mathbb{D}^2 \cap \mathbb{R}^{\leq 0} \times \mathbb{R}^{\geq 0}\right)$ and $\partial_{+,R} \mathbb{D}_M= \iota_M \left( \partial \mathbb{D}^2 \cap \mathbb{R}^{\geq 0} \times \mathbb{R}^{\geq 0}\right)$. Write $\{p^+\}=\partial_{+,L} \mathbb{D}_M \cap \partial_{+,R} \mathbb{D}_M$. Isotope $\phi_1$ and $\phi_2$ slightly in the neighborhood of $\partial \mathbb{D}_M$ such that $(1)$ $\phi_1$ maps $a$ to $\partial_{+,L} \mathbb{D}_M$ and $\phi_2$ maps $a$ to $\partial_{+,R} \mathbb{D}_M$ and $(2)$ the intersection of the images of $\phi_1$ and $\phi_2$ is $\{p^+\}$. Define an  embedding $\phi: \overline{\Sigma}\wedge \Sigma \hookrightarrow \partial M$ by pushing slightly $\partial_{+,L} \mathbb{D}_M$  and $\partial_{+,R} \mathbb{D}_M$ inside $\partial M$ and considering an embedding of the  triangle $T$ with edges $\partial_+ \mathbb{D}_M$,  $\partial_{+,L} \mathbb{D}_M$  and $\partial_{+,R} \mathbb{D}_M$. This embedding together with $\phi_1, \phi_2$ glue together to define $\phi$.
Consider the cylinder $\mathbf{\Sigma}\times I$ and define an embedding $\psi: \Sigma \wedge \overline{\Sigma} \hookrightarrow \partial (\Sigma\times I)$ as follows. Write $\partial_- \mathbb{D}_{\mathbf{\Sigma}}= \partial_{-, L} \mathbb{D}_{\mathbf{\Sigma}} \cup \partial_{-, R} \mathbb{D}_{\mathbf{\Sigma}}$ as before. Isotope the embedding $\Sigma \cong \Sigma\times \{+1\} \hookrightarrow \partial (\Sigma\times I)$ to get an embedding $\psi_1: \Sigma \hookrightarrow \partial (\Sigma\times I)$ sending $a$ to $\partial_{-, R} \mathbb{D}_{\mathbf{\Sigma}}$. Isotope the embedding $\overline{\Sigma} \cong \Sigma \times \{-1\} \hookrightarrow \partial \Sigma \times I$ to get an embedding $\psi_2: \overline{\Sigma} \hookrightarrow \partial (\Sigma \times I)$ sending $a$ to $\partial_{-, L} \mathbb{D}_{\mathbf{\Sigma}}$. Like before, push slightly $\partial_{-, L} \mathbb{D}_{\mathbf{\Sigma}}$ and $\partial_{+, L} \mathbb{D}_{\mathbf{\Sigma}}$ inside and define an embedding of the triangle with edges $\partial_- \mathbb{D}_{\mathbf{\Sigma}}$, $\partial_{-, L} \mathbb{D}_{\mathbf{\Sigma}}$ and $\partial_{-, R} \mathbb{D}_{\mathbf{\Sigma}}$. This embedding together with $\psi_1$, $\psi_2$ define the embedding $\psi: \Sigma \wedge \overline{\Sigma} \hookrightarrow \partial (\Sigma\times I)$.

An illustration of the self-gluing operation where $\mathbf{M}= \mathbb{D}_2\times I$, the attaching map $\phi_1$ is given by a braid and $\phi_2$ is given by the identity. The resulting manifold $\mathbf{M}_{\phi_1\# \phi_2}$ is the exterior of the link obtained by closing the braid (here the Hopf link) with a ball removed.

The marked $3$ manifold $\mathbf{M}_{\phi_1 \# \phi_2} \in \mathcal{M}_{\con}^{(1)}$ is the gluing $\mathbf{M}_{\phi_1 \# \phi_2} := (\mathbf{\Sigma}\times I)\cup_{\overline{\Sigma}\wedge \Sigma} \mathbf{M}$ defined by the two attaching maps $\phi$ and $\psi$. We denote by $\Psi: \mathbf{M} \to \mathbf{M}_{\phi_1 \# \phi_2}$ the natural embedding.

While gluing the cylinder $\mathbf{\Sigma}\times I$ to $\mathbf{M}$ using the gluing maps $\phi_1$ and $\psi_2$ we get a marked $3$-manifold $(\mathbf{\Sigma}\times I)\cup_{\Sigma}\mathbf{M}$ isomorphic to $\mathbf{M}$, so the gluing map defines an embedding
$\nabla_{\phi_1}^L : (\mathbf{\Sigma}\times I) \wedge \mathbf{M} \to \mathbf{M}$. Similarly, while gluing $\mathbf{M}$ with the cylinder $\mathbf{\Sigma}\times I$ using $\phi_2$ and $\psi_1$, we also get a marked $3$-manifold $(\mathbf{\Sigma}\times I)\cup_{\Sigma}\mathbf{M}$ isomorphic to $\mathbf{M}$, so the gluing map defines an embedding  $\nabla_{\phi_2}^R : \mathbf{M} \wedge (\mathbf{\Sigma}\times I)\to \mathbf{M}$.

Let $(\mathcal{C}, \otimes, c_{\cdot, \cdot}, \theta_{\cdot})$ be a braided balanced category, $A\in \mathcal{C}$ an algebra object and $M \in \mathcal{C}$ an $A$-bimodule object with left and right modules maps $\nabla^L : A\otimes M \to M$ and $\nabla^R: M\otimes A \to M$. The $0$-th Hochschild cohomology group $\mathrm{HH}^0(A,M)$  is  (when it exists) the coequalizer

$$ \begin{tikzcd}
A\otimes M
\ar[rr,shift left=.75ex,"\nabla^L"]
\ar[rr,shift right=.75ex,swap,"\nabla^R \circ c_{A, M} \circ (\theta_A \otimes \id_M)"]
&{}& {M}
\ar[r]
\end{tikzcd}$$

Note that if $(M, \nabla^R)$ is a right $A$-module object in $\mathcal{C}$ then $(M, \nabla^R \circ c_{A,M} \circ (\theta_A \otimes \id_M) )$ is a left $A$-module object.

The maps $\nabla_{\phi_1}^L$ and $\nabla_{\phi_2}^R$ give $\mathbf{M}$ the structure of a bimodule over $\mathbf{\Sigma}\times I$ in $\mathcal{M}_{\con}^{(1)}$. Composing with $P: \mathcal{M}_{\con}^{(1)}\to \widehat{\BT}$, we obtain that $P_M$ is a $P_{\Sigma}$ bimodule in $\widehat{\BT}$. Similarly composing with $\Rep_q^G$, we obtain that $\Rep_q^G(\mathbf{M})$ is a bimodule over $\Rep_q^G(\mathbf{\Sigma})$ in $\overline{\mathcal{C}_q^G}$.

* One has $P_{M_{\phi_1\# \phi_2}}= \mathrm{HH}^0(P_{\Sigma}, P_M)$.

* One has $\Rep_q^G(\mathbf{M}_{\phi_1\# \phi_2}) = \mathrm{HH}^0( \Rep_q^G(\mathbf{\Sigma}), \Rep_q^G(\mathbf{M}))$ and  $\mathcal{S}_q(\mathbf{M}_{\phi_1\# \phi_2}) = \mathrm{HH}^0( \mathcal{S}_q(\mathbf{\Sigma}), \mathcal{S}_q(\mathbf{M}))$.

By the quantum Van Kampen Theorem <ref>, the image by $P: \mathcal{M}_{\con}^{(1)}\to \widehat{\BT}$ of the following sequence is a coequalizer (here and henceforth we write $\mathbf{\Sigma}$ instead of $\mathbf{\Sigma}\times I$ for simplicity):

$$ \begin{tikzcd}
\mathbf{\Sigma} \wedge( \overline{\mathbf{\Sigma}} \wedge \mathbf{\Sigma} ) \wedge \mathbf{M}
\ar[rr,shift left=.75ex,"\nabla_1\wedge \id"]
\ar[rr,shift right=.75ex,swap,"\id \wedge \nabla_2"]
&{}& \mathbf{\Sigma} \wedge \mathbf{M}
\ar[r, "\Psi'"]
&\mathbf{M}_{\phi_1\# \phi_2}.
\end{tikzcd}$$

We want to deduce from this fact that the image by $P$ of the following sequence is a coequalizer too:

$$ \begin{tikzcd}
\mathbf{\Sigma}  \wedge \mathbf{M}
\ar[rr,shift left=.75ex,"\nabla^L_{\phi_1}"]
\ar[rr,shift right=.75ex,swap,"\nabla_{\phi_2}^R \circ \Psi_{\Sigma, M}"]
&{}&  \mathbf{M}
\ar[r, "\Psi"]
&\mathbf{M}_{\phi_1\# \phi_2}.
\end{tikzcd}$$

To analyze the morphisms $\nabla_1$ and $\nabla_2$, let us introduce the homeomorphism $\hT_{\Sigma}: \overline{\Sigma}\times I \xrightarrow{\cong} \Sigma \times I$ which consists in performing a half-twist in a tubular neighborhood of the boundary disc $\mathbb{D}_{\Sigma}$ while leaving the rest invariant. The morphism $\hT_{\Sigma}$ does not define a morphism $\overline{\mathbf{\Sigma}} \to \mathbf{\Sigma}$ in $\mathcal{M}_{\con}^{(1)}$ because it reverses the height order of the boundary disc instead of preserving it. However, for any other marked $3$-manifold $\mathbf{N}$ one can define an isomorphism $\id\wedge \hT_{\Sigma} : \mathbf{N} \wedge \overline{\mathbf{\Sigma}} \xrightarrow{\cong} \mathbf{N}\wedge \mathbf{\Sigma}$ in $\mathcal{M}_{\con}^{(1)}$ which consists in performing a half-twist in a neighborhood of $\mathbb{D}_{\overline{\Sigma}}$. Then by definition of the gluing maps, the morphisms $\nabla_1$, $\nabla_2$ decompose as

$$ \nabla_1: \mathbf{\Sigma}\wedge \overline{\mathbf{\Sigma}} \wedge \mathbf{\Sigma} \xrightarrow{ \id \wedge \hT_{\Sigma}^{-1} \wedge \id} {\mathbf{\Sigma}}\wedge {\mathbf{\Sigma}} \wedge \mathbf{\Sigma}  \xrightarrow{\psi_{\Sigma, \Sigma} \wedge \id} {\mathbf{\Sigma}}\wedge {\mathbf{\Sigma}} \wedge \mathbf{\Sigma} \xrightarrow{\mu^{(2)}_{\Sigma}} \mathbf{\Sigma}$$

$$ \nabla_2 : \overline{\mathbf{\Sigma}} \wedge \mathbf{\Sigma} \wedge \mathbf{M}  \xrightarrow{ \id \wedge \nabla_{\phi_1}^L} \overline{\mathbf{\Sigma}}\wedge \mathbf{M} \xrightarrow{\psi_{\overline{\Sigma}, M}} \mathbf{M}\wedge \overline{\mathbf{\Sigma}} \xrightarrow{\id\wedge \hT_{\Sigma}} \mathbf{M}\wedge \mathbf{\Sigma} \xrightarrow{\nabla_{\phi_2}^R} \mathbf{M}.

Define a morphism $j_1: \mathbf{\Sigma} \wedge (\overline{\mathbf{\Sigma}}\wedge \mathbf{\Sigma}) \wedge \mathbf{M} \to \mathbf{\Sigma} \wedge \mathbf{M}$ as the composition:

\begin{multline*}
j_1: \mathbf{\Sigma} \wedge (\overline{\mathbf{\Sigma}}\wedge \mathbf{\Sigma}) \wedge \mathbf{M}
\xrightarrow{ \id \wedge \hT_{\Sigma}^{-1} \wedge \id \wedge \id}
\mathbf{\Sigma} \wedge {\mathbf{\Sigma}}\wedge \mathbf{\Sigma} \wedge \mathbf{M} \\
\xrightarrow{ \psi_{\Sigma, \Sigma}\wedge \id \wedge \id}
\mathbf{\Sigma} \wedge {\mathbf{\Sigma}}\wedge \mathbf{\Sigma} \wedge \mathbf{M}
\xrightarrow{\id \wedge \mu_{\Sigma} \wedge \id}
{\mathbf{\Sigma}}\wedge \mathbf{\Sigma} \wedge \mathbf{M}
\xrightarrow{\id \wedge \nabla_{\phi_1}^L}
\mathbf{\Sigma} \wedge \mathbf{M},
\end{multline*}

and set $j_2:= \nabla_{\phi_1}^L$. Then in the diagram

\begin{equation}\label{big_diagram}
\begin{tikzcd}
\mathbf{\Sigma} \wedge( \overline{\mathbf{\Sigma}} \wedge \mathbf{\Sigma} ) \wedge \mathbf{M}
\ar[rr,shift left=.75ex,"\nabla_1\wedge \id"]
\ar[dd, "j_1"]
\ar[rr,shift right=.75ex,swap,"\id \wedge \nabla_2"]
&{}& \mathbf{\Sigma} \wedge \mathbf{M}
\ar[dd, "j_2"]
\ar[rd, "\Psi'"]
&{} \\
\mathbf{M}_{\phi_1\# \phi_2} \\
\mathbf{\Sigma}  \wedge \mathbf{M}
\ar[rr,shift left=.75ex,"\nabla^L_{\phi_1}"]
\ar[rr,shift right=.75ex,swap,"\nabla_{\phi_2}^R \circ \Psi_{\Sigma, M}\circ (\theta_{\Sigma}\wedge \id_M)"]
&{}&  \mathbf{M}
\ar[ru, "\Psi"]
\end{tikzcd}
\end{equation}

the three following subdiagrams commute (up to isotopy):

\begin{equation}\label{3_diagrams}
\begin{tikzcd}
\mathbf{\Sigma} \wedge( \overline{\mathbf{\Sigma}} \wedge \mathbf{\Sigma} ) \wedge \mathbf{M}
\ar[r,"\nabla_1\wedge \id"]
\ar[d, "j_1"]
& \mathbf{\Sigma} \wedge \mathbf{M}
\ar[d, "j_2"]
\\
\mathbf{\Sigma}  \wedge \mathbf{M}
\ar[r,"\nabla^L_{\phi_1}"]
&  \mathbf{M}
\end{tikzcd}
\quad
\begin{tikzcd}
\mathbf{\Sigma} \wedge( \overline{\mathbf{\Sigma}} \wedge \mathbf{\Sigma} ) \wedge \mathbf{M}
\ar[r,"\id \wedge \nabla_2"]
\ar[d, "j_1"]
& \mathbf{\Sigma} \wedge \mathbf{M}
\ar[d, "j_2"]
\\
\mathbf{\Sigma}  \wedge \mathbf{M}
\ar[r,"\nabla_{\phi_2}^R \circ \Psi_{\Sigma, M}\circ (\theta_{\Sigma}\wedge \id_M)"]
&  \mathbf{M}
\end{tikzcd}
\quad
\begin{tikzcd}
\mathbf{\Sigma} \wedge \mathbf{M}
\ar[dd, "j_2"] \ar[rd, "\Psi'"]
&{} \\
\mathbf{M}_{\phi_1\# \phi_2}\\
\mathbf{M}
\ar[ru, "\Psi"]
\end{tikzcd}
\end{equation}

Let $\mathcal{C}$ a  category and consider a diagram

\begin{tikzcd}
\ar[r,shift left=.75ex,"f"]
\ar[r,shift right=.75ex,swap,"g"]
\ar[dd, "j_1"] &
B \ar[dd, "j_2"] \ar[rd, "h"] &{} \\
{}&{}&D \\
\ar[r,shift left=.75ex,"f'"]
\ar[r,shift right=.75ex,swap,"g'"]
& B'  \ar[ru, "h'"]&{}
\end{tikzcd}

such that:
$(i)$  the following diagrams are commutative:

\begin{tikzcd}
\ar[r,"f"]
\ar[dd, "j_1"] &
B \ar[dd,   "j_2"]  \\
{}&{} \\
\ar[r,"f'"]
& B'
\end{tikzcd}
\quad
\begin{tikzcd}
\ar[r,"g"]
\ar[dd, "j_1"] &
B \ar[dd,   "j_2"]  \\
{}&{} \\
\ar[r,"g'"]
& B'
\end{tikzcd}
\quad
\begin{tikzcd}
B \ar[dd, "j_2"] \ar[rd, "h"] &{} \\
{} & D \\
B' \ar[ru, "h'"] &{}
\end{tikzcd};

$(ii)$ $j_1$ and $j_2$  are epimorphisms

and $(iii)$ the following diagram is a coequalizer:

\begin{tikzcd}
\ar[r,shift left=.75ex,"f"]
\ar[r,shift right=.75ex,swap,"g"] &
B  \ar[r, "h"] & D
\end{tikzcd}.

Then the following diagram is a coequalizer:

\begin{tikzcd}
\ar[r,shift left=.75ex,"f'"]
\ar[r,shift right=.75ex,swap,"g'"] &
B' \ar[r, "h'"] & D
\end{tikzcd}.

First, the equality $h'\circ f'= h'\circ g'$ follows from the fact that $j_1$ is an epimorphism together with the commutativity of the diagrams in $(i)$ and $(iii)$.
Let $k: B' \to E$ be a morphism such that $k\circ f' = k\circ g'$ and let us prove that there exists a unique morphism $m: D \to E$ such that $k=m\circ h'$. Since $k\circ f' \circ j_1= k\circ g'\circ j_1$ by commutativity of the first two diagrams in Hypothesis $(i)$, one has $(k\circ j_2 )\circ f=(k\circ j_2)\circ g$. By $(iii)$, there exists a unique $m:D\to E$ such that $k\circ j_2= m\circ h$. By commutativity of the third diagram in $(i)$, the last identity is equivalent to $k\circ j_2= m\circ h' \circ j_2$ and since $j_2$ is an epimorphism $(ii)$, this is equivalent to $k=m\circ h'$. This proves the claim.

We apply Lemma <ref> to the image by $P$ of Diagram <ref>. The commutativity of Diagrams <ref> implies hypothesis $(i)$. Hypothesis $(ii)$ follows from the fact that, for $k=1,2$,  the codomain of  $j_k$ retracts by deformation on its image, therefore for every $n\geq 0$, $P_n(j_k) $ is surjective, so $P(j_k)$ is an epimorphism.  Hypothesis $(iii)$ follows from the quantum Van Kampen Theorem <ref>. Therefore $P_{M_{\phi_1\# \phi_2}}= \mathrm{HH}^0(P_{\Sigma}, P_M)$. The corresponding results for $\Rep_q^G$ and $\mathcal{S}_q$ are deduced from the fact that $\bullet \otimes_{\BT} Q_{B_qG}$ preserves right exact sequences exactly in the same manner than in the proof of Theorem <ref> so we leave the details to the reader.

§ QUANTUM REPRESENTATION SPACES OF MAPPING TORI

§.§ The quantum adjoint coaction

Let $\mathbf{\Sigma}=(\Sigma, \{a\}) \in \MS^{(1)}_{\con}$ be a connected $1$-marked surface let $\ad_{\Sigma} : \mathbf{\Sigma}\times I \to (\mathbf{\Sigma}\times I) \wedge \mathbf{H}_1$ be the embedding whose image by the quantum fundamental group functor is the morphism $P(\ad_{\Sigma})$ illustrated in Figure <ref>. More precisely, consider the marked surface $\mathbf{\Sigma}\wedge {\mathbb{D}_1}$ obtained from $\mathbf{\Sigma}\bigsqcup \mathbb{D}_1\bigsqcup (\mathbb{D}^2, \{e_1,e_2,e_3\})$ by gluing $e_1$ with $a$ and $e_2$ with the boundary arc of the annulus $\mathbb{D}_1$. In particular, $ (\mathbf{\Sigma}\times I) \wedge \mathbf{H}_1$ is the image by the functor $\cdot \times I : \MS \to \mathcal{M}$ of $\mathbf{\Sigma}\wedge \mathbf{\mathbb{D}_1}$.
Define an embedding $\ad^0: \Sigma \to \Sigma \wedge \mathbb{D}_1$ whose restriction outside a collar neighborhood $N(a) \cong [0,1]^2$of $a$ is the identity and sending $N(a)$ to a band between $a$ and $e_3$ making one turn around the hole of the annulus $\mathbb{D}_1$. Then $\ad_{\Sigma}$ is the image of $\ad^0$ by  $\cdot \times I : \MS \to \mathcal{M}$ of $\mathbf{\Sigma}\wedge \mathbf{\mathbb{D}_1}$.

An illustration of the quantum adjoint coaction.

The quantum adjoint coactions are the morphisms $\Ad_{\Sigma}: \Rep_q^G(\mathbf{\Sigma}) \to \Rep_q^G(\mathbf{\Sigma}) \overline{\otimes} B_qG$ and $\Ad_{\Sigma}: \mathcal{S}_q(\mathbf{\Sigma}) \to \mathcal{S}_q(\mathbf{\Sigma}) \overline{\otimes} B_q\SL_2$  obtained by taking the images of $\ad_{\Sigma}$ by $\Rep_q^G$ and $\mathcal{S}_q$ respectively and using the lax monoidality.

The quantum adjoint coaction equips $\mathcal{S}_q(\mathbf{\Sigma})$  (resp. $\Rep_q^G(\mathbf{\Sigma})$) with a structure of $B_q\SL_2$ (resp. $B_qG$) right comodule object in the braided category $\overline{\mathcal{C}_q^{\SL_2}}$ (resp. $\overline{\mathcal{C}_q^G}$).
In the particular case where $\mathbf{\Sigma}= \mathbf{H}_1$, we have proved in Theorem <ref> that  $\Ad_{\mathbf{H}_1}$ coincides with Majid's braided adjoint coaction. In particular, $\mathcal{S}_q(\mathbf{\Sigma})$ is an $\mathcal{O}_q[\SL_2]$ right comodule in $\Mod_k$ and a $B_q\SL_2$ right comodule in $\overline{\mathcal{C}_q^{\SL_2}}$. Recall from Section <ref> the isomorphism $f:B\mathcal{S}_q(\mathbb{B}) \xrightarrow{\cong} \mathcal{S}_q(\mathbf{H}_1)$ where $B\mathcal{S}_q(\mathbb{B})$ is equal to $\mathcal{S}_q(\mathbb{B})$ as a co-algebra.

The following diagram (in $\Mod_k$) commutes:

$$ \begin{tikzcd}
{} & \mathcal{S}_q(\mathbf{\Sigma})\otimes \mathcal{S}_q(\mathbb{B}) \ar[dd, "\id\otimes f"] \\
\mathcal{S}_q(\mathbf{\Sigma}) \ar[ru, "\Delta^R_a"] \ar[rd, "\Ad_{\Sigma}"] & {} \\
{}& \mathcal{S}_q(\mathbf{\Sigma})\overline{\otimes}\mathcal{S}_q(\mathbf{H}_1)
\end{tikzcd}

In particular, the subspace of $\mathcal{O}_q[\SL_2]$ coinvariant vectors of $\mathcal{S}_q(\mathbf{\Sigma})$ coincides with its subspace of $B_q\SL_2$ coinvariant vectors.

Let $[T,s] =  \adjustbox{valign=c}{\includegraphics[width=2cm]{QAdPf1.eps}}\in \mathcal{S}_q(\mathbf{\Sigma})$ be the class of an arbitrary stated tangle in $\mathbf{\Sigma}$. Applying the skein relation in Remark <ref>, one finds:

$$ \mathcal{S}_q(\ad_{\Sigma}) ([T,s]) =\adjustbox{valign=c}{\includegraphics[width=2.5cm]{QAdPf2.eps}} = \sum_{i_1, \ldots, i_n} \adjustbox{valign=c}{\includegraphics[width=3cm]{QAdPf3.eps}} \xrightarrow[\cong]{\Psi} \sum_{i_1, \ldots, i_n} \adjustbox{valign=c}{\includegraphics[width=4cm]{QAdPf4.eps}},  $$

where $\Psi: \mathcal{S}_q(\mathbf{\Sigma}\wedge \mathbf{H}_1) \xrightarrow{\cong} \mathcal{S}_q(\mathbf{\Sigma})\overline{\otimes} \mathcal{S}_q(\mathbf{H}_1)$ is the isomorphism of Section <ref>.
Applying $\id \otimes f^{-1}$, we obtain:

$$ (\id\otimes f^{-1}) \Ad_{\Sigma} ([T,s]) = \sum_{i_1, \ldots, i_n} \adjustbox{valign=c}{\includegraphics[width=4cm]{QAdPf5.eps}} =\Delta^R_a ([T,s]).$$

§.§ Mapping tori

Let $\mathbf{\Sigma}=(\Sigma, \{a\}) \in \MS^{(1)}_{\con}$ be a connected $1$-marked surface and denote by $\partial$ the connected component of $\partial \Sigma$ containing $a$. Let $\phi: \Sigma \to \Sigma$ be an oriented diffeomorphism whose restriction to $\partial$ is the identity. Let $M_{\phi}:= \quotient{\Sigma \times I}{\left((\phi(x), +1)\sim (x,-1), x\in \Sigma\right)}$ be the associated mapping torus and $\pi: \Sigma\times I \to M_{\phi}$ the quotient map. Consider a parametrized disc $\mathbb{D}_{M_{\phi}} \subset \partial M_{\phi}$ contained in the image of $\partial \times I$ and let $\mathbf{M}_{\phi} := (M_{\phi}, \{\mathbb{D}_{M_{\phi}}\}) \in \mathcal{M}^{(1)}_{\con}$. The goal of this section is to describe $\Rep_q^G(\mathbf{M}_{\phi})$.

Let $\mathbf{M}:= (\mathbf{\Sigma}\times I) \wedge \mathbf{H}_1= (\Sigma\wedge \mathbb{D}_1)\times I$ and consider two attaching maps $\phi_1: \Sigma \hookrightarrow \partial M$ and $\phi_2: \overline{\Sigma} \hookrightarrow \partial M$ defined as follows.

* The embedding $\phi_2: \overline{\Sigma} \to \partial M_1$ is the composition $\phi_2: \overline{\Sigma}\xrightarrow{ \overline{\ad_{\Sigma}^0}} \overline{\Sigma \wedge \mathbb{D}_1} \cong (\Sigma\wedge \mathbb{D}_1)\times \{-1\} \subset \partial M$.

* The embedding $\phi_1: \Sigma \to \partial M_1$ is the composition $\phi_1: \Sigma \xrightarrow{\phi} \Sigma \xrightarrow{\iota_1} \Sigma\wedge \mathbb{D}_1 \cong (\Sigma\wedge \mathbb{D}_1)\times \{+1\} \subset \partial M$.

Then $\mathbf{M}_{\phi_1 \# \phi_2}$ is isomorphic to $\mathbf{M}_{\phi}$, so Theorem <ref> implies that $\Rep_q^G(\mathbf{M}_{\phi}) \cong \mathrm{HH}^0( \Rep_q^G(\mathbf{\Sigma}), \Rep_q^G(\mathbf{M}))$. Note that, since $\mathbf{M}$ is a thickened surface, it has an algebra structure with product $\mu_M$. Write $\mu_M^{top}:= \mu_M \circ \psi_{M, M} \circ (\theta_M\wedge \id_M): M\wedge M \to M$ the twisted opposite product. By Remark <ref>, $\mu_M^{top}$ is not really a product since it is not associative but rather equips $M$ with a structure of left module over itself in the sense that one has the equality

$$ \mu^{top}_M (\mu_M \otimes \id_M) = \mu^{top}_M(\id_M \otimes \mu^{top}_M).$$

By definition,  the module morphisms $\nabla_{\phi_1}^L$ and $\nabla_{\phi_2}^R$ decompose as $ \nabla_{\phi_1}^L= \mu_M^{top} \circ (\ad_{\Sigma} \wedge \id)$ and $\nabla_{\phi_2}^R= \mu_M \circ (\iota_1 \phi \wedge \id)$. Therefore we have proved the

The following sequence is exact:

$$ \Rep_q^G(\mathbf{\Sigma}) \overline{\otimes} (\Rep_q^G(\mathbf{\Sigma})\overline{\otimes} B_qG) \xrightarrow{ \mu \circ (\iota_1\phi_*\otimes \id) - \mu^{top} \circ (\Ad_{\Sigma} \otimes \id)} \Rep_q^G(\mathbf{\Sigma})\overline{\otimes} B_qG \to \Rep_q^G(\mathbf{M}_{\phi})\to 0.$$

See Figure <ref> for an illustration.

An illustration of the equality
$P \left(\mu^{top} \circ (\ad_{\Sigma} \wedge \id)\right) (\alpha \otimes \gamma) = P\left( \mu \circ(\iota_1 \beta_* \otimes \id) \right) (\alpha \otimes \gamma)$
in $P_{M_{\beta}}$ where $\Sigma=\mathbb{D}_2$,  $\beta$ a $2$ strands braid, $\alpha \in  P_1(\mathbb{D}_2)$ and $\beta \in P_1(\mathbb{D}_2\wedge \mathbf{H}_1)$.

§ QUANTUM REPRESENTATION SPACES OF LINKS EXTERIOR

§.§ Links exterior

Let $\beta \in B_n$ be a braid, seen as a mapping class of $\mathbb{D}_n$, and let $L\subset S^3$ be the link obtained by closing $\beta$.  Since $\mathbf{H}_n=\mathbb{D}_n\times I$, the mapping class $\beta$ induces a morphism (still denoted by the same letter) $\beta: \mathbf{H}_n \to \mathbf{H}_n$ in $\mathcal{M}^{(1)}_{\con}$.
Let $\mathbf{M}_L\in \mathcal{M}^{(1)}_{\con}$ be the marked $3$ manifold where $M_L= S^3 \setminus(\mathring{N}(L) \cup \mathring{B}^3)$ is obtained by removing from $S^3$ the union of the  interior of a tubular neighborhood of $L$ and an open ball, and with a single boundary disc $\mathbb{D}_L \subset \partial \mathbb{B}^3$. The product $\mu_{\mathbb{D}_n} : \mathbb{D}_n \wedge \mathbb{D}_n \to \mathbb{D}_n$ defines a product $\mu: \Rep_q^G(\mathbb{D}_n)^{\overline{\otimes} 2} \to \Rep_q^G(\mathbb{D}_n)$. Set $\mu^{top}:= \mu \circ \psi \circ (\theta\otimes \id)$.

One has an exact sequence in $\overline{\mathcal{C}_q^G}$:

$$ \Rep_q^G(\mathbb{D}_n)\overline{\otimes} \Rep_q^G(\mathbb{D}_n) \xrightarrow{ \mu^{top} - \mu \circ (\beta_*\otimes \id)} \Rep_q^G(\mathbb{D}_n) \to \Rep_q^G(\mathbf{M}_{L}) \to 0.$$

Said differently, identifying $\Rep_q^G(\mathbb{D}_n)$ with $(B_qG)^{\overline{\otimes}n}$, one has

$$ \Rep_q^G(\mathbf{M}_L) \cong \quotient{ (B_qG)^{\overline{\otimes}n}}{ \left( \mu^{top}(x\otimes y) - \beta_*(x)y \right)}.$$

The marked $3$-manifold $\mathbf{M}_L$ is obtained from $\mathbf{M}:= \mathbb{D}_n\times I$ by self-gluing using the attaching maps $\phi_1: \mathbb{D}_n \hookrightarrow \partial M$, $\phi_1: \mathbb{D}_n \xrightarrow{\beta} \mathbb{D}_n \cong \mathbb{D}_n \times \{ +1\} \subset \partial (\mathbb{D}_n \times I)$ and $\phi_2 : \overline{\mathbb{D}_n} \hookrightarrow \partial M$, $\phi_2: \overline{\mathbb{D}_n} \cong \mathbb{D}_n \times\{-1\} \subset \partial (\mathbb{D}_n \times I)$. The left module map $\nabla_{\phi_1}^L$ is the composition $\mu \circ(\beta_* \otimes \id)$ and the right module map $\nabla_{\phi_2}^R$ is just the product $\mu_{\mathbb{D}_n}$ so the results follows from Theorem <ref>.

Let $M_L = S^3 \setminus N(L) \in \mathcal{M}^{(0)}$ and $\mathcal{S}_q^{rat}(M_L)$ be the standard (non stated) skein module. Then

$$ \mathcal{S}_q^{rat}(M_L) \cong \quotient{ (B_q\SL_2 ^{rat})^{\overline{\otimes}n}}{ \left( \mu^{top}(x\otimes y) - \beta_*(x)y \right)}.$$

§.§ Comparison with the constructions in [59]

We preserve the notations of the last subsection. The following was introduced by the second author together with Van-der-Veen in  [59]:

Let $\mathcal{I}_{\beta} \subset (B_q\SL_2)^{\overline{\otimes} n}$ be the right ideal generated by elements $\beta_*(x) - x$. We set

$$ \mathcal{A}_{\beta} := \quotient{ (B_q\SL_2)^{\overline{\otimes} n} }{\mathcal{I}_{\beta}}.$$

It is proved in  [59] that $\mathcal{I}_{\beta}$ is a bilateral ideal preserved by the $\mathcal{O}_q\SL_2$ coaction so $\mathcal{A}_{\beta}$ is an algebra in $\overline{\mathcal{C}_q^{\SL_2}}$. Moreover the adjoint coaction $\Ad :  (B_q\SL_2)^{\overline{\otimes} n} \to  (B_q\SL_2)^{\overline{\otimes} n} \overline{\otimes} B_q\SL_2$ sends $\mathcal{I}_{\beta}$ inside $\mathcal{I}_{\beta}\otimes B_q\SL_2$ so induces a comodule morphism (still denoted by the same letter) $\Ad: \mathcal{A}_{\beta} \to \mathcal{A}_{\beta} \otimes B_q\SL_2$ which is an algebra morphism.

The subalgebra $\mathcal{A}_{\beta}^{coinv} \subset \mathcal{A}_{\beta}$ is the subalgebra of coinvariant vectors for the $\Ad$ coaction.

The main result of  [59] is the fact that if $\beta$ and $\beta'$ are two braids which admit the same Markov closure, then $\mathcal{A}_{\beta}\cong \mathcal{A}_{\beta'}$ in $\Alg(\overline{\mathcal{C}_q^{\SL_2}})$ therefore the isomorphism classes of both $\mathcal{A}_{\beta}$ and $\mathcal{A}_{\beta}^{coinv}$ only depends on the link $L$. In  [59], the algebra $\mathcal{A}_{\beta}$ was called quantum representation variety whereas $\mathcal{A}_{\beta}^{coinv}$ was named quantum character variety of $L$. The initial motivation for the present paper was to relate these algebras to Habiro's quantum representation and character varieties and to (stated) skein modules. As can be shown by taking a simple example (such as the trivial knot) the two modules $\Rep_q^{\SL_2}(\mathbf{M}_L) $ and $\mathcal{A}_{\beta}$ are not isomorphic. However Theorem <ref> shows that they are very similar; indeed one has

$$ \mathcal{A}_{\beta}= \quotient{(B_q\SL_2)^{\overline{\otimes} n}}{ \left( \mu(x\otimes y) - \beta_*(x)y\right)} \quad \mbox{ and }\quad \Rep_q^{\SL_2}(\mathbf{M}_L)\cong  \quotient{(B_q\SL_2)^{\overline{\otimes} n}}{ \left( \mu^{top}(x\otimes y) - \beta_*(x)y\right)} $$

so we just have changed $\mu$ to $\mu^{top}$ in the quotient. Moreover, since taking the coinvariant vectors for the $\mathcal{O}_q[\SL_2]$ coaction than for the braided $B_q\SL_2$ quantum coaction by Lemma <ref>, the skein module of $M_L$ and $\mathcal{A}_{\beta}^{coinv}$ are very similar. We expect that the computational techniques developed in [58] can be adapted to compute the peripheral ideal of links exteriors using this analogy.

§ FINITE PRESENTATION OF THE CATEGORY OF BOTTOM TANGLES

In this appendix, we explain how the works of Kerler [37, 38] and Bobtcheva-Piergallini [8]  permit to prove the following reformulation of Theorem <ref>:

The category $\BT$ is presented by the generators $(\mu, \eta, \Delta, \epsilon, S^{\pm 1}, \theta)$ of Figure <ref> and the relations of Figure <ref>.

Let us fix the terminology used here. Let $\mathcal{C}$ be a PROP, i.e. a monoidal category with set of objects $\mathbb{N}$ and such that $a\otimes b=a+b$, and consider a set $G$ of morphisms in $\mathcal{C}$. For a morphism $\mu$ of $\mathcal{C}$, we write $s(\mu), t(\mu)\in \mathbb{N}$ the integers (source and target) such that $\mu: s(\mu)\to t(\mu)$. The set $G$ is said to generate $\mathcal{C}$ if for every $a,b\in \mathbb{N}$, every morphism in $\mathcal{C}(a,b)$ can be expressed as a composition of elements of the form $\mathds{1}_x \otimes g \otimes \mathds{1}_y$ with $g\in G$. Let $\mathcal{F}_G$ be the PROP freely generated by $G$. By definition, the morphisms in $\mathcal{F}_G(a,b)$ are formal expressions of the form $\mu_1\ldots \mu_n$ where $\mu_i= \mathds{1}_{x_i} \otimes g_i \otimes \mathds{1}_{y_i}$ with $g_i \in G$,  such that $s(\mu_i)=t(\mu_{i+1})$, $s(\mu_1)=a$, $t(\mu_n)=b$. Composition is the concatenation of words. Equivalently, we can represent the morphisms in $\mathcal{F}_G(a,b)$ by graphs with $a$ leaves on the top and $b$ leaves on the bottom and possibly some $(s(g), t(g))$ coupons for each $g\in G$. For instance, letting $G$ be the set of morphisms of $\BT$ of Figure <ref>, the morphism $\mu (\mathds{1}\otimes S) \Delta \in \mathcal{F}_G(1,1)$ is represented by the graph $\adjustbox{valign=c}{\includegraphics[width=0.5cm]{tree_antipode.eps}}$. There is a unique monoidal functor $F: \mathcal{F}_G \to \mathcal{C}$ sending $g\in \mathcal{F}_G(s(g), t(g))$ to $g\in \mathcal{C}(s(g),t(g))$ and $G$ is a generating set if and only if $F$ is essentially surjective.

An equivalence relation on a PROP $\mathcal{F}$ is the data for each $a,b\in \mathbb{N}$ of an equivalence relation $\sim$ on the set $\mathcal{F}(a,b)$ in such a way that $\mu\sim \nu$ implies that $\eta\otimes \mu \otimes \kappa \sim \eta\otimes \nu \otimes \kappa$ and that $\mu\sim \nu$ implies that $\mu' \circ \mu \circ \mu'' \sim \mu' \circ \nu \circ \mu''$ (i.e. $\sim$ is preserved by composition and tensor product). In this case we can define a new PROP $\quotient{\mathcal{F}}{\sim}$ by $\quotient{\mathcal{F}}{\sim}(a,b):= \quotient{\mathcal{F}(a,b)}{\sim}$. For instance, if $G$ is a set of morphisms in $\mathcal{C}$ and $F: \mathcal{F}_G\to \mathcal{C}$ the associated functor, we define an equivalence $\sim_{\mathcal{C}}$ in $\mathcal{F}_G$ by setting $\mu \sim_{\mathcal{C}} \nu$ if $F(\mu)=F(\nu)$. When $G$ generates $\mathcal{C}$, then $F: \quotient{\mathcal{F}_G}{\sim_{\mathcal{C}}} \to \mathcal{C}$ is an equivalence of categories. A relation in $\mathcal{C}$ is a pair $(\mu, \nu)$ of morphisms in $\mathcal{F}_G$ with the same target and source and such that $F(\mu)=F(\nu)$. For instance $\left( \mu(\mu\otimes \mathds{1}) ,  \mu(\mathds{1}\otimes \mu) \right)$ is the relation in $\BT$ corresponding to the fact that $\mu$ is associative. Define a poset structure $\leq$ on the set $ER$ of equivalence relations on $\mathcal{F}_G$ by setting $\sim_1 \leq \sim_2$ if $\mu \sim_1 \nu$ implies $\mu \sim_2\nu$. The poset $(ER, \leq)$ is clearly filtrant: given $\sim_1$ and $\sim_2$ the relation $\sim$ defined by $\mu \sim \nu$ if we have both $\mu \sim_1 \nu$ and $\mu\sim_2\nu$ clearly satisfies $\sim \leq \sim_1$ and $\sim\leq \sim_2$. Therefore, by Zorn lemma, given a set $Rel$ of relations, we can speak of the smallest equivalence relation $\sim_{Rel}$ containing $Rel$; we say that $Rel$ is a complete set of relations if $\sim_{Rel}=\sim_{\mathcal{C}}$.

For $\mathcal{C}$ a PROP, $G$ a set of morphisms and $Rel$ a set of relations, we say that $(G,Rel)$ is a presentation of $\mathcal{C}$ if $G$ generates $\mathcal{C}$ and if $Rel$ is a complete set of relations.

(Habiro <cit.>) The set $G:=\{\mu, \eta, \Delta, \epsilon, S^{\pm 1}, \theta\}$  of morphisms of Figure <ref> generates $\BT$.

Recall from Remark <ref> the faithful braided functor

$$\partial : \BT \to \mathcal{C}^{CYK}$$

sending $\mathbf{H}_g$ to $\Sigma_{g,1}$ and sending an embedding $\mu : \mathbf{H}_a \to \mathbf{H}_b$ to the cobordism $\mathbf{H}_b \setminus \mu(\mathbf{H}_a)$. Not every cobordism in $\mathcal{C}^{CYK}(a,b)$ is homeomorphic to a cobordism of the form $\mathbf{H}_b \setminus \mu(\mathbf{H}_a)$ (so $\partial$ is not full), however every element of $\mathcal{C}^{CYK}(a,b)$ can be obtained from a cobordism $\mathbf{H}_b \setminus \mu(\mathbf{H}_a)$ by performing some surgery  along a closed framed link $L\subset \mathbf{H}_b \setminus \mu(\mathbf{H}_a)$. We can thus represent pictorially a morphism in $\mathcal{C}^{CYK}(a,b)$ by a $a$-bottom tangle $T$ in $\mathbf{H}_b$ entangled with a closed framed link $L \subset H_b \setminus T$ and think of $T\cup L$ as the cobordism obtained from $\mathbf{H}_b \setminus \mu_T(\mathbf{H}_a)$ by performing a surgery along $L$. The pairs $(T,L)$ are thus only considered up to some Kirby moves (see [37, 38, 8] for a precise definition of the Kirby moves involved). For instance, one can consider the two morphisms:

$$ \lambda := \adjustbox{valign=c}{\includegraphics[width=1cm]{BT_integral.eps}} \in \mathcal{C}^{CYK}(1,0) \quad \mbox{ and }\quad \Lambda := \adjustbox{valign=c}{\includegraphics[width=1cm]{BT_cointegral.eps}} \in \mathcal{C}^{CYK}(0,1), $$

which do not belong to the image of $\partial$ (we perform the surgery along the green curve here). They satisfy some relations drawn in Figure <ref> which justify the terminology integral and cointegral for $\lambda$ and $\Lambda$ respectively.

The relations satisfied by the integral and cointegral in $\mathcal{C}^{CYK}$.

By abuse of notations, we identify $\BT$ with its image by $\partial$ inside $\mathcal{C}^{CYK}$ so we consider the set $G$ of morphisms of Figure <ref> both as a set of morphisms in $\BT$ and $\mathcal{C}^{CYK}$. Similarly,  the set $Rel$ of relations defined by Figure <ref> is seen as a set of relations for $\mathcal{C}^{CYK}$ as well.

(Bobtcheva-Piergallini <cit.>) The category $\mathcal{C}^{CYK}$ is presented by the set of generators $G \bigsqcup \{ \lambda, \Lambda\}$ and the set of relations which is the union of $Rel$ with the relations of Figure <ref>.

That the set $G \bigsqcup \{ \lambda, \Lambda\}$ generates $\mathcal{C}^{CYK}$ and that they satisfy the above relations (with the exception of the two bottom "BP" relations in Figure <ref>) was proved by Kerler in [37] who conjectured that they form a presentation of $\mathcal{C}^{CYK}$. The two additional BP relations were found by Bobtcheva-Piergallini in [8].

Let $\mathcal{D}$ be the category presented by $(G,Rel)$ and $f: \mathcal{D}\to \mathcal{C}^{CYK}$ be the inclusion functor. It suffices to prove that $f$ is faithful; indeed, since $\partial: \BT \to \mathcal{C}^{CYK}$ is faithful and since $\BT$ is generated by $G$ by Theorem <ref>, this will prove that $\BT$ is equivalent to $\mathcal{D}$ and conclude the proof. Write $G':= G\bigsqcup \{\lambda, \Lambda\}$, let $Rel''$ be the set of relations (in $\mathcal{F}_{G'}$) of Figure <ref> and $Rel':= Rel \bigsqcup Rel''$ so that $\mathcal{C}^{CYK}$ is presented by $(G', Rel')$ (in virtue of Theorem <ref>).
The faithfulness of $f$ follows from the following remark: for each relation in $Rel''$ of Figure <ref>, if we replace $\lambda$ by the counit $\epsilon$ and $\Lambda$ by the unit $\eta$, then get a relation in $Rel$. Therefore the quotient category $\quotient{ \mathcal{C}^{CYK}}{\left( \adjustbox{valign=c}{\includegraphics[width=1.5cm]{Integral_quotient.eps}} \ \right)}$ is equivalent to $\mathcal{D}$ and the quotient functor

$$g : \mathcal{C}^{CYK} \to  \quotient{ \mathcal{C}^{CYK}}{\left( \adjustbox{valign=c}{\includegraphics[width=1.5cm]{Integral_quotient.eps}} \right)} \cong \mathcal{D}$$

satisfies $g\circ f = \id_{\mathcal{D}}$. This proves that $f$ is faithful and concludes the proof.

[1]
A.Y. Alekseev, H. Grosse, and V. Schomerus, Combinatorial quantization of
the Hamiltonian Chern-Simons theory I, Communications in
Mathematical Physics 172 (1995), no. 2, 317–358.

[2]
, Combinatorial quantization of the Hamiltonian Chern-Simons
theory II, Communications in Mathematical Physics 174 (1996),
no. 3, 561–604.

[3]
A.Y. Alekseev and A.Z. Malkin, Symplectic structures associated to
Lie-Poisson groups, Communications in Mathematical Physics 162
(1994), no. 1, 147–173.

[4]
J.W. Barett, Skein spaces and spin structures, Math. Proc. Cambridge
126 (1999), no. 2, 267–275.

[5]
B. Bartlett, C.L. Douglas, Schommer-Pries C.J., and J. Vicary, Extended
3-dimensional bordism as the theory of modular objects, arXiv:1411.0945.

[6]
A. Beliakova and M. De Renzi, Kerler–Lyubashenko functors on
4-dimensional 2-handlebodies, Int. Math. Res. Not. 39 (2022), 55.

[7]
W. Bloomquist and T.T.Q. Le, The Chebyshev-Frobenius homomorphism
for stated skein modules of $3$-manifolds, arXiv:2011.02130[math.GT].

[8]
I. Bobtcheva and R. Piergallini, On 4-dimensional 2-handlebodies and
3-manifolds, J. Knot Theory Ramifications 21 (2012), no. 12, 230.

[9]
E. Buffenoir and Ph. Roche, Two-dimensional lattice gauge theory based on
a quantum group, Comm. Math. Phys. 170 (1995), no. 3, 669–698.

[10]
, Link invariants and combinatorial quantization of Hamiltonian
Chern-Simons theory, Comm. Math. Phys. 181 (1996), no. 2,

[11]
D. Bullock, Rings of $sl_2(\mathbb{C})$-characters and the Kauffman
bracket skein module, Comentarii Math. Helv. 72 (1997), no. 4,

[12]
F. Bonahon and H. Wong, Quantum traces for representations of surface
groups in $\mathrm{SL}_2(\mathbb{C})$, Geom. Topol. 15 (2011),

[13]
D. Ben-Zvi, A. Brochier, and D. Jordan, Integrating quantum groups over
surfaces, J.Topology 11 (2018), no. 4, 873–916.

[14]
F. Costantino, J. Korinman, and T.Q.T. Le, Stated skein algebras for
Tannakian ribbon categories, In preparation.

[15]
F. Costantino and T.T.Q. Lê, Stated skein algebras of surfaces,
ArXiv:1907.11400, 2019.

[16]
, Stated skein modules of $3$ manifolds and TQFT,
arXiv:2206.10906, 2022.

[17]
L. Charles and J. Marché, Multicurves and regular functions on the
representation variety of a surface in SU(2), Comentarii Math. Helv.
87 (2009), 409–431.

[18]
J. Cooke, Excision of skein categories and factorisation homology,
arXiv:1910.02630, 2019.

[19]
L. Crane and D. Yetter, On algebraic structures implicit in topological
quantum field theories, J. Knot Theory and Ramifications 8 (1999),

[20]
B. Day, Construction of biclosed categories, Ph.D. thesis, University of
New South Wales, 2070, url: http://web.science.mq.edu.au/ street/DayPhD.pdf.

[21]
M. Faitg, Holonomy and (stated) skein algebras in combinatorial
quantization, arXiv:2003.08992, 2020.

[22]
C. Frohman, R. Gelca, and W. Lofaro, The A-polynomial from the
noncommutative viewpoint, Trans. Amer. Math. Soc. 354 (2002),
no. 2, 735–747.

[23]
Z. Fiedorowicz, The symmetric bar construction, url:
https://people.math.osu.edu/fiedorowicz.1/symbar.ps.gz., 1996.

[24]
B. Fresse, Homotopy of operads & Grothendieck-Teichmüller groups,
Mathematical Surveys and Monographs, vol. 217, AMS, 2017.

[25]
S. Garoufalidis, On the characteristic and deformation varieties of a
knot, Proceedings of the Casson Fest, Geom. Topol. Monogr. 7
(2004), 291–309.

[26]
S. Gunningham, D. Jordan, and P. Safronov, The finiteness conjecture for
skein modules, arXiv:1908.05233[math.QA].

[27]
S. Garoufalidis and T.T.Q. Lê, The colored Jones function is
q-holonomic, Geom. Topol. 9 (2005), 1253–1293.

[28]
K. Habiro, Bottom tangles and universal invariants, , Algebr. Geom.
Topol. 6 (2006), 1113–1214.

[29]
, A note on quantum fundamental groups and quantum representation
varieties for $3$-manifolds, RIMS Kokyuroku 1777 (2012), 21–30,

[30]
B. Haioun, Relating stated skein algebras and internal skein algebras,
SIGMA 18 (2022), no. 042, 39.

[31]
A. Hatcher, Algebraic topology, Cambridge Univ. Press, Cambridge,

[32]
J. Hempel, $3$-manifolds, AMS Chelsea Pub, Providence, U.S., 2004.

[33]
V. Higgins, Triangular decomposition of $sl_3$ skein algebras,

[34]
J. Hoste and J. Przytycki, A survey of skein modules of $3$-manifolds,
Knots 90 (Osaka, 1990), de Gruyter, Berlin, 1992, pp. 363–379. 1177433

[35]
C. Kassel, Quantum groups, Springer-Verlag, New York, 1995, Graduate
Texts in Mathematics, No. 155.

[36]
T. Kerler, Genealogy of non-perturbative quantum invariants of
$3$-manifolds: the surgical family, Lecture Notes in Pure and Applied Math.
184 (1997), 503–547.

[37]
, Bridged links and tangle presentations of cobordism categories,
Adv. Math. 141 (1999), 207–281.

[38]
, Towards an algebraic characterization of 3-dimensional
cobordisms, diagrammatic morphisms and applications, Contemp. Math.
318 (2003), 141–173.

[39]
M. Kapovitch and J.J. Millson, On representation varieties of
$3$-manifolds groups, Geometry and Topology 21 (2017), no. 4,

[40]
J. Korinman, Finite presentations for stated skein algebras and
lattice gauge field theory, Accepted in Algebraic & Geometric Topology.
arXiv:2012.03237, 2020.

[41]
, Stated skein algebras and their representations,
arXiv:2105.09563, 2021.

[42]
, Mapping class group representations derived from stated skein
algebras, SIGMA 18 (2022), no. 64, 35, arXiv:2201.07649.

[43]
J. Korinman and A. Quesney, Classical shadows of stated skein
representations at odd roots of unity, arXiv:1905.03441, 2019.

[44]
, The quantum trace map as a quantum non-abelianization, J.
Knot theory & its ramifications 31 (2022), no. 06, 2250032.

[45]
A.N. Kirillov and N.Y. Reshetikhin, q-Weyl group and a multiplicative
formula for universal R-matrices, Comm. Math. Phys. 134 (1990),
no. 2, 421–431.

[46]
M. Kashiwara and P. Schapira, Categories and sheaves, Springer-Verlag,
2006, Grundlehren der Math. Wiss. 332.

[47]
J. Kamnitzer and P. Tingley, The crystal commutor and Drinfelds
unitarized R-matrix, Journal of Algebraic Combinatorics 29
(2009), no. 3, 315.

[48]
T.T.Q. Le, The colored Jones polynomial and the A-polynomial of
knots, Adv. Math. 207 (2006), no. 2, 782–804.

[49]
, Triangular decomposition of skein algebras, Quantum Topology
9 (2018), 591–632.

[50]
T.T.Q. Le and A. Sikora, Stated $\mathrm{SL}(n)$ stated skein modules
and algebras, arXiv:2201.00045.

[51]
S.Z. Levandorskii and Y.S. Soibelman, The quantum Weyl group and a
multiplicative formula for the R-matrix of a simple Lie algebra, Funct.
Annal. Appl. 25 (1991), no. 2, 143–145.

[52]
G. Lusztig, Introduction to quantum groups, Progress in Mathematics,
vol. 110, Birkhäuser Boston, 1993.

[53]
V. Lyubashenko, Modular transformations for tensor categories, Journal
of Pure and Applied Algebra 98 (1995), 279–327.

[54]
S. Majid, Rank of quantum groups and braided groups in dual form, Proc.
of the Euler Institute, St. Petersberg 1570 (1990), 79–89.

[55]
, Algebras and Hopf algebras in braided categories, Advances in
Hopf Algebras, Lec. Notes Pure and Applied Maths 158 (1994),

[56]
, Foundations of quantum group theory, Cambridge University
Press, 1995.

[57]
J. Milnor, Spin structure on manifolds, Enseign. Math. 9
(1963), 198–203.

[58]
J. Murakami, On quantum character varieties of knots, RIMS Kokyuroku
2191 (2021).

[59]
J. Murakami and R. Van der Veen, Quantized SL(2) representations of
knot groups, ArXiv:1812.09539[math.GT].

[60]
C. Negron, Log-modular quantum groups at even roots of unity and the
quantum Frobenius i, arXiv:1812.02277[math.QA], 2018.

[61]
J.H Przytycki and S. Sikora, On skein algebras and
$\mathrm{SL}_2(\mathbb{C})$-character varieties, Topology 39
(2000), no. 1, 115–148.

[62]
S.F. Sawin, Quantum groups at roots of unity and modularity, J. of Knot
Theory and Ramifications 15 (2006), no. 10, 1245–1277.

[63]
N. Snyder and P. Tingley, The half-twist for $u_q (g)$ representations,
Algebra and number theory 3 (2009), no. 7, 809–834.

[64]
P. Salvatore and N. Wahl, Framed discs operads and Batalin-Vilkovisky
algebras, Q. J. Math. 54 (2003), no. 2, 213–231.

[65]
M.E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A.
Benjamin, Inc., 1969.

[66]
P. Tingley, A minus sign that used to annoy me but now I know why it is
there, arXiv:1002.0555.

[67]
V. G. Turaev, The Conway and Kauffman modules of a solid torus, Zap.
Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167
(1988), no. Issled. Topol. 6, 79–89, 190. 964255 (90f:57012)

[68]
V.G. Turaev, Quantum invariants of knots and 3-manifolds, revised ed.,
de Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin,
2010. 2654259 (2011f:57023)