such ideals.

###### Proof..

Claim (1) is clear, since multiplication in $\mathcal{F}_{q}(N\backslash G)$
only increases weights in the ordering. For Claim (2), we note that every
$U_{q}(\mathfrak{g})\otimes U_{q}(\mathfrak{t})$-subspace of
$\mathcal{F}_{q}(N\backslash G)$ is a direct sum of its full isotypic
components, because $\mathcal{F}_{q}(N\backslash G)$ is multiplicity-free.
Moreover, we recall that the multiplication in $\mathcal{F}_{q}(N\backslash
G)$ is such that $m(V_{\lambda}\otimes V_{\mu})=V_{\lambda+\mu}$, so that once
$V_{\lambda}$ occurs in the ideal, all its dominant translates do as well. ∎

######  Corollary 3.44.

An object $M$ of $\mathcal{F}_{q}(N\backslash
G)\mathrm{-mod}_{G\times\overline{T}}$ is torsion-free if, and only if all its
compact subobjects have zero annihilator ideals.

###### Proof..

If we suppose $M$ is torsion, clearly all its compact subobjects have non-
trivial annihilator ideals. Conversely, as soon as the annihilator ideal of
any compact subobject is non-empty, then it is of the form
$I=\sum_{i}I_{\lambda_{i}}$ described in Lemma 3.43, which implies that it is
torsion, since we may take $\lambda=\lambda_{i}$ for any $i$ in the definition
of a torsion module. ∎

######  Theorem 3.45.

The comparison functor

$\displaystyle\underline{\operatorname{Hom}}_{G\times\overline{T}}(\mathds{1}_{B},-)\colon\mathrm{Rep}_{q}(B)$
$\displaystyle\longrightarrow\mathcal{F}_{q}(N\backslash
G)\mathrm{-mod}_{G\times\overline{T}}$ $\displaystyle V$
$\displaystyle\longmapsto\left(V\otimes\mathcal{F}_{q}(G\times
T)\right)^{U_{q}(\mathfrak{b})},$

defines a fully faithful embedding, realizing $\mathrm{Rep}_{q}(B)$ as the
open subcategory $\mathrm{Tors}^{\perp}$.

###### Proof..

Thanks to 3.41, we may apply Theorem 3.16 to conclude that
$\underline{\operatorname{Hom}}_{G\times\overline{T}}(\mathds{1}_{B},-)$
yields a fully faithful embedding of $\mathrm{Rep}_{q}(B)$ as an open
subcategory of $\mathcal{F}_{q}(N\backslash
G)\mathrm{-mod}_{G\times\overline{T}}$. So by 3.19, all that remains is to
identify the kernel of its left adjoint $\widetilde{L}$ with $\mathrm{Tors}$.
In complete analogy with the classical parabolic induction functor from
Example 3.32, this left adjoint is given by

$\widetilde{L}\colon\mathcal{F}_{q}(N\backslash
G)\mathrm{-mod}_{G\times\overline{T}}\longrightarrow\mathrm{Rep}_{q}{B},\qquad
M\longmapsto M/(\ker\varepsilon)M,$

where $\ker\varepsilon$ is the 2-sided ideal in $\mathcal{F}_{q}(N\backslash
G)$ given by the kernel of the evaluation morphism
$\varepsilon\colon\mathcal{F}_{q}(N\backslash G)\rightarrow\mathds{1}_{B}$,
and the $U_{q}(\mathfrak{b})$-action on $M/(\ker\varepsilon)M$ is inherited
from that of $U_{q}(\mathfrak{g})\otimes U_{q}(\mathfrak{t})$ on $M$.

Now suppose that $M$ is an object of $\mathrm{Tors}$.

Recall the principal minors $\\{f_{\lambda}\\}_{\lambda\in\Lambda_{+}}$
defined in Remark 3.35, and observe that
$f_{\lambda}\in\mathcal{F}_{q}(N\backslash G)$, while
$\varepsilon(f_{\lambda})=1$ for all $\lambda\in\Lambda_{+}$. So for each
$m\in M$, choosing $\lambda$ sufficiently large that $f_{\lambda}\cdot m=0$,
we have $m=1\cdot m=(1-f_{\lambda})m\in(\ker\varepsilon)M$, and thus
$\widetilde{L}(M)=0$.

To conclude the proof, we must show that if $M$ is torsion-free, then
$M/\ker\varepsilon M\neq 0$. For this, let us first suppose that $M$ is
finitely generated as an $\mathcal{F}_{q}(N\backslash G)$–module. Given a
weight $\lambda$, denote by $M[\lambda]$ the corresponding weight space of $M$
with respect to the $U_{q}(\mathfrak{t})$-action. Let $\lambda_{0}$ be the
highest weight such that $M[\lambda_{0}]\neq 0$. Since for any weight
$\lambda$

$\ker(\varepsilon)M[\lambda]\subseteq\oplus_{\mu\leqslant\lambda}M[\mu],$

the condition $M=\ker(\varepsilon)M$ implies
$M[\lambda_{0}]\subseteq\ker(\varepsilon)M[\lambda_{0}]$. Now let
$\left\\{m_{1},\dots,m_{n}\right\\}$ be a basis of $M[\lambda_{0}]$. Then any
element $m\in M[\lambda_{0}]$ can be written as

$m=\sum_{i}h_{i}m_{i}$ (3.6)

for some $h_{i}\in\ker\varepsilon$. Note that by comparing the $T$-weights of
both sides, we may assume that the $h_{i}$ are elements of the (commutative)
polynomial ring $\mathbb{C}(q)[\Lambda_{+}]$ generated by the principal minors
$f_{\lambda}$. We may thus make a similar argument to that used to prove the
classical Nakayama lemma: expand

$\displaystyle m_{i}=\sum_{j}f_{ij}m_{j}$ (3.7)

and form the $n\times n$–matrix $\delta_{ij}-f_{ij}$ with entries in
$\mathbb{C}(q)[\Lambda_{+}]$. Multiplying Eq. 3.7 through by its adjugate
matrix, we find that $\det(\delta_{ij}-f_{ij})$ annihilates all $m_{i}$, and
is of the form $1+\ker\varepsilon$. The annihilator of the $m_{i}$ is thus
nonzero, which by Proposition 3.44 contradicts the assumption that $M$ is
torsion-free.

To treat the general case, we note that $\widetilde{L}$ is exact on torsion
free modules. Hence, if $\widetilde{L}(M)=0$, then for any finitely generated
submodule $M^{\prime}\hookrightarrow M$ we must also have
$\widetilde{L}(M^{\prime})=0$. By the preceding discussion this implies
$M^{\prime}$ is torsion. Hence $M$ is torsion, as required. ∎

###  3.6. Monadic reconstruction: $\mathcal{Z}(\mathbb{D}_{n})$ and
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$

######  Definition 3.46.

We define the braided tensor category
$\mathrm{Rep}_{q}(G\times\overline{T}^{\>\\!n})$ by

$\mathrm{Rep}_{q}(G\times\overline{T}^{\>\\!n}):=\mathrm{Rep}_{q}(G)\boxtimes\overline{\mathrm{Rep}_{q}(T)}^{\>\\!\boxtimes
n}.$

######  Definition 3.47.

We endow the categories $\mathcal{Z}(\mathbb{D}_{n})$ with an action of the
monoidal category $\mathrm{Rep}_{q}(G\times\overline{T}^{\>\\!n}),$ by
choosing a single interval in $\partial S\cap S_{G}$, and one in each
connected component of $\partial S\cap S_{T}$, as indicated in Fig. 7. We let
$\mathrm{Dist}$ denote the distinguished object of the category
$\mathcal{Z}(\mathbb{D}_{n})$.

$\mathbb{D}_{G}$$\mathbb{D}_{T}$$\mathbb{D}_{T}$$\mathbb{D}_{T}$$\mathbb{D}_{T}$$\mathbb{D}_{T}$
Figure 7. A gated pentagon $\mathbb{D}_{5}$ with a depiction of the action by
disk insertions.

######  Definition 3.48.

For $n\geq 1$, the _quantum coordinate algebra of $n$ framed flags_ is the
$n$-fold braided tensor product,

$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n}):=\mathcal{F}_{q}(N\backslash
G)\otimes\cdots\otimes\mathcal{F}_{q}(N\backslash G).$

By definition, this means that as an object it is the tensor product in
$\mathrm{Rep}_{q}(G\times\overline{T}^{n})$, and that multiplication within
each factor is given by the algebra structure of $\mathcal{F}_{q}(N\backslash
G)$ and that relations between the factors are given by

$b^{(j)}a^{(i)}=\left(\mathcal{R}_{(2)}\rightharpoonup
a^{(i)}\right)\left(\mathcal{R}_{(1)}\rightharpoonup b^{(j)}\right),$ (3.8)

where $1\leqslant i<j\leqslant n$ and $x^{(i)}$ denotes the element $x$ placed
in the $i$-th braided tensor factor, where $\mathcal{R}$ denotes the universal
$R$-matrix of $U_{q}(\mathfrak{g})$. We note that each subalgebra
$\mathcal{F}_{q}(N\backslash G)$ occupies a _distinct_ $\mathrm{Rep}_{q}(T)$
factor, while sharing the same $\mathrm{Rep}_{q}(G)$ factor, hence the braided
tensor product involves only the universal $\mathcal{R}$-matrix for
$U_{q}(\mathfrak{g})$.

######  Theorem 3.49.

Let $\mathrm{Dist}_{\mathbb{D}_{n}}\in\mathcal{Z}(\mathbb{D}_{n})$ denote the
distinguished object. We have an isomorphism

$\underline{\operatorname{End}}_{G\times\overline{T}^{\>\\!n}}(\mathrm{Dist}_{\mathbb{D}_{n}})\cong\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n}).$

###### Proof..

The case $n=1$ has been already treated in Example 3.39. For $n>1$ we only
need to apply excision. By induction we can write
$\mathbb{D}_{n}=\mathbb{D}_{1}\sqcup_{\mathbb{D}_{G}}\cdots\sqcup_{\mathbb{D}_{G}}\mathbb{D}_{1}$,
hence we obtain an equivalence of categories,

$\mathcal{Z}(\mathbb{D}_{n})\simeq\mathrm{Rep}_{q}(B)\mathop{\boxtimes}_{\mathrm{Rep}_{q}(G)}\cdots\mathop{\boxtimes}_{\mathrm{Rep}_{q}(G)}\mathrm{Rep}_{q}(B),$

and therefore applying [BBJ18][Corollary 4.13], we have an isomorphism,

$\underline{\operatorname{End}}_{G\times\overline{T}^{\>\\!n}}(\mathrm{Dist}_{\mathbb{D}_{n}})\simeq\underline{\operatorname{End}}_{G\times\overline{T}}(\mathds{1}_{B})\widetilde{\otimes}\cdots\widetilde{\otimes}\underline{\operatorname{End}}_{G\times\overline{T}}(1_{B}),$

where $\widetilde{\otimes}$ denotes the braided tensor product of algebras in
$\mathrm{Rep}_{q}(G)$, as required.

∎

######  Definition 3.50.

For $n\geq 1$, we denote by
$\mathrm{Tors}_{n}\subset\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})\mathrm{-mod}_{G\times
T^{n}}$ the full Serre subcategory consisting of modules whose restriction to
some $\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{1})$ subfactor is torsion.

######  Proposition 3.51.

For any $n\geq 1$, the functor
$\underline{\operatorname{Hom}}_{G\times\overline{T}^{n}}(\mathrm{Dist}_{\mathbb{D}_{n}},-)$
induces an equivalence of categories,

$\mathcal{Z}(\mathbb{D}_{n})\simeq\mathrm{Tors}_{n}^{\perp}\subset\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})\mathrm{-mod}_{G\times\overline{T}^{n}}.$

###### Proof..

Let $\widetilde{L}_{n}$ denote the left adjoint. By 3.19 we need to identify
the kernel of $\widetilde{L}$ with $\mathrm{Tors}_{n}$. The case $n=1$ is
precisely Theorem 3.45. For $n>1$, we have have the equivalence,

$Z(\mathbb{D}_{n})\simeq\mathrm{Rep}_{q}(B)\underset{\mathrm{Rep}_{q}(G)}{\boxtimes}\cdots\underset{\mathrm{Rep}_{q}(G)}{\boxtimes}\mathrm{Rep}_{q}(B).$

Under this equivalence $\widetilde{L}$ identifies with the functor,
$\widetilde{L}_{1}\underset{\mathrm{Rep}_{q}(G)}{\boxtimes}\cdots\underset{\mathrm{Rep}_{q}(G)}{\boxtimes}\widetilde{L}_{1}$.
The canonical functor from
$\widetilde{L}_{1}\boxtimes\cdots\boxtimes\widetilde{L}_{1}$ to the relative
tensor product is conservative, hence we have
$\ker(\widetilde{L}_{n})=\ker(\widetilde{L}_{1})\underset{\mathrm{Rep}_{q}(G)}{\boxtimes}\cdots\underset{\mathrm{Rep}_{q}(G)}{\boxtimes}\ker(\widetilde{L}_{1})=\mathrm{Tors}_{n}$.

∎

###  3.7. Further computations of
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$ for $G=\operatorname{SL}_{2}$

For $G=SL_{2}$, the algebra $\mathcal{F}_{q}(G)$ is generated by matrix
coefficients of the first fundamental representation, and thus can be written
as

$\mathcal{F}_{q}(G)\simeq\mathbb{C}(q)\left\langle
a_{ij}\right\rangle\qquad\text{with}\qquad i,j={1,2},$

where

$a_{11}a_{12}=qa_{12}a_{11},\qquad a_{11}a_{21}=qa_{21}a_{11},\qquad
a_{12}a_{21}=a_{21}a_{12},$

and

$a_{11}a_{22}-qa_{12}a_{21}=1.$

Spelling out the above equality, and setting $x_{1}=a_{11}$ and $x_{2}=a_{12}$
in each factor, we arrive at the following Proposition.

######  Proposition 3.52.

The algebra $\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$ is an associative
algebra over $\mathbb{C}(q)$ with generators

$x^{(i)}_{1},x^{(i)}_{2},\quad 1\leqslant i\leqslant n,$

and relations

$x^{(i)}_{1}x^{(i)}_{2}=qx^{(i)}_{2}x^{(i)}_{1}\quad\text{for all}\quad
1\leqslant i\leqslant n,$

and

$\displaystyle x_{1}^{(j)}x_{1}^{(i)}$
$\displaystyle=q^{-\frac{1}{2}}x_{1}^{(i)}x_{1}^{(j)},$ $\displaystyle
x_{1}^{(j)}x_{2}^{(i)}$ $\displaystyle=q^{\frac{1}{2}}x_{2}^{(i)}x_{1}^{(j)},$
$\displaystyle x_{2}^{(j)}x_{1}^{(i)}$
$\displaystyle=q^{\frac{1}{2}}\left(x_{1}^{(i)}x_{2}^{(j)}+(q^{-1}-q)x_{2}^{(i)}x_{1}^{(j)}\right),$
(3.9) $\displaystyle x_{2}^{(j)}x_{2}^{(i)}$
$\displaystyle=q^{-\frac{1}{2}}x_{2}^{(i)}x_{2}^{(j)}.$

for all $1\leqslant i<j\leqslant n$.

All the above identities in 3.52 follow from viewing functions on $N\backslash
G$ as a $G$-representation, and using the $R$-matrix to braid the factors in
$\mathcal{F}_{q}((N\backslash G)^{n})$ before multiplying.

######  Definition 3.53.

For any pair $1\leqslant i,j\leqslant n$ such that $i\neq j$, we define
elements $D_{ij}\in\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$ by

$D_{ij}=x_{1}^{(i)}x_{2}^{(j)}-qx_{2}^{(i)}x_{1}^{(j)}.$

######  Lemma 3.54.

For $1\leqslant i<j\leqslant n$ we have

$D_{ji}=-q^{\frac{3}{2}}D_{ij}.$

###### Proof..

The proof follows immediately from relation (3.9) in 3.52:

$\displaystyle D_{ji}$
$\displaystyle=x_{1}^{(j)}x_{2}^{(i)}-qx_{2}^{(j)}x_{1}^{(i)}$
$\displaystyle=q^{\frac{1}{2}}x_{2}^{(i)}x_{1}^{(j)}-q^{\frac{3}{2}}\left(x_{1}^{(i)}x_{2}^{(j)}+(q^{-1}-q)x^{(i)}_{2}x_{1}^{(j)}\right)$
$\displaystyle=-q^{\frac{3}{2}}D_{ij}.$

∎

######  Proposition 3.55.

Elements $D_{ij}$ are invariant with respect to the (left coregular)
$U_{q}(\mathfrak{sl}_{2})$-action, that is
$D_{ij}\in\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})^{U_{q}(\mathfrak{sl}_{2})}$.
Moreover, for all $1\leqslant i<j\leqslant n$, and $a=1,2$, they satisfy
commutation relations

$D_{ij}x_{a}^{(k)}=\begin{cases}q^{\frac{1}{2}}x_{a}^{(i)}D_{ij},&\text{if}\;\;k=i,\\\
q^{-\frac{1}{2}}x_{a}^{(j)}D_{ij},&\text{if}\;\;k=j,\\\
x_{a}^{(k)}D_{ij},&\text{if}\;\;k<i\;\;\text{or}\;\;k>j,\end{cases}$

as well as

$\displaystyle D_{ij}x_{a}^{(k)}$
$\displaystyle=q^{-\frac{1}{2}}x_{a}^{(j)}D_{ik}+q^{\frac{1}{2}}x_{a}^{(i)}D_{kj},$
$\displaystyle x_{a}^{(k)}D_{ij}$
$\displaystyle=q^{\frac{1}{2}}D_{ik}x_{a}^{(j)}+q^{-\frac{1}{2}}D_{kj}x_{a}^{(i)}$

if $i<k<j$.

###### Proof..

Note that we have

$\displaystyle E\rightharpoonup x_{1}$ $\displaystyle=0,$ $\displaystyle
K\rightharpoonup x_{1}$ $\displaystyle=qx_{1},$ $\displaystyle
F\rightharpoonup x_{1}$ $\displaystyle=x_{2},$ $\displaystyle E\rightharpoonup
x_{2}$ $\displaystyle=x_{1},$ $\displaystyle K\rightharpoonup x_{2}$
$\displaystyle=q^{-1}x_{2},$ $\displaystyle F\rightharpoonup x_{2}$
$\displaystyle=0.$

The proof of the invariance of degree 2 element $D_{ij}$ then follows from a
direct calculation using the module algebra structure. For example, we have

$\displaystyle E\rightharpoonup D_{ij}$ $\displaystyle=\left((E\rightharpoonup
x_{1}^{(i)})x_{2}^{(j)}+(K\rightharpoonup x_{1}^{(i)})(E\rightharpoonup
x_{2}^{(j)})\right)$ $\displaystyle-q\left((E\rightharpoonup
x_{2}^{(i)})x_{1}^{(j)}+(K\rightharpoonup x_{2}^{(i)})(E\rightharpoonup
x_{1}^{(j)})\right)$
$\displaystyle=qx_{1}^{(i)}x_{1}^{(j)}-qx_{1}^{(i)}x_{1}^{(j)}=0.$

As for the commutation relations, we shall only prove the top equality here,
with the other cases being proved in a similar fashion. Substituting $a=1,2$
we obtain

$\displaystyle D_{ij}x_{2}^{(i)}$
$\displaystyle=x_{1}^{(i)}x_{2}^{(j)}x_{2}^{(i)}-qx_{2}^{(i)}x_{1}^{(j)}x_{2}^{(i)}$
$\displaystyle=q^{\frac{1}{2}}x_{2}^{(i)}x_{1}^{(i)}x_{2}^{(j)}-q^{\frac{3}{2}}x_{2}^{(i)}x_{2}^{(i)}x_{1}^{(j)}$
$\displaystyle=q^{\frac{1}{2}}x_{2}^{(i)}D_{ij}$

and

$\displaystyle D_{ij}x_{1}^{(i)}$
$\displaystyle=x_{1}^{(i)}x_{2}^{(j)}x_{1}^{(i)}-qx_{2}^{(i)}x_{1}^{(j)}x_{1}^{(i)}$
$\displaystyle=q^{\frac{1}{2}}x_{1}^{(i)}\left(x_{1}^{(i)}x_{2}^{(j)}+(q^{-1}-q)x_{2}^{(i)}x_{1}^{(j)}\right)-q^{-\frac{1}{2}}x_{1}^{(i)}x_{2}^{(i)}x_{1}^{(j)}$
$\displaystyle=q^{\frac{1}{2}}x_{2}^{(i)}\left(x_{1}^{(i)}x_{2}^{(j)}-)x_{2}^{(i)}x_{1}^{(j)}\right)$
$\displaystyle=q^{\frac{1}{2}}x_{2}^{(i)}D_{ij}.$

∎

As an immediate corollary of 3.55, we obtain the following commutation
relations between the $D_{ij}$.

######  Corollary 3.56.

For $i<k<j$, we have

$\displaystyle D_{ik}D_{kj}$ $\displaystyle=q^{-\frac{1}{2}}D_{kj}D_{ik},$
$\displaystyle D_{kj}D_{ij}$ $\displaystyle=q^{-\frac{1}{2}}D_{ij}D_{kj},$
$\displaystyle D_{ij}D_{ik}$ $\displaystyle=q^{-\frac{1}{2}}D_{ik}D_{ij}.$

We also obtain the following exchange relation:

######  Corollary 3.57.

For $i<k<j<l$, we have

$D_{ij}D_{kl}=q^{-\frac{1}{2}}D_{ik}D_{jl}+q^{\frac{1}{2}}D_{kj}D_{il}.$

Next, we will show that certain sets of functions $D_{ij}$ are multiplicative
Ore sets, which will allow us to localize the ring
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$.

We will say that an $n$-gon $\mathbb{D}_{n}$ is _based_ if its vertices are
labelled 1 to $n$ in a counterclockwise order. For any $1\leqslant k<n$, a
_step $k$ diagonal_ $e_{ij}$ in a based $n$-gon $\mathbb{D}_{n}$ is a segment
connecting vertices $i$ and $j$ with $j-i=k$. In particular, all step 1
diagonals are sides of $\mathbb{D}_{n}$, as well as the only step $(n-1)$
diagonal $e_{1n}$. In what follows, diagonals of step 2 will be referred to as
_short diagonals_. We say that diagonals $e_{ij}$ and $e_{kl}$ are _crossing_
if either of the following two inequalities holds:

$i<k<j<l\qquad\text{or}\qquad k<i<l<j.$

A _triangulation_ of $\mathbb{D}_{n}$ is a maximal collection of its non-
crossing diagonals. Note that every triangulation contains all sides of
$\mathbb{D}_{n}$ and at least one short diagonal. We will call any collection,
not necessarily maximal, of non-crossing diagonals of $\mathbb{D}_{n}$ its
_partial triangulation._

To an algebra $\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$ let us associate
an $n$-gon $\mathbb{D}_{n}$, whose vertices represent tensor factors of
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$ and are labelled accordingly.
A partial triangulation ${\underline{\Delta}}$ of $\mathbb{D}_{n}$ gives rise
to the multiplicative set
$S_{{\underline{\Delta}}}\subset\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$
generated by elements $D_{ij}$ such that $e_{ij}\in{\underline{\Delta}}$. Note
that by 3.56, all elements of $S_{{\underline{\Delta}}}$ skew-commute.

######  Lemma 3.58.

For any element $r\in\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$ and any
triangulation ${\underline{\Delta}}$ of $\mathbb{D}_{n}$, there exist elements
$s_{r},s^{\prime}_{r}\in S_{{\underline{\Delta}}}$ such that both products

$s_{r}r\qquad\text{and}\qquad rs^{\prime}_{r}$

can be expressed as sums of monomials in $x_{a}^{(1)}$, $x_{a}^{(n)}$, and
$D_{ij}\in S_{{\underline{\Delta}}}$.

###### Proof..

We shall prove the Lemma by induction on $n$. For $n=2$ the statement is
immediate. Note that by 3.52, any element
$r\in\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$ can be uniquely written as
a sum

$r=\sum_{\ell=1}^{m}c_{\ell}\left(x_{1}^{(1)}\right)^{\alpha_{\ell,1}}\left(x_{2}^{(1)}\right)^{\beta_{\ell,1}}\dots\left(x_{1}^{(n)}\right)^{\alpha_{\ell,n}}\left(x_{2}^{(n)}\right)^{\beta_{\ell,n}},$

where $c_{\ell}\in\mathbb{C}(q)$ and $m$ is the number of monomials in the
expression. Let us set

$d_{i}=\max_{\ell}\left(\alpha_{\ell,i}+\beta_{\ell,i}\right).$

Now, consider a triangulation ${\underline{\Delta}}$ of $\mathbb{D}_{n}$ and
any of its short diagonals $e=e_{i-1,i+1}\in{\underline{\Delta}}$. By 3.55 and
3.56, we have relations

$\displaystyle D_{e}x_{a}^{(i)}$
$\displaystyle=q^{-\frac{1}{2}}x_{a}^{(i+1)}D_{i-1,i}+q^{\frac{1}{2}}x_{a}^{(i-1)}D_{i,i+1},$
$\displaystyle x_{a}^{(i)}D_{e}$
$\displaystyle=q^{\frac{1}{2}}D_{i-1,i}x_{a}^{(i+1)}+q^{-\frac{1}{2}}D_{i,i+1}x_{a}^{(i-1)}$

and know that $D_{e}$ skew-commutes with all monomials in the right hand side
of the above equalities. This in turn implies that the products

$D_{e}^{d_{i}}\cdot r\qquad\text{and}\qquad r\cdot D_{e}^{d_{i}}$

can be expressed as sums of monomials in elements $D_{i-1,i}$, $D_{i,i+1}$,
and $x_{a}^{(j)}$ with $j\neq i$.

Now, let $\mathbb{D}_{n-1}$ be the $(n-1)$-gon obtained from $\mathbb{D}_{n}$
by erasing the sides $e_{i-1,i}$ and $e_{i,i+1}$. Choosing a short diagonal
$e^{\prime}=e_{j-1,j+1}$ of $\mathbb{D}_{n-1}$ and repeating the above
reasoning, we derive that the products

$D_{e^{\prime}}^{d^{\prime}_{j}}D_{e}^{d_{i}}\cdot r\qquad\text{and}\qquad
r\cdot D_{e}^{d_{i}}D_{e^{\prime}}^{d^{\prime}_{j}}$

can be expressed as sums of monomials in elements $D_{i-1,i}$, $D_{i,i+1}$,
$D_{j-1,j}$, $D_{j,j+1}$, and $x_{a}^{(k)}$ with $k\neq i,j$. The rest of the
proof is done by induction on the number $n$ of sides in $\mathbb{D}_{n}$. ∎

######  Corollary 3.59.

For any triangulation ${\underline{\Delta}}$ of $\mathbb{D}_{n}$, the
collection $S_{{\underline{\Delta}}}$ forms a multiplicative (left and right)
Ore set in the ring $R=\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$: for all
$r\in\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$ and $s\in
S_{\underline{\Delta}}$ one has

$S_{\underline{\Delta}}r\cap Rs\neq\emptyset\qquad\text{and}\qquad
rS_{\underline{\Delta}}\cap sR\neq\emptyset.$

###### Proof..

By Lemma 3.58, for any $r\in R$ there exist elements $s_{r},s^{\prime}_{r}\in
S_{\underline{\Delta}}$ so that the products $s_{r}r$ and $rs^{\prime}_{r}$
can be expressed as sums of monomials in $x_{a}^{(1)}$, $x_{a}^{(n)}$, and
$D_{ij}\in S_{\underline{\Delta}}$. Since elements of $S_{\underline{\Delta}}$
skew-commute with each other, as well as with $x_{a}^{(1)}$ and $x_{a}^{(n)}$,
we conclude that

$ss_{r}r=r^{\prime}s\qquad\text{and}\qquad rs^{\prime}_{r}s=sr^{\prime\prime}$

for some $r^{\prime},r^{\prime\prime}\in R$. ∎

######  Definition 3.60.

Let ${\underline{\Delta}}$ be a partial triangulation of $\mathbb{D}_{n}$. We
will denote by
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[S_{\underline{\Delta}}^{-1}]$
the Ore localization of $\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$ at the
multiplicative Ore set $S_{\underline{\Delta}}$. In particular, we will write
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[{e_{ij}}^{-1}]$ for
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[D_{ij}^{-1}]$.

######  Lemma 3.61.

For $n\geqslant 2$ and $1\leqslant i<j\leqslant n$, there is an injective map
of $U_{q}(\mathfrak{g})$-module algebras

$\eta_{ij}\colon\mathcal{F}_{q}(G)\longrightarrow\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[{e_{ij}}^{-1}]$

defined by

$\displaystyle a_{11}\longmapsto x_{1}^{(i)},\qquad\qquad a_{21}\longmapsto
x_{1}^{(j)}D_{ij}^{-1},$ $\displaystyle a_{12}\longmapsto
x_{2}^{(i)},\qquad\qquad a_{22}\longmapsto x_{2}^{(j)}D_{ij}^{-1}.$

###### Proof..

The statement of the Lemma is easily verified with the help of relations
established in 3.52 and 3.55. ∎

###  3.8. Cluster charts on $\mathcal{Z}(\mathbb{D}_{n})$ for
$G=\operatorname{SL}_{2}$

We have the algebra $\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$ along with
its localization
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[S_{\underline{\Delta}}^{-1}]$.
We would like to relate the localization with $\mathcal{Z}(\mathbb{D}_{n})$.
By construction the two inclusions indicated by solid arrows below define open
subcategories: we collect the embedding functors and their left adjoints in
the diagram below.

$\mathcal{Z}(\mathbb{D}_{n})$$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[S_{\underline{\Delta}}^{-1}]\mathrm{-mod}_{G\times\overline{T}^{\>\\!n}}$$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})\mathrm{-mod}_{G\times\overline{T}^{\>\\!n}}$

We define the dotted arrows as the evident compositions. In this section we
will show that the dotted arrows realize
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[S_{\underline{\Delta}}^{-1}]\mathrm{-mod}_{G\times
T^{n}}$ as an open subcategory of $\mathcal{Z}(\mathbb{D}_{n})$. Having
already identified $\mathcal{Z}(\mathbb{D}_{n})$ as the subcategory of
torsion-free modules, it remains only to show that localizing to
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[S_{\underline{\Delta}}^{-1}]$
kills all torsion modules.

######  Theorem 3.62.

The dotted arrows realize
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[S_{\underline{\Delta}}^{-1}]\mathrm{-mod}_{G\times
T^{n}}$ as an open subcategory of $\mathcal{Z}(\mathbb{D}_{n})$.

###### Proof..

Let us identify the weight lattice of $\operatorname{SL}_{2}$ with
$\mathbb{Z}$ via $\omega_{1}\mapsto 1,\alpha_{1}\mapsto 2$. As an algebra
object in $\mathrm{Rep}_{q}(G\times\overline{T}^{n})$,
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$ is graded by the torus
$T^{n}$; let us write $\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})^{(i;k)}$
for the set of elements of weight $\geq k$ with respect to the $i$th
$T$-factor, and
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})^{k\epsilon_{i}}$ for the subset
of elements whose weight with respect to $T^{n}$ is the vector $k\epsilon_{i}$
with a $k$ in position $i$ and zeros elsewhere. Then it is evident from the
homogeneity of the relations (3.9) that we have

$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})^{(i;k)}=\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})^{k\epsilon_{i}}.$
(3.10)

Suppose some $\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$-module $M$ admits
a non-zero morphism from a torsion module. Then there must exist an element
$m\in M$, an index $1\leqslant k\leqslant n$, and a natural number $N$ such
that

$r\geqslant
N\implies\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})^{r\epsilon_{k}}m=0.$

Now any $Z_{ij}$ such that $k\in\\{i,j\\}$ is an element of
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})^{k;1}$. So by Eq. 3.10, we have
$Z_{ij}^{r}\in\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})^{k;r}=\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})^{r\epsilon_{k}}$.
Therefore $\left(Z_{ij}\right)^{r}m=0$, which shows that the element $Z_{ij}$
does not act invertibly on such a module $M$. Thus we have
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[S_{\underline{\Delta}}^{-1}]\mathrm{-mod}_{G\times
T^{n}}\subset\mathrm{Tors}^{\perp}$. ∎

######  Theorem 3.63.

For any partial triangulation ${\underline{\Delta}}\neq\emptyset$ of a based
$n$-gon associated to $\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})$, the
functor of taking $U_{q}(g)$-invariants,

$\mathcal{G}\colon\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})\mathrm{-mod}_{G\times\overline{T}^{\>\\!n}}\longrightarrow\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})^{U_{q}(\mathfrak{g})}\mathrm{-mod}_{\overline{T}^{\>\\!n}},$

restricts to an equivalence of categories,

$\mathcal{G}^{\prime}\colon\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[S_{\underline{\Delta}}^{-1}]\mathrm{-mod}_{G\times\overline{T}^{\>\\!n}}\longrightarrow\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[S_{\underline{\Delta}}^{-1}]^{U_{q}(\mathfrak{g})}\mathrm{-mod}_{\overline{T}^{\>\\!n}}.$

###### Proof..

Because the category $\mathrm{Rep}_{q}(G)$ is semisimple, the functor
$\mathcal{G}^{\prime}$ of taking $U_{q}(\mathfrak{g})$-invariants is exact.
Thus to prove that it is an equivalence, it suffices to show that it is also
conservative. Since the partial triangulation ${\underline{\Delta}}$ is not
empty, there is an edge $e_{ij}\in{\underline{\Delta}}$ and hence an injective
homomorpshim
$\eta_{ij}\colon\mathcal{F}_{q}(G)\rightarrow\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[S_{\underline{\Delta}}^{-1}]$
by Lemma 3.61. This in turn allows us to include the functor
$\mathcal{G}^{\prime}$ into the commutative diagram

$\textstyle{\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[S_{\underline{\Delta}}^{-1}]\mathrm{-mod}_{G\times\overline{T}^{\>\\!n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{G}^{\prime}}$$\scriptstyle{\mathcal{F}^{\prime\prime}}$$\textstyle{\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{n})[S_{\underline{\Delta}}^{-1}]^{U_{q}(\mathfrak{g})}\mathrm{-mod}_{\overline{T}^{\>\\!n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}^{\prime}}$$\textstyle{\mathcal{F}_{q}(G)\mathrm{-mod}_{G\times\overline{T}^{\>\\!n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{G}^{\prime\prime}}$$\textstyle{\overline{\mathrm{Rep}_{q}}(T^{\>\\!n})}$

where $\mathcal{F}^{\prime}$ and $\mathcal{F}^{\prime\prime}$ are forgetful
functors, and $\mathcal{G}^{\prime\prime}$ is again the functor of taking
$U_{q}(\mathfrak{g})$-invariants. Now, it is enough to show that the
composition $\mathcal{G}^{\prime\prime}\circ\mathcal{F}^{\prime\prime}$ is
conservative. Being a forgetful functor, $\mathcal{F}^{\prime\prime}$ is
obviously conservative, while $\mathcal{G}^{\prime\prime}$ is in fact an
equivalence with inverse given by $V\mapsto\mathcal{F}_{q}(G)\otimes V$. ∎

##  4\. Charts and flips on $\mathcal{Z}(\mathbb{S})$ via excision

In this section we will compute with an arbitrary simple decorated surface
$\mathbb{S}$. We will describe a special family of open subcategories of the
category $\mathcal{Z}(\mathbb{S})$, which we regard as open subvarieties –
charts – on the quantum decorated character stack of $\mathbb{S}$. We have one
such chart for each isotopy class of triangulation of the $\mathbb{S}$, and
each chart comes equipped with a distinguished compact projective generator
$\mathrm{Dist}_{\underline{\Delta}}$, and an identification as the category of
modules for a quantum torus, resembling the notion of a cluster chart.

The basic mechanism to obtain these charts is to start from the monadic
description of $\mathcal{Z}(\mathbb{D}_{3})$ obtained in the previous section,
and then apply excision with respect to gluing triangles along a common digon
$\mathbb{D}_{2}$. The charts so constructed admit pairwise transition maps
resembling those in the quantum $\mathcal{A}$-variety variety of
Fock–Goncharov – we call the whole structure a _quantum
$\widehat{\mathcal{A}}$-variety_, and we compare it to the quantum cluster
$\mathcal{A}$-variety of Fock–Goncharov. Taking $T$-invariants yields a
version of the quantum cluster ensemble map, giving a reconstruction of
cluster $\mathcal{X}$-varieties.

###  4.1. Opening $T$-gates

If $\mathbb{S}$ is a decorated surface with a collection $\mathcal{G}$ of
gates, recall that the category $\mathcal{Z}(\mathbb{S})$ inherits the
structure of a module category for the braided tensor category
$\mathrm{Rep}_{q}(\mathcal{G})=\mathrm{Rep}_{q}(G^{m}\boxtimes\overline{T}^{n})$,
where $m$ and $n$ denote the number of $G$\- and $T$-gates in $\mathcal{G}$.
Given some object $X\in\mathbb{S}$ – typically in examples $X$ will be some
$\mathrm{Dist}_{\underline{\Delta}}$ – different choices of $\mathcal{G}$
clearly produce distinct internal endomorphism algebras
$\underline{\operatorname{End}}_{\mathcal{G}}(X)$.

However, the category
$\underline{\operatorname{End}}_{\mathcal{G}}(X)\mathrm{-mod}_{\mathcal{G}}$
of $\underline{\operatorname{End}}_{\mathcal{G}}(X)$-modules in
$\mathrm{Rep}_{q}(\mathcal{G})$ does not change when we open gates.

######  Lemma 4.1.

Let $\mathcal{G}$ be some configuration of $T$-gates, and let
$\mathcal{G}^{\prime}$ be obtained from $\mathcal{G}$ by replacing any
$T$-gate with a pair of adjacent gates. Then the functor of taking invariants
at either new gate yields an equivalence of categories:

$\underline{\operatorname{End}}_{\mathcal{G}}(\mathrm{Dist}_{\mathbb{S}})\mathrm{-mod}_{\mathcal{G}}\simeq\underline{\operatorname{End}}_{\mathcal{G}^{\prime}}(\mathrm{Dist}_{\mathbb{S}})\mathrm{-mod}_{\mathcal{G}^{\prime}}.$

###### Proof..

Indeed, in this case the $\mathcal{G}^{\prime}$-module category
$\underline{\operatorname{End}}_{\mathcal{G}}^{\prime}(\mathrm{Dist}_{\mathbb{S}})\mathrm{-mod}_{\mathrm{Rep}_{q}(\mathcal{G}^{\prime})}$
is simply the pullback of
$\underline{\operatorname{End}}_{\mathcal{G}}(\mathrm{Dist}_{\mathbb{S}})\mathrm{-mod}_{\mathrm{Rep}_{q}(\mathcal{G})}$
along the tensor functor
$\mathrm{Rep}_{q}(\overline{T})\boxtimes\mathrm{Rep}_{q}(\overline{T})\to\mathrm{Rep}_{q}(\overline{T})$.
∎

The specification of $T$-gates, as well as the presentation for the resulting
algebras of internal endomorphisms will be encoded combinatorially in the data
of a _fencing graph_ associated to the triangulation.

######  Definition 4.2.

Let ${\underline{\Delta}}$ be a triangulation of a simple decorated surface.
Then the boundary of the $G$-region of every digon or triangle in
${\underline{\Delta}}$ consists of interlacing _short_ and _long_ , where the
former lie in $\mathbb{S}_{T}$, the latter in $\mathbb{S}_{G}$.

Thus each triangle or digon gives rise to a hexagon or rectangle respectively
with alternating long and short edges.

######  Definition 4.3.

The _fencing graph_ $\Gamma_{\underline{\Delta}}$ has one vertex for each
corner of a hexagon appearing in ${\underline{\Delta}}$. The edge set consists
of a directed arrow connecting opposite vertices of each short edge of either
a hexagon or a rectangle, with the direction being compatible with the
clockwise orientation within $T$-region, and a bivalent edge connecting the
opposite pair of vertices of each long edge.

An example of a fencing graph for the punctured triangle
$\mathbb{D}_{3}^{\circ}$ is illustrated in Fig. 8. Clearly, there are $6h$
vertices in $\Gamma_{\underline{\Delta}}$, where $h$ is the number of
triangles in ${\underline{\Delta}}$. We associate to the fencing graph
$\Gamma_{{\underline{\Delta}}}$ a collection of $T$-gates in $\mathbb{S}$ by
declaring that each vertex of $\Gamma_{\underline{\Delta}}$ lies in a unique
$T$-gate. Thus a notched triangulation ${\underline{\Delta}}_{\bullet}$
determines in each $T$-region a _distinguished vertex_ of the fencing graph
$\Gamma_{{\underline{\Delta}}}$, for convenience we assume that as we traverse
the boundary of any $T$-disk in a counterclockwise direction, we encounter the
distinguished vertex first. It also induces a total ordering on the edge-ends
incident to vertices at each $T$-region: we number edge-ends as we traverse
the $T$-region clockwise, starting with the one incident to the distinguished
vertex.

This in turn determines a _notched ideal triangulation_ , where we identif all
vertices in each $T$-region of the fencing graph, collapse parallel long
edges, and collapse all short edges but one whose head is the distinguished
vertex. The remaining short edge becomes a loop, which we call a _notch_ ;
this defines a total ordering on the edge-ends of the notched ideal
triangulation, see Fig. 8.

We will be considering the following configurations of gates associated to a
notched triangulation ${\underline{\Delta}}_{\bullet}$:

  * •

$\mathcal{T}_{\Gamma}$ has a single gate located at each vertex of the fencing
graph.

  * •

$\mathcal{T}_{P}$ contains only the distinguished gate at each puncture.

  * •

$\mathcal{T}_{P^{c}}$ contains only the non-distguished gates at each
puncture.

  * •

$\mathcal{T}_{M}$ contains only the gates at each marked point.

†

†

$\boldsymbol{2}$$\boldsymbol{1}$$\boldsymbol{3}$ Figure 8. Presentation of a
punctured triangle $\mathbb{D}_{3}^{\circ}$ as in Eq. 1.1 and Eq. 4.1, the
corresponding fencing graph with a distinguished vertex, and the notched ideal
triangulation with the total numbering on the edge-ends adjacent to the
puncture. The location of the distinguished gate in $\mathbb{D}_{3}^{\circ}$,
and the distinguished vertex of the fencing graph, are marked with

$\Cross$

.

Hence, we have a decomposition

$\mathcal{T}_{\Gamma}=\mathcal{T}_{P^{c}}\boxtimes\mathcal{T}_{P}\boxtimes\mathcal{T}_{M}$

######  Remark 4.4.

Let us discuss the rationale for introducing additional $T$-gates. We will see
below that opening gates produces quantum tori
$\widetilde{\zeta}({\underline{\Delta}}_{\bullet})$ with more generators than
the tori $\zeta({\underline{\Delta}}_{\bullet})$, in which we are ultimately
interested. However, the larger tori
$\widetilde{\zeta}({\underline{\Delta}}_{\bullet})$ have simpler commutation
relations which are expressed more locally in the triangulation and hence
easier to compute, as in Propositions 4.7 and 4.6 below. So we compute with
the algebra $\widetilde{\zeta}({\underline{\Delta}}_{\bullet})$ while
performing excision, and close the additional gates only at the end.

###  4.2. Charts on $\mathcal{Z}(\mathbb{D}_{2})$ and
$\mathcal{Z}(\mathbb{D}_{3})$

Let us first consider the two cases when $\mathbb{S}$ is a triangle or digon.
Recall from Theorem 3.62 the open subcategories
$\mathcal{Z}(\underline{\mathbb{D}_{3}})\subset\mathcal{Z}(\mathbb{D}_{3})$
and
$\mathcal{Z}(\underline{\mathbb{D}_{3}})\subset\mathcal{Z}(\mathbb{D}_{3})$
defined, respectively, by the condition that elements of
$S_{\underline{\mathbb{D}_{2}}}$ and $S_{\underline{\mathbb{D}_{3}}}$ act
invertibly. We consider the restriction of the distinguished object to each
subcategory, and define

$\displaystyle\zeta(\underline{\mathbb{D}_{2}})$
$\displaystyle:=\underline{\operatorname{End}}_{T^{2}}(\mathrm{Dist}_{\underline{\mathbb{D}_{2}}})\cong\mathcal{F}_{q}\left(\operatorname{Conf}^{fr}_{2}[\underline{\mathbb{D}_{2}}^{-1}]\right)^{U_{q}(\mathfrak{g})},$
$\displaystyle\zeta(\underline{\mathbb{D}_{3}})$
$\displaystyle:=\underline{\operatorname{End}}_{T^{3}}(\mathrm{Dist}_{\underline{\mathbb{D}_{3}}})\cong\mathcal{F}_{q}\left(\operatorname{Conf}^{fr}_{3}[\underline{\mathbb{D}_{3}}^{-1}]\right)^{U_{q}(\mathfrak{g})},$

where the $T$-actions are induced by introducing one $T$-gate in each
$T$-region. By Theorem 3.63, we have equivalences of categories

$\mathcal{Z}(\underline{\mathbb{D}_{2}})\simeq\zeta(\underline{\mathbb{D}_{2}})\mathrm{-mod}_{\overline{T}^{2}},\quad\mathcal{Z}(\underline{\mathbb{D}_{3}})\simeq\zeta(\underline{\mathbb{D}_{3}})\mathrm{-mod}_{\overline{T}^{3}}.$

It is evident that we have
$\zeta(\underline{\mathbb{D}_{2}})\simeq\mathbb{C}(q)\left\langle D^{\pm
1}\right\rangle$, and the following Lemma shows that the algebra
$\zeta(\underline{\mathbb{D}_{3}})$ is also a quantum torus.

######  Lemma 4.5.

The algebra $\zeta(\underline{\mathbb{D}_{3}})$ is generated by the elements
$D_{ij}^{\pm 1}$ for $1\leqslant i<j\leqslant 3$ given in the 3.53.

###### Proof..

The algebra
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{3})^{U_{q}(\mathfrak{g})}$ is the
direct sum of its weight subspaces with respect to
$U_{q}(\mathfrak{t})^{\otimes 3}$:

$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{3})^{U_{q}(\mathfrak{g})}=\bigoplus_{l,m,n\geq
0}\left(\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{3})^{U_{q}(\mathfrak{g})}\right)^{l,m,n},$

and each of these subspaces is at either 1-dimensional if $l+m+n$ is even and
there exists a Euclidean triangle with side lengths $l,m,n$, or zero-
dimensional otherwise. But it is straightforward to see that the submodule of
$\mathbb{Z}^{3}$ generated by the weights $(1,1,0),(1,0,1),(0,1,1)$ of the
$D_{ij}$ is exactly the lattice of all vectors with even coordinate sum. Thus
for each triple $(l,m,n)$ such that the corresponding subspace of
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{3})^{U_{q}(\mathfrak{g})}$ is non-
zero, there exists a triple of integers $n_{1},n_{2},n_{3}\in\mathbb{Z}$ such
that

$D_{12}^{n_{3}}D_{13}^{n_{2}}D_{23}^{n_{1}}\in\left(\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{3})^{U_{q}(\mathfrak{g})}\right)^{\alpha,\beta,\gamma}.$

∎

Opening an additional gate at each $T$-region equips
$\mathcal{Z}(\mathbb{D}_{2})$ and $\mathcal{Z}(\mathbb{D}_{3})$ with $T^{4}$\-
and $T^{6}$-actions, respectively, over which the restricted distinguished
objects are still relative generators. Thus writing

$\widetilde{\zeta}(\underline{\mathbb{D}_{2}}):=\underline{\operatorname{End}}_{\overline{T}^{\>\\!4}}\left(\mathrm{Dist}_{\underline{\mathbb{D}_{2}}}\right),\quad\widetilde{\zeta}(\underline{\mathbb{D}_{3}}):=\underline{\operatorname{End}}_{\overline{T}^{\>\\!6}}\left(\mathrm{Dist}_{\underline{\mathbb{D}_{3}}}\right),$

we obtain equivalences of categories

$\mathcal{Z}(\underline{\mathbb{D}_{2}})\simeq\widetilde{\zeta}(\underline{\mathbb{D}_{2}})\mathrm{-mod}_{T^{4}},\quad\mathcal{Z}(\underline{\mathbb{D}_{3}})\simeq\widetilde{\zeta}(\underline{\mathbb{D}_{3}})\mathrm{-mod}_{T^{6}}.$

######  Lemma 4.6.

The algebra $\widetilde{\zeta}(\underline{\mathbb{D}_{2}})$ is generated by
elements $A_{1},A_{2}$ in correspondence with the long edges of the fencing
graph for $\underline{\mathbb{D}_{2}}$, as well as generators $a_{1},a_{2}$ in
correspondence with its short edges. These generators are subject to the
relations

$[A_{1},A_{2}]=[a_{1},a_{2}]=0,\qquad
a_{i}A_{j}=q^{\frac{1}{2}}A_{j}a_{i},\qquad\text{and}\qquad
a_{1}A_{2}=a_{2}A_{1}.$

###### Proof..

The Lemma is a special case of Example 3.38, where we open one additional gate
in each $T$-region. Introducing the weights
$\varepsilon_{1}=(1,0),\varepsilon_{2}=(0,1)$ for the
$\mathrm{Rep}_{q}(T^{2})$–action on $\zeta(\underline{\mathbb{D}}_{2})$ and
setting
$\mathbb{C}_{\lambda,\mu}=\mathbb{C}_{\lambda}\boxtimes\mathbb{C}_{\mu}\in\mathrm{Rep}_{q}(T^{2})\boxtimes\mathrm{Rep}_{q}(T^{2})$,
we take the generators $A_{i},a_{i}$ to be

$\displaystyle A_{1}$
$\displaystyle=D\otimes\chi_{\varepsilon_{1},\varepsilon_{2}},\qquad
a_{1}=1\otimes\chi_{\varepsilon_{1},-\varepsilon_{1}},$ $\displaystyle A_{2}$
$\displaystyle=D\otimes\chi_{\varepsilon_{2},\varepsilon_{1}},\qquad
a_{2}=1\otimes\chi_{\varepsilon_{2},-\varepsilon_{2}}.$

The asserted relations then follow from the general multiplication rule
(3.38). ∎

Similarly, in the case of $\mathbb{D}_{3}$ we have

######  Lemma 4.7.

The algebra $\widetilde{\zeta}(\underline{\mathbb{D}_{3}})$ is a quantum torus
with generators $A_{12},A_{13},A_{23}$ in correspondence with the long edges
of the fencing graph for $\underline{\mathbb{D}_{3}}$, as well as generators
$a_{1},a_{2},a_{3}$ in correspondence with its short edges. The relations
among these generators are

$[A_{ij},A_{kl}]=[a_{i},a_{j}]=0\qquad\text{and}\qquad
a_{k}A_{ij}=q^{\frac{\delta_{ik}+\delta_{kj}}{2}}A_{ij}a_{k}.$

###### Proof..

The proof is identical to that of Lemma 4.6. The generators are given by

$A_{12}=D_{12}\otimes\chi_{\varepsilon_{2},\varepsilon_{1}},\qquad
A_{23}=D_{23}\otimes\chi_{\varepsilon_{3},\varepsilon_{2}},\qquad
A_{13}=D_{13}\otimes\chi_{\varepsilon_{1},\varepsilon_{3}},$

and

$a_{k}=1\otimes\chi_{\varepsilon_{k},-\varepsilon_{k}},\qquad\text{for}\qquad
k=1,2,3,$

with the asserted relations between them following immediately from formula
(3.38). ∎

The above computations were all for $G=\operatorname{SL}_{2}$. However, we
have the following parallel descriptions for $G=\operatorname{PGL}_{2}$. The
proof in this case is identical except that one requires all representations
which appear to have exclusively even weights.

######  Corollary 4.8.

We have the following:

  * •

Each algebra $\zeta_{\operatorname{PGL}_{2}}(\underline{\mathbb{D}_{2}})$ is a
quantum sub-torus of
$\zeta_{\operatorname{SL}_{2}}(\underline{\mathbb{D}_{2}})$ generated by
$D^{\pm 2}$.

  * •

Each algebra $\zeta_{\operatorname{PGL}_{2}}(\underline{\mathbb{D}_{3}})$ is a
quantum sub-torus of
$\zeta_{\operatorname{SL}_{2}}(\underline{\mathbb{D}_{2}})$ generated by

$D_{12}^{\pm 2},D_{13}^{\pm 2},D_{23}^{\pm 2},(D_{12}D_{13}D_{23})^{\pm 1}.$

  * •

Each algebra
$\widetilde{\zeta}_{\operatorname{PGL}_{2}}(\underline{\mathbb{D}_{2}})$ is a
quantum sub-torus of
$\zeta_{\operatorname{SL}_{2}}(\underline{\mathbb{D}_{2}})$ generated by

$A_{1}^{\pm 2},A_{2}^{\pm 2},a_{1}^{\pm 2},a_{2}^{\pm 2}.$

  * •

Each algebra
$\widetilde{\zeta}_{\operatorname{PGL}_{2}}(\underline{\mathbb{D}_{3}})$ is a
quantum sub-torus of
$\zeta_{\operatorname{SL}_{2}}(\underline{\mathbb{D}_{3}})$ generated by

$A_{12}^{\pm 2},A_{13}^{\pm 2},A_{23}^{\pm
2},(A_{12}A_{23}A_{13}a_{1}a_{2}a_{3})^{\pm 1},a_{1}^{\pm 2},a_{2}^{\pm
2},a_{3}^{\pm 2}.$

.

######  Remark 4.9.

We remark that a monomial subalgebra of a quantum torus is necessarily a
quantum torus, so that even though there are relations above the generators
listed above one can always (non-canonically) choose free generators. For
instance, in the second list one may take
$D_{12}^{2},D_{13}^{2},D_{12}D_{13}D_{23}$.

######  Remark 4.10.

Note that the orientation of $\mathbb{D}_{3}$ defines a cyclic ordering on its
$T$-regions. In what follows, we regard the generator $A_{ij}$ as being in
correspondence with the long edge of the fencing graph
$\Gamma_{\mathbb{D}_{3}}$ connecting the $T$-regions labelled $i$ and $j$, and
the generator $a_{k}$ as being in correspondence with the short edge of
$\Gamma_{\mathbb{D}_{3}}$ sitting on the boundary of the $k$-th $T$-region.

###  4.3. Charts on $\mathcal{Z}(\mathbb{S})$ from triangulations

Suppose now that ${\underline{\Delta}}$ is a triangulation of a general
decorated surface $\mathbb{S}$ consisting of $t$ triangles and $l$ digons. To
organize computations, we shall fix a framing of $\mathbb{S}$, in which all
triangles are drawn in the plane of the blackboard with their long edges
parallel to its $x$-axis. Each digon we represent as a ribbon connecting a
pair of long edges of the triangles. In particular, such a framing of
$\mathbb{S}$ fixes for each long edge of the fencing graph
$\Gamma_{{\underline{\Delta}}}$ an ordering on the vertices at its endpoints.
An example of such a framing in the case that $\mathbb{S}$ is a punctured
torus is drawn in Fig. 9. We wish to emphasize that this additional choice of
a framing is _immaterial_ to the determination of the charts we will
construct, but is merely _convenient_ for giving presentations, and formulas
for mutations. See Section 4.9 for more details.

$2$$6$$4$$5$$1$$2$$3$$4$$1$$2$$3$$4$$5$$6$ Figure 9. The once-punctured torus
with its fencing graph overlaid, and its 2-framing as presented by a pair of
jellyfish.

The triangulation ${\underline{\Delta}}$ is a presentation of $\mathbb{S}$ as

$\mathbb{S}=(\mathbb{D}_{3})^{\sqcup t}\bigsqcup_{(\mathbb{D}_{2})^{\sqcup
2\ell}}(\mathbb{D}_{2})^{\sqcup\ell},$

where the $2l$ digons over which we glue are obtained as small neighborhoods
of the long edges of the fencing graph $\Gamma_{{\underline{\Delta}}}$ as
depicted in Fig. 9. Applying excision, we thus obtain an equivalence of
categories

$\displaystyle\mathcal{Z}(\mathbb{S})\simeq\mathcal{Z}(\mathbb{D}_{3})^{t}\bigboxtimes_{\mathcal{Z}(\mathbb{D}_{2})^{2l}}\mathcal{Z}(\mathbb{D}_{2})^{l}$
(4.1)

which allows us to define a open subcategory
$\mathcal{Z}({\underline{\Delta}})$ of $\mathcal{Z}(\mathbb{S})$ by

$\displaystyle{\mathcal{Z}({\underline{\Delta}})}:={\mathcal{Z}(\underline{\mathbb{D}_{3}})}^{t}\bigboxtimes_{{\mathcal{Z}(\underline{\mathbb{D}_{2}})}^{2l}}\mathcal{Z}(\underline{\mathbb{D}_{2}})^{l}.$
(4.2)

By construction, $\widetilde{\zeta}(\underline{\mathbb{D}_{2}})^{\otimes l}$
is a commutative algebra object in the braided tensor category
$\mathcal{T}_{\Gamma}$. Moreover, in view of Lemmas 4.7 and 4.6 we have
equivalences

$\mathcal{Z}(\underline{\mathbb{D}_{3}})^{t}\simeq\widetilde{\zeta}(\underline{\mathbb{D}_{3}})^{\otimes
t}\mathrm{-mod}_{\mathcal{T}_{\Gamma}},\qquad\mathcal{Z}(\underline{\mathbb{D}_{2}})^{2l}\simeq\widetilde{\zeta}(\underline{\mathbb{D}_{2}})^{\otimes
2l}\mathrm{-mod}_{\mathcal{T}_{\Gamma}},$

where $\widetilde{\zeta}(\underline{\mathbb{D}_{3}})^{\otimes t}$ and
$\widetilde{\zeta}(\underline{\mathbb{D}_{2}})^{\otimes 2l}$ are both
$\widetilde{\zeta}(\underline{\mathbb{D}_{2}})^{\otimes l}$–algebras. We thus
find ourselves in the setting of Lemma 1.17, which we may apply to obtain the
following description of the open subcategory corresponding to the
triangulation ${\underline{\Delta}}$:

$\mathcal{Z}({\underline{\Delta}})\simeq\widetilde{\zeta}({\underline{\Delta}})\mathrm{-mod}_{\mathcal{T}_{\Gamma}},$

where

$\widetilde{\zeta}({\underline{\Delta}}):=\underline{\operatorname{End}}_{\mathbf{T}_{\underline{\Delta}}}(\mathrm{Dist}_{\underline{\Delta}})\cong\widetilde{\zeta}(\underline{\mathbb{D}_{3}})^{\otimes
t}\otimes_{{\zeta}(\underline{\mathbb{D}_{2}})^{\otimes
2l}}\widetilde{\zeta}(\underline{\mathbb{D}_{2}})^{\otimes l}.$

The quantum torus $\widetilde{\zeta}({\underline{\Delta}})$ can be described
explicitly with the help of the relations from Lemmas 4.7 and 4.6. Let us
write $E(\Gamma_{\underline{\Delta}})$ for the set of edges of the fencing
graph $\Gamma_{\underline{\Delta}}$, so that

$E(\Gamma_{\underline{\Delta}})=E_{l}(\Gamma_{\underline{\Delta}})\sqcup
E_{s}(\Gamma_{\underline{\Delta}}),$

where $E_{l}(\Gamma_{\underline{\Delta}})$ and
$E_{s}(\Gamma_{\underline{\Delta}})$ denote the subsets of long and short
edges respectively. Recall that the union of all short edges coincides with
the set of walls $\mathcal{C}$ between $G$ and $T$-regions. Let us now orient
the short edges so that they travel around $T$-regions in a clockwise manner.
Given a short edge $e\in E_{s}(\Gamma_{\underline{\Delta}})$, we define its
_head_ $h(e)$ and its _tail_ $t(e)$, to be its endpoints so that $e$ is
oriented from $t(e)$ to $h(e)$.

######  Proposition 4.11.

The algebra object $\widetilde{\zeta}({\underline{\Delta}})$ in the category
$\mathcal{T}_{\Gamma}$ is a quantum torus with generators $A_{\ell}^{\pm 1}$,
$\ell\in E_{l}(\Gamma_{\underline{\Delta}})$, $a_{e}^{\pm 1}$, $e\in
E_{s}(\Gamma_{\underline{\Delta}})$. These generators are subject to the
following _fencing graph relations_ which hold for all long edges
$\ell,\ell^{\prime}$ and short edges $e,e^{\prime}$:

$A_{\ell}A_{\ell^{\prime}}=A_{\ell^{\prime}}A_{\ell},\qquad
a_{1}A_{2}=a_{2}A_{1},$ (4.3)

if $a_{1},a_{2},A_{1},A_{2}$ are the four edges in a digon as in Lemma 4.6,

$\displaystyle a_{e}A_{\ell}$
$\displaystyle=\begin{cases}q^{\frac{1}{2}}A_{\ell}a_{e}&\text{if}\quad\ell\cap
e\neq\emptyset,\\\ A_{\ell}a_{e}&\text{otherwise};\end{cases},$ $\displaystyle
a_{e}a_{e^{\prime}}$
$\displaystyle=\begin{cases}q^{\frac{1}{2}}a_{e^{\prime}}a_{e}&\text{if}\quad
h(e)=t(e^{\prime})\quad\text{and}\quad t(e)\neq h(e^{\prime}),\\\
a_{e^{\prime}}a_{e}&\text{otherwise}.\end{cases},$

The $\mathbf{T}_{\underline{\Delta}}$-equivariant structure is as follows:

$\operatorname{wt}_{i}(A_{\ell})=\begin{cases}1&\text{if}\quad i\in\ell,\\\
0&\text{if}\quad
i\notin\ell,\end{cases}\qquad\operatorname{wt}_{i}(a_{e})=\begin{cases}1&\text{if}\quad
i=t(e),\\\ -1&\text{if}\quad i=h(e),\\\ 0&\text{otherwise},\end{cases}$

where $\operatorname{wt}_{i}(x)\in\Lambda(T)$ indicates weight of an element
$x\in\widetilde{\zeta}({\underline{\Delta}})$ with respect to the torus at the
$i$-th vertex in $\Gamma_{\underline{\Delta}}$.

######  Corollary 4.12.

The algebra object
$\widetilde{\zeta}_{\operatorname{PGL}_{2}}({\underline{\Delta}})$ is the
subalgebra of
$\widetilde{\zeta}_{\operatorname{SL}_{2}}({\underline{\Delta}})$ generated by
$(a_{i})^{\pm 2}$ for each short edge, $A_{\ell}^{\pm 2}$ for each long edge,
and $(A_{\ell_{1}}A_{\ell_{2}}A_{\ell_{3}}a_{1}a_{2}a_{3})^{\pm 1}$ for each
triangle.

###  4.4. Closing $T$-gates

######  Definition 4.13.

If ${\underline{\Delta}}_{\bullet}$ is a notched triangulation of a decorated
surface $\mathbb{S}$, we define the quantum torus
$\zeta({\underline{\Delta}}_{\bullet})$ to be the subalgebra of
$\mathcal{T}_{P^{c}}$-invariants in $\widetilde{\zeta}({\underline{\Delta}})$.

By Lemma 4.1, the functor of taking $\mathcal{T}_{P^{c}}$ invariants is
conservative. Hence we have an equivalence of categories,

$\zeta({\underline{\Delta}}_{\bullet})\mathrm{-mod}_{\mathcal{T}_{P\cup
M}}\simeq\widetilde{\zeta}({\underline{\Delta}})\mathrm{-mod}_{\mathcal{T}_{\Gamma}}.$

The generators of $\zeta({\underline{\Delta}}_{\bullet})$ are labelled by the
edges and the notch of the notched ideal triangulations. We shall now describe
these generators and their relations, and express them in terms of generators
of $\widetilde{\zeta}({\underline{\Delta}})$. We will make heavy use of the
Weyl ordering on the quantum torus $\widetilde{\zeta}({\underline{\Delta}})$.

######  Definition 4.14.

Suppose that $\zeta$ is a quantum torus with a basis of generators
$\\{X_{i}\\}$. Then the _Weyl ordering_ with respect to the generating set
$(X_{i})$ is defined recursively by declaring $\hskip
1.99997pt\boldsymbol{:}X_{i}\boldsymbol{:}\hskip 1.99997pt=X_{i}$ for all $i$,
and if $Y$ and $Z$ are two monomial elements satisfying $YZ=q^{2r}ZY$,

$q^{-r}\\!\\!\hskip 1.99997pt\boldsymbol{:}Y\boldsymbol{:}\hskip
1.99997pt\\!\hskip 1.99997pt\boldsymbol{:}Z\boldsymbol{:}\hskip
1.99997pt=\hskip 1.99997pt\boldsymbol{:}YZ\boldsymbol{:}\hskip
1.99997pt=q^{r}\\!\\!\hskip 1.99997pt\boldsymbol{:}Z\boldsymbol{:}\hskip
1.99997pt\\!\hskip 1.99997pt\boldsymbol{:}Y\boldsymbol{:}\hskip 1.99997pt.$

Let $s$ be an edge in the notched triangulation
${\underline{\Delta}}_{\bullet}$, and pick a long edge $l$ in the fencing
graph corresponding to $s$. Then the endpoints of $l$ are vertices
$w_{1},w_{2}$ of the fencing graph contained in regions whose distinguished
vertices are $v_{1}$ and $v_{2}$. Denote by $a(l_{1})$ the set of short edges
in the minimal path in the fencing graph starting at distinguished vertex
$v_{1}$ and following the short edges clockwise to the vertex $w_{1}$.
Similarly we define $a(l_{2})$ to be the set of short edges in the minimal
path in the fencing graph starting at the distinguished vertex $v_{2}$ and
travelling clockwise to the vertex $w_{2}$. Then we define an element $Z_{s}$
of $\zeta({\underline{\Delta}}_{\bullet})$ by

$Z_{s}:=\hskip
1.99997pt\boldsymbol{:}a(l_{1})a(l_{2})A_{l}\boldsymbol{:}\hskip 1.99997pt.$

It follows from the relations (4.3) in 4.11 that the element $Z_{s}$ does not
depend on the choice of long edge $l$ projecting to the edge $s$ in the
underlying notched triangulation.

Given a puncture $p\in\widehat{S}$, let $v\in\Gamma_{\underline{\Delta}}$ be
the corresponding distinguished vertex, and $\gamma_{p}$ be the path which
connects $v$ to itself and winds once around the corresponding $T$-region.
Then we define another element of $\zeta({\underline{\Delta}}_{\bullet})$ as
the Weyl-ordered product

$\alpha_{p}:=\hskip
1.99997pt\boldsymbol{:}\prod_{s\in\gamma_{p}}a_{s}\boldsymbol{:}\hskip
1.99997pt,$

where the product of short edges is taken in clockwise order starting at any
vertex in $\gamma_{p}$. We now describe the weights and commutation relations
between the edge variables $Z_{s}$. In order to do this, we will make use of
the two edge-ends $s_{1},s_{2}$ associated to each edge in a notched
triangulation. Each edge-end $s_{i}$ has a a boundary vertex $a=\partial
s_{i}$. Note that the weights of the generators $Z_{s}$ and $\alpha_{p}$ with
respect to the $i$-th factor of $\mathcal{T}_{P\cup M}$ are as follows:

$\operatorname{wt}_{i}(Z_{s})=\\#\left\\{\text{edge--ends of $s$ lying in the
region
$i$}\right\\}\qquad\text{and}\qquad\operatorname{wt}_{i}(\alpha_{p})=0.$ (4.4)

######  Definition 4.15.

Let $s$ be an edge-end in a notched triangulation
${\underline{\Delta}}_{\bullet}$, and $p=\partial s$ be the vertex at which
$s$ terminates. We write $|s|$ for the number of $s$ in the total ordering on
the set of edge-ends terminating at the vertex $p$ defined by the
distinguished gate.

For a pair of edge-ends $s_{1}$ and $s_{2}$ we now define

$\omega(s_{1},s_{2})=\begin{cases}-1&\text{if $p:=\partial s_{1}=\partial
s_{2}$ and $|s_{1}|<|s_{2}|$},\\\ 1&\text{if $p:=\partial s_{1}=\partial
s_{2}$ and $|s_{2}|<|s_{1}|$},\\\ 0&\text{otherwise}.\end{cases}$

Now let $s,t\in{\underline{\Delta}}_{\bullet}$ be a pair of edges with edge-
ends $s_{1},s_{2}$ and $t_{1},t_{2}$ respectively. We define $\omega(s,t)$ by
the following formula:

$\omega(s,t)=\sum_{i,j=1}^{2}\omega(s_{i},t_{j}).$

Then from the presentation of $\widetilde{\zeta}({\underline{\Delta}})$
described in 4.11 we deduce:

######  Theorem 4.16.

For any notching ${\underline{\Delta}}_{\bullet}$ of a triangulation
${\underline{\Delta}}$, we have an equivalence of categories

$\mathcal{Z}({\underline{\Delta}})\simeq\zeta({\underline{\Delta}}_{\bullet})\mathrm{-mod}_{\mathcal{T}_{P\cup
M}}.$

The algebra $\zeta({\underline{\Delta}}_{\bullet})$ is a quantum torus with
generators $Z_{s}^{\pm 1}$ labelled by the edges $s$ of the notched ideal
triangulation ${\underline{\Delta}}_{\bullet}$, and by an invertible generator
$\alpha_{p}$ for each puncture $p$ of $\mathbb{S}$. The relations between
these generators are as follows:

  * •

for a pair of edges $s,t\in{\underline{\Delta}}_{\bullet}$ we have

$Z_{s}Z_{t}=q^{\frac{1}{2}\omega(s,t)}Z_{t}Z_{s};$

  * •

for any puncture $p$ and an edge $s$ with vertices $\partial s_{1},\partial
s_{2}$ we have

$\alpha_{p}Z_{s}=q^{\delta_{p,\partial s_{1}}+\delta_{p,\partial
s_{2}}}Z_{s}\alpha_{p}.$

Recall now the subalgebra $\chi\left({\underline{\Delta}}\right)$ of
$\zeta({\underline{\Delta}}_{\bullet})$ obtained by closing the distinguished
gate at each puncture:

$\chi\left({\underline{\Delta}}\right)=\zeta({\underline{\Delta}}_{\bullet})^{\mathcal{T}_{P}}.$

######  Corollary 4.17.

Suppose that either $G=\operatorname{SL}_{2}$ and we have at least one marked
point, or that $G=\operatorname{PGL}_{2}$. Then the functor of taking
$\mathcal{T}_{P}$ invariants defines an equivalence,

$\mathcal{Z}({\underline{\Delta}})\simeq\chi({\underline{\Delta}})\mathrm{-mod}_{\mathcal{T}_{M}}.$

###### Proof..

First we treat the case $G=\operatorname{SL}_{2}$. Since the functor of taking
$\mathcal{T}_{P}$-invariants is exact, we need only show that it is
conservative, i.e. that no object is sent to zero, equivalently that for any
$\Lambda(\mathcal{T}_{P\cup M})$-graded
$\zeta({\underline{\Delta}}_{\bullet})$-module, the degree-zero subspace with
respect to the $\Lambda(\mathcal{T}_{P})$-grading is non-zero. For this,
simply note that the invertible elements $D_{ij}$ emanating from any puncture
have weight $1$ at the puncture. Hence, taking ratios $D_{ij}D_{jk}^{-1}$ for
any triangle with vertices $ijk$, where $i$ is the puncture, we can shift the
degree to vertex $k$. In this way, we can move any non-vector to another non-
zero vector with degree supported exclusively over the non-empty set of marked
points.

In the case of $\operatorname{PGL}_{2}$, we do not need to assume existence of
a marked point. In this case, we have generators $D_{ij}^{2}$ and
$D_{ij}D_{jk}D_{kl}$, and by definition we consider only modules of even total
degree at each gate. For any triangle with vertices $ijk$, the invertible
element $(D_{ij}D_{jk}D_{ki})^{2}/D_{jk}^{2}$ has degree $2$ at vertex $i$ and
degree zero at all other vertices, so it can be used directly to reduce the
degree at any vertex $i$, in particular at the puncture. ∎

######  Corollary 4.18.

Let $G=\operatorname{PGL}_{2}$, and let
$\eta({\underline{\Delta}})=\chi({\underline{\Delta}})^{\mathcal{T}_{M}}$.
Then we have an equivalence of categories,

$\mathcal{Z}_{\operatorname{PGL}_{2}}({\underline{\Delta}})\simeq\eta({\underline{\Delta}})\mathrm{-mod}.$

######  Remark 4.19.

For $\operatorname{SL}_{2}$, the functor of taking $T$-invariants at the final
gate is not an equivalence for obvious parity reasons (owing to existence of a
center of $\operatorname{SL}_{2}$). However, one can show that instead of
$\mathrm{Dist}_{\underline{\Delta}}$ being a generator of each chart, we have
instead that the sum
$\mathrm{Dist}_{\underline{\Delta}}\oplus\mathrm{Dist}_{\underline{\Delta}}[1]$
is a generator, where the latter denotes a parity shift applied to
$\mathrm{Dist}_{\underline{\Delta}}$ is a generator, so that one has an
equivalence:

$\mathcal{Z}_{\operatorname{PGL}_{2}}({\underline{\Delta}})\simeq\left(\eta_{\operatorname{SL}_{2}}({\underline{\Delta}})\oplus\eta_{\operatorname{SL}_{2}}({\underline{\Delta}})\right)\mathrm{-mod}.$

###  4.5. Comparison of $\operatorname{PGL}_{2}$ and $\operatorname{SL}_{2}$
charts

Let us consider the commuting diagram of inclusions of quantum tori, where
below each one we recall which $T$-gates remain open when defining that
algebra. Hence $M$ bijects with the marked points, $P$ bijects with punctures,
and $\widetilde{P}$ bijects with the set of incidences $(v,T)$, where $v$ is a
vertex of ${\underline{\Delta}}$, $T$ is a triangle of ${\underline{\Delta}}$,
and $v$ is a vertex of $T$.

${\eta_{\operatorname{SL}_{2}}({\underline{\Delta}})}$${\chi_{\operatorname{SL}_{2}}({\underline{\Delta}})}$${\zeta_{\operatorname{SL}_{2}}({\underline{\Delta}}_{\bullet})}$${\widetilde{\zeta}_{\operatorname{SL}_{2}}({\underline{\Delta}})}$${\eta_{\operatorname{PGL}_{2}}({\underline{\Delta}})}$${\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}})}$${\zeta_{\operatorname{PGL}_{2}}({\underline{\Delta}}_{\bullet})}$${\widetilde{\zeta}_{\operatorname{PGL}_{2}}({\underline{\Delta}})}$${\emptyset}$${M}$${M\cup
P}$${M\cup\widetilde{P}}$$\hookleftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$$\hookleftarrow$$\rightarrow$

Let $H_{1}(S,X)$ denote the homology of $S$ relative to $X$, with
$\mathbb{Z}/2\mathbb{Z}$ coefficients.

######  Proposition 4.20.

We have:

  1. (1)

The algebra $\eta_{\operatorname{SL}_{2}}({\underline{\Delta}})$ is a free
module over $\eta_{\operatorname{PGL}_{2}}({\underline{\Delta}})$ of rank
$|H_{1}(S)|$.

  2. (2)

The algebra $\chi_{\operatorname{SL}_{2}}({\underline{\Delta}})$ is a free
module over $\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}})$ of rank
$|H_{1}(S,M)|$.

  3. (3)

The algebra $\zeta_{\operatorname{SL}_{2}}({\underline{\Delta}}_{\bullet})$ is
a free module over $\eta_{\operatorname{PGL}_{2}}({\underline{\Delta}})$ of
rank $|H_{1}(S,M\cup P)|$.

  4. (4)

The algebra
$\widetilde{\zeta}_{\operatorname{SL}_{2}}({\underline{\Delta}}_{\bullet})$ is
a free module over
$\widetilde{\zeta}_{\operatorname{PGL}_{2}}({\underline{\Delta}})$ of rank
$|H_{1}(S,M\cup\widetilde{P})|$.

Moreover in each case, a free basis can be taken consisting of
$M_{1}^{\varepsilon_{1}}\cdots M_{N}^{\varepsilon_{N}}$, for some monomials
$M_{1},\ldots,M_{N}$, with $\varepsilon_{i}\in\\{0,1\\}$, and such that
$M_{i}^{2}$ lies in the $\operatorname{SL}_{2}$ quantum torus.

###### Proof..

We prove (4) first; each of the other statements will follow by computing the
appropriate subalgebras of invariants. First, an elementary computation with
the long exact sequence in relative homology yields
$|H_{1}(S,M\cup\widetilde{P})|=2^{N}$, where $N=E-F$ denotes the number of
(long and short) edges in the fencing graph, and $F$ denotes the number of
(hexagonal or rectangular) faces. Here, we regard the fencing graph as giving
a simplicial decomposition of the surface. Then the rank asserted in formula
(4) and the required basis in that case follows immediately from 4.12: a basis
may be given by monomials of degree either one or zero with respect to each
(long and short) edge, modulo an equivalence relation identifying any edge
with the product of all other edges on the same face.

Each time we close a gate by taking $T$-invariants at some vertex, the parity
requirement there becomes vacuous, and elements of our basis become redundant.
It is then elementary to verify each remaining formula (3), (2), and (1). ∎

###  4.6. Comparison of $\mathcal{X}^{q}_{{\underline{\Delta}}}$ and
$\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}})$

Let $s$ be an edge in the triangulation ${\underline{\Delta}}$, and $s_{1}$,
$s_{2}$ be its edge ends incident to vertices $v_{1}$ and $v_{2}$. Denote by
$s_{i}^{+}$ and $s_{i}^{-}$ the edge-ends incident to $v_{i}$ which
immediately follow and precede $s_{i}$ in the clockwise order. We then define
an element
$\widetilde{X}_{s}\in\chi_{\operatorname{SL}_{2}}({\underline{\Delta}})$ as
follows:

  * •

if $s$ is not a boundary edge and is not the internal edge of a self-folded
triangle,

$\widetilde{X}_{s}=\hskip
1.99997pt\boldsymbol{:}Z_{s_{1}^{-}}Z_{s_{1}^{+}}^{-1}Z_{s_{2}^{-}}Z_{s_{2}^{+}}^{-1}\boldsymbol{:}\hskip
1.99997pt$

  * •

if $s$ is a boundary edge, and $v_{2}$ follows $v_{1}$ as we traverse the
corresponding boundary component in a counterclockwise direction,

$\widetilde{X}_{s}=\hskip
1.99997pt\boldsymbol{:}Z_{s}Z_{s_{1}^{-}}Z_{s_{2}^{+}}^{-1}\boldsymbol{:}\hskip
1.99997pt$

  * •

if $s$ is the internal edge of a self-folded triangle with the outer edge
$s^{\prime}$, we set

$\widetilde{X}_{s}=\widetilde{X}_{s^{\prime}}$

Given a puncture $p$ and an edge $s$ in ${\underline{\Delta}}_{\bullet}$, we
say that $s$ is _adjacent to the notch_ at $p$ if an edge-end of $s$ is either
the first or the last one at $v_{i}$. We say that $s$ is _opposing the notch_
at $p$ if there exist edges $s^{\prime}$ and $s^{\prime\prime}$, both adjacent
to notch at $p$, such that $\left\\{s,s^{\prime},s^{\prime\prime}\right\\}$
form a triangle in ${\underline{\Delta}}_{\bullet}$, possibly self-folded or
with some of the vertices coinciding. We then set

$\epsilon_{p}(s)=\begin{cases}1,&\text{if $s$ is adjacent to the notch at
$p$},\\\ -1,&\text{if $s$ is opposing the notch at $p$},\\\
0,&\text{otherwise}.\end{cases}$

If $s$ is not the internal edge of a self-folded triangle, we now define

$\alpha_{s}=\prod_{p\in P}\alpha_{p}^{\epsilon_{p}(s)},$

whereas if $s$ is the internal edge of a self-folded triangle with the
internal puncture $p$ and the outer edge $s^{\prime}$, we set

$\alpha_{s}=\alpha_{s^{\prime}}\alpha_{p}^{2}.$

Now, let ${\underline{\Delta}}$ be the ideal triangulation underlying
${\underline{\Delta}}_{\bullet}$. Denote by
$\mathcal{X}^{q}_{\underline{\Delta}}$ the quantum chart of the cluster
Poisson variety defined by ${\underline{\Delta}}$, see [GS19], and set $X_{s}$
to be the quantum cluster $\mathcal{X}$-variable labelled by the edge $s$. We
have:

######  Theorem 4.21.

We have a well-defined homomorphism,

$\displaystyle\iota_{{\underline{\Delta}}_{\bullet}}\colon\mathcal{X}^{q}_{\underline{\Delta}}$
$\displaystyle\longrightarrow\zeta_{\operatorname{SL}_{2}}({\underline{\Delta}}_{\bullet}),$
$\displaystyle X_{s}$ $\displaystyle\longmapsto\hskip
1.99997pt\boldsymbol{:}\widetilde{X}_{s}\alpha_{s}\boldsymbol{:}\hskip
1.99997pt.$

Moreover, $\iota_{{\underline{\Delta}}_{\bullet}}$ defines an isomorphism
$\mathcal{X}^{q}_{\underline{\Delta}}\cong\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}}).$

###### Proof..

The map $\iota_{{\underline{\Delta}}_{\bullet}}$ is a minor modification of
the (quantum) cluster ensemble map, and can be shown to be well-defined by a
straightforward calculation with the powers in the $q$-commutation relations.
It is also straightforward to see that
$\iota_{{\underline{\Delta}}_{\bullet}}(\mathcal{X}^{q}_{\underline{\Delta}})\subset\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}})$.

Let us now show that $\iota_{{\underline{\Delta}}_{\bullet}}$ is surjective
onto $\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}})$. In what follows we
will abuse notation and denote $\iota_{{\underline{\Delta}}_{\bullet}}(X_{s})$
by $X_{s}$. Given a puncture $p$ and a triangle $\Delta$ in
${\underline{\Delta}}_{\bullet}$, we set

$\upsilon_{p}(\Delta)=\begin{cases}1,&\text{if two sides of $\Delta$ are
adjacent to the notch at $p$},\\\ 0,&\text{otherwise}.\end{cases}$

Then, for a triangle $\Delta$ with the sides $s_{1},s_{2},s_{3}$ (two of which
coincide if $\Delta$ is self-folded) we set

$M_{\Delta}=\hskip
1.99997pt\boldsymbol{:}Z_{s_{1}}Z_{s_{2}}Z_{s_{3}}\prod_{p\in
P}\alpha_{p}^{\upsilon_{p}(\Delta)}\boldsymbol{:}\hskip 1.99997pt.$

From 4.8 and 4.20, we see that
$\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}})$ is generated by
$\alpha_{p}^{\pm 2}$ for all punctures $p$, and the products

$\prod_{s}Z_{s}^{2n_{s}}\prod_{\Delta}M_{\Delta}^{n_{\Delta}},\quad
n_{s},n_{\Delta}\in\mathbb{Z},$

which have zero weight with respect to the torus action at each puncture. At
the same time, for any pair of triangles $\Delta,\Delta^{\prime}$ which share
a common edge $s$, we have

$M_{\Delta^{\prime}}=X_{s}M_{\Delta}\cdot\prod_{s^{\prime}}Z_{s^{\prime}}^{2n_{s^{\prime}}}\prod_{p}\alpha_{p}^{2n_{p}},$

for some $n_{s^{\prime}},n_{p}\in\mathbb{Z}$. Therefore, for any fixed
triangle $\Delta$ in ${\underline{\Delta}}$, the quantum torus
$\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}})$ is generated by $X_{s}$
for all edges $s$, $\alpha_{p}^{\pm 2}$ for all punctures, and the products

$M_{\Delta}^{n_{\Delta}}\prod_{s}Z_{s}^{2n_{s}},\quad
n_{s},n_{\Delta}\in\mathbb{Z},$

invariant under the $T$-action at punctures. Note however, that the
$T$-invariance condition forces $n_{\Delta}$ to be even, and $M_{\Delta}^{2}$
itself is a product of elements of the form $Z_{s}^{2}$ and $\alpha_{p}^{2}$.

Now, let $\gamma=(s_{1},\dots,s_{2n})$ be an _even path in
${\underline{\Delta}}$_, that is an ordered collection of edges, such that
$s_{i}$ and $s_{i+1}$ share a common vertex $v_{i}$ for any
$i\in\mathbb{Z}/2n\mathbb{Z}$. Define a product

$M_{\gamma}=\prod_{i=1}^{2n}{Z_{s_{i}}^{(-1)^{i}}}.$

Then the above discussion yields that
$\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}})$ is generated by $X_{s}$
for all edges $s$, $\alpha_{p}^{\pm 2}$ for all punctures $p$, and monomials
$M_{\gamma}^{\pm 2}$ for all even paths $\gamma$. If $p$ is not the internal
puncture of a self-folded triangle, one can check that

$\alpha_{p}^{2}=\hskip 1.99997pt\boldsymbol{:}\prod_{p\in\partial
s}X_{s}\boldsymbol{:}\hskip 1.99997pt,$

where the product is taken over all edges $s$ incident to $p$. Otherwise, if
$p$ is the internal puncture of the self-folded triangle with the inner edge
$s$ and the outer one $s^{\prime}$, we get

$\alpha_{p}^{2}=X_{s}X_{s^{\prime}}^{-1}.$

Similarly, we have

$M_{\gamma}^{2}=\prod_{i=1}^{2n}\prod_{e_{i}<e<e_{i+1}}X_{e}^{(-1)^{i}},$

where the product is taken over all edges $e$ incident to the vertex $v_{i}$
and sitting between $e_{i}$ and $e_{i+1}$ with respect to the clockwise cyclic
order at $v_{i}$. This completes the proof of the surjectivity of
$\iota_{{\underline{\Delta}}_{\bullet}}$.

Finally, let us prove that $\iota_{{\underline{\Delta}}_{\bullet}}$ is
injective. We note first that as a monomial map between quantum tori, the
kernel is flat in $q$. Hence it suffices to show at $q=1$ that
$\operatorname{Spec}(\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}}))$ is
an algebraic torus of dimension $|E|$, where $|E|$ is the number of edges in
${\underline{\Delta}}$, and is the dimension of
$\operatorname{Spec}(\mathcal{X}^{q}_{\underline{\Delta}})$. Indeed, we know
that
$\operatorname{Spec}(\zeta_{\operatorname{SL}_{2}}({\underline{\Delta}}_{\bullet}))$
has dimension $|P|+|E|$, where $|P|$ is the number of punctures. Since the
action of $T$ on
$\operatorname{Spec}(\zeta_{\operatorname{SL}_{2}}({\underline{\Delta}}_{\bullet}))$
is free at every puncture, the dimension of
$\operatorname{Spec}(\chi_{\operatorname{SL}_{2}}({\underline{\Delta}}))$
equals $|E|$, and so does the dimension of
$\operatorname{Spec}(\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}}))$. ∎

###  4.7. The flip on $\mathcal{Z}(\mathbb{D}_{4})$

The study of flips between triangulations of $\mathbb{S}$ starts with the
quadrilateral $\mathbb{D}_{4}$ and its two triangulations
${\underline{\Delta}}$ and ${\underline{\Delta}}^{\prime}$. An arbitrary flip
can be understood in terms of the one between
${\underline{\Delta}},{\underline{\Delta}}^{\prime}$, using excision to
isolate the pair of adjacent triangles where the flip takes place. In what
follows, we retain our convention from Section 3.8 of enumerating the
$T$–regions $\\{1,2,3,4\\}$ of $\mathbb{D}_{4}$ in counter-clockwise order.

Recall from 3.51 that the category $\mathcal{Z}(\mathbb{D}_{4})$ is equivalent
to the orthogonal complement in
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{4})\mathrm{-mod}_{G\times T^{4}}$
of the localizing subcategory of torsion modules. By Theorem 3.62, we have
full reflective subcategories
$\mathcal{Z}({\underline{\Delta}}),\mathcal{Z}({\underline{\Delta}}^{\prime})$
of $\mathcal{Z}(\mathbb{D}_{4})$, each equivalent to the full subcategory of
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{4})$–modules on which all elements
$\\{\Delta_{e}\\}$ associated to the edges of the corresponding triangulation
act invertibly. By the same argument used to prove Theorem 3.62, there is
another open reflective subcategory

$\mathcal{Z}({\underline{\Delta}},{\underline{\Delta}}^{\prime})\subset\mathcal{Z}(\mathbb{D}_{4})\hookrightarrow\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{4})\mathrm{-mod}_{G\times
T^{4}},$

equivalent to the full subcategory of
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{4})$–modules on which all elements
$D_{ij}$ with $1\leqslant i<j\leqslant 4$ act invertibly.

To describe concretely the category
$\mathcal{Z}({\underline{\Delta}},{\underline{\Delta}}^{\prime})$ we open four
gates in each $T$-region, thereby endowing $\mathbb{D}_{4}$ with a
$T^{16}$-action. We denote by
$\mathrm{Dist}_{{\underline{\Delta}},{\underline{\Delta}}^{\prime}}$ the
restriction of the distinguished object, and we set

$\widetilde{\zeta}({\underline{\Delta}},{\underline{\Delta}}^{\prime})=\underline{\operatorname{End}}_{T^{16}}(\mathrm{Dist}_{{\underline{\Delta}},{\underline{\Delta}}^{\prime}}),$

giving an equivalence,

$\mathcal{Z}({\underline{\Delta}},{\underline{\Delta}}^{\prime})\simeq\widetilde{\zeta}({\underline{\Delta}},{\underline{\Delta}}^{\prime})\mathrm{-mod}_{T^{16}}$

and the following:

######  Proposition 4.22.

The transition functor between charts $\mathcal{Z}({\underline{\Delta}})$ and
$\mathcal{Z}({\underline{\Delta}}^{\prime})$ is given by:

$\displaystyle\mu_{{\underline{\Delta}},{\underline{\Delta}}^{\prime}}\colon\mathcal{Z}({\underline{\Delta}})$
$\displaystyle\longrightarrow\mathcal{Z}({\underline{\Delta}}^{\prime})$
$\displaystyle m$
$\displaystyle\longmapsto\zeta({\underline{\Delta}}_{\bullet},{\underline{\Delta}}_{\bullet}^{\prime})\underset{\zeta({\underline{\Delta}}_{\bullet})}{\otimes}m.$

This transition functor can be described completely explicitly as follows.
Consider the ideal tetrahedron whose four triangular faces are partitioned
into two copies of $\mathbb{D}_{4}$. On the boundary of this tetrahedron we
have a fencing graph
$\Gamma_{{\underline{\Delta}},{\underline{\Delta}}^{\prime}}$ illustrated in
Fig. 10.

$\boldsymbol{a_{1}^{(1)}}$$\boldsymbol{a_{2}^{(1)}}$$\boldsymbol{a_{3}^{(1)}}$$\boldsymbol{a_{1}^{(2)}}$$\boldsymbol{a_{2}^{(2)}}$$\boldsymbol{a_{3}^{(2)}}$$\boldsymbol{a_{1}^{(3)}}$$\boldsymbol{a_{2}^{(3)}}$$\boldsymbol{a_{3}^{(3)}}$$\boldsymbol{a_{1}^{(4)}}$$\boldsymbol{a_{2}^{(4)}}$$\boldsymbol{a_{3}^{(4)}}$$\boldsymbol{1}$$\boldsymbol{2}$$\boldsymbol{3}$$\boldsymbol{4}$$A_{12}$$A_{23}$$A_{34}$$A_{14}$$A^{\prime}_{13}$$A^{\prime\prime}_{13}$$A^{\prime}_{24}$$A^{\prime\prime}_{24}$
Figure 10. The fencing graph
$\Gamma_{{\underline{\Delta}},{\underline{\Delta}}^{\prime}}$ in 4.23

######  Proposition 4.23.

The algebra
$\widetilde{\zeta}({\underline{\Delta}},{\underline{\Delta}}^{\prime})$ has
generators $A_{\ell}^{\pm 1}$ with $\ell\in
E_{l}(\Gamma_{{\underline{\Delta}},{\underline{\Delta}}^{\prime}})$ and
$a_{e}^{\pm 1}$ with $e\in
E_{s}(\Gamma_{{\underline{\Delta}},{\underline{\Delta}}^{\prime}})$, in
correspondence with the edges of the fencing graph
$\Gamma_{{\underline{\Delta}},{\underline{\Delta}}^{\prime}}$ on the surface
of the ideal tetrahedron shown in Fig. 10. These generators satisfy the
fencing graph relations described in 4.11, and in addition to these the
_exchange relation_

$\displaystyle
Z_{1,3}Z_{2,4}=q^{-\frac{1}{2}}Z_{1,2}Z_{3,4}+q^{\frac{1}{2}}Z_{2,3}Z_{1,4},$
(4.5)

where

$\displaystyle Z_{1,2}$ $\displaystyle=\hskip
1.99997pt\boldsymbol{:}a^{(1)}_{1}a^{(1)}_{2}a^{(1)}_{3}A_{1,2}\boldsymbol{:}\hskip
1.99997pt,$ $\displaystyle Z_{3,4}$ $\displaystyle=\hskip
1.99997pt\boldsymbol{:}a^{(3)}_{1}a^{(3)}_{2}a^{(3)}_{3}A_{3,4}\boldsymbol{:}\hskip
1.99997pt,$ $\displaystyle Z_{1,3}$ $\displaystyle=\hskip
1.99997pt\boldsymbol{:}a^{(1)}_{1}A_{1,3}^{\prime}a_{1}^{(3)}a_{2}^{(3)}\boldsymbol{:}\hskip
1.99997pt,$ $\displaystyle Z_{2,3}$ $\displaystyle=\hskip
1.99997pt\boldsymbol{:}a^{(2)}_{1}a^{(2)}_{2}a^{(2)}_{3}A_{2,3}\boldsymbol{:}\hskip
1.99997pt,$ $\displaystyle Z_{1,4}$ $\displaystyle=\hskip
1.99997pt\boldsymbol{:}a^{(4)}_{1}a^{(4)}_{2}a^{(4)}_{3}A_{1,4}\boldsymbol{:}\hskip
1.99997pt,$ $\displaystyle Z_{2,4}$ $\displaystyle=\hskip
1.99997pt\boldsymbol{:}a^{(2)}_{1}a^{(2)}_{2}A_{2,4}^{\prime}a_{1}^{(4)}\boldsymbol{:}\hskip
1.99997pt.$

In particular, we have injective algebra homomorphisms

$\widetilde{\zeta}({\underline{\Delta}})\to\widetilde{\zeta}({\underline{\Delta}},{\underline{\Delta}}^{\prime}),\qquad\widetilde{\zeta}({\underline{\Delta}}^{\prime})\to\widetilde{\zeta}({\underline{\Delta}},{\underline{\Delta}}^{\prime}),$

corresponding at level of generators to the inclusions of fencing graphs
$\Gamma_{{\underline{\Delta}}}\subset\Gamma_{{\underline{\Delta}},{\underline{\Delta}}^{\prime}}\supset\Gamma_{{\underline{\Delta}}^{\prime}}$.

###### Proof..

The generators of
$\widetilde{\zeta}({\underline{\Delta}},{\underline{\Delta}}^{\prime})$ are
defined in complete analogy with Propositions 4.6 and 4.7. For instance, if we
enumerate the tensor factors in the $i$–th $T$-region of $\mathbb{D}_{4}$ in
clockwise order, then for boundary long edges $\overline{ij}$ we have

$A_{i,j}=D_{i,j}\otimes\chi^{(i)}_{(0,0,0,1)}\boxtimes\chi^{(j)}_{(1,0,0,0)},\qquad\text{while}\qquad
a^{(i)}_{j}=\chi^{(i)}_{\varepsilon_{j}-\varepsilon_{j+1}}.$

The diagonal long edge variables are

$\displaystyle A^{\prime}_{1,3}$
$\displaystyle=D_{1,3}\otimes\chi^{(1)}_{(0,1,0,0)}\boxtimes\chi^{(3)}_{(0,0,1,0)},$
$\displaystyle A^{\prime\prime}_{1,3}$
$\displaystyle=D_{1,3}\otimes\chi^{(1)}_{(0,0,1,0)}\boxtimes\chi^{(3)}_{(0,1,0,0)},$
$\displaystyle A^{\prime}_{2,4}$
$\displaystyle=D_{2,4}\otimes\chi^{(2)}_{(0,0,1,0)}\boxtimes\chi^{(4)}_{(0,1,0,0)},$
$\displaystyle A^{\prime\prime}_{2,4}$
$\displaystyle=D_{2,4}\otimes\chi^{(2)}_{(0,1,0,0)}\boxtimes\chi^{(4)}_{(0,0,1,0)}.$

The fencing graph relations easily follow from this description, the
$q$-commutation relations among the $D_{i,j}$ in Corollary (3.56), and the
multiplication rule (3.38). The exchange relation follows by observing that

$Z_{i,j}=q^{-\frac{3}{4}}D_{i,j}\otimes\chi^{(i)}_{(1,0,0,0)}\boxtimes\chi^{(j)}_{(1,0,0,0)},$

and then applying the three-term relation in
$\mathcal{F}_{q}(\operatorname{Conf}^{fr}_{4})$

$D_{1,3}D_{2,4}=q^{-\frac{1}{2}}D_{1,2}D_{3,4}+q^{\frac{1}{2}}D_{2,3}D_{1,4}$

established in 3.57. ∎

######  Remark 4.24.

The exchange relation (4.5) can be rewritten in terms of Weyl ordered quantum
torus elements as

$\displaystyle Z_{1,3}$ $\displaystyle=~{}\hskip
1.99997pt\boldsymbol{:}Z_{2,4}^{-1}Z_{1,2}Z_{3,4}\boldsymbol{:}\hskip
1.99997pt+\hskip
1.99997pt\boldsymbol{:}Z_{2,4}^{-1}Z_{2,3}Z_{1,4}\boldsymbol{:}\hskip
1.99997pt,$ (4.6)

or alternatively as

$\displaystyle Z_{2,4}$ $\displaystyle=~{}\hskip
1.99997pt\boldsymbol{:}Z_{1,3}^{-1}Z_{1,2}Z_{3,4}\boldsymbol{:}\hskip
1.99997pt+\hskip
1.99997pt\boldsymbol{:}Z_{1,3}^{-1}Z_{2,3}Z_{1,4}\boldsymbol{:}\hskip
1.99997pt.$

###  4.8. Flips on $\mathcal{Z}(\mathbb{S})$

Now let us consider a general decorated surface $\mathbb{S}$, and two notched
triangulations ${\underline{\Delta}}_{\bullet}$ and
${\underline{\Delta}}_{\bullet}^{\prime}$ differing by the flip of a single
edge. Let us denote by $\widetilde{\mathbb{S}}$ the decorated surface obtained
as union of all but the two triangles sharing the edge, and by
$\widetilde{{\underline{\Delta}}}$ the resulting triangulation of
$\widetilde{\mathbb{S}}$. Then we have a decomposition

$\mathbb{S}=\widetilde{\mathbb{S}}\sqcup_{\mathbb{D}_{2}^{\sqcup
4}}\mathbb{D}_{4}.$

By excision, the flip functor
$\mu_{{\underline{\Delta}},{\underline{\Delta}}}$ thus admits a natural
description as

$\mu_{{\underline{\Delta}},{\underline{\Delta}}^{\prime}}=\operatorname{id}_{\widetilde{\mathbb{S}}}\boxtimes~{}\mu^{\mathbb{D}_{4}},$

where $\mu^{\mathbb{D}_{4}}$ is the functor between open subcategories of
$\mathcal{Z}(\mathbb{D}_{4})$ from 4.22 induced by changing its triangulation.
The only subtlety arises when we wish to describe this functor at the level of
generators and relations for the algebras
$\zeta({\underline{\Delta}}_{\bullet}),\zeta({\underline{\Delta}}_{\bullet}^{\prime})$.
Recall that if $\mathbb{S}$ has punctures, it is the the data of the notching
that singles out a distinguished set of generators for the quantum torus
$\zeta({\underline{\Delta}}_{\bullet})$. Keeping track of the effect of flips
on the notch necessitates a small amount of additional bookkeeping, which we
shall now describe.

Let us again enumerate the (possibly coincident) $T$-regions of the 4-gon
$\\{1,2,3,4\\}$ in counterclockwise order. Then we have edges
$\left\\{e_{1,2},e_{2,3},e_{3,4},e_{4,1}\right\\}$ common to both
triangulations
${\underline{\Delta}}_{\bullet},{\underline{\Delta}}_{\bullet}^{\prime}$,
where $\partial e_{i,j}=\left\\{i,j\right\\}$ and we allow for the possibility
that some of the $e_{i,j}$ are in fact identified. Using the notchings, we
assign to each edge–end $e_{i}$ of a diagonal of $\mathbb{D}_{4}$ an integer
$\gamma_{i}$ defined by

$\displaystyle\gamma_{i}=\begin{cases}1&\text{if}\quad|e_{i}|\text{ is
minimal,}\\\ -1&\text{if}\quad|e_{i}|\text{ is maximal},\\\
0&\text{otherwise}.\end{cases}$ (4.7)

More informally, we declare $\gamma_{i}=1$ if $e_{i}$ is the first edge-end
with respect to the total order from the notched triangulation in which
$e_{i}$ appears, while $\gamma_{i}=-1$ if $e_{i}$ is the last edge-end with
respect to the notching; in all other cases we set $\gamma_{i}=0$. Then we
have the following exchange relation between the distinguished generating sets
of the quantum tori
$\zeta({\underline{\Delta}}_{\bullet}),\zeta({\underline{\Delta}}_{\bullet}^{\prime})$
corresponding to the notched triangulations
${\underline{\Delta}}_{\bullet},{\underline{\Delta}}_{\bullet}^{\prime}$:

######  Proposition 4.25.

Suppose that
${\underline{\Delta}}_{\bullet},{\underline{\Delta}}_{\bullet}^{\prime}$ are
two notched triangulations whose underlying triangulations differ by flipping
a single edge. Let
${Z}_{i,j}\in\zeta({\underline{\Delta}}_{\bullet},{\underline{\Delta}}_{\bullet}^{\prime})$
denote the element corresponding to the diagonal $e_{i,j}$ with edge-ends
$e_{i}$ and $e_{j}$, and let $(\gamma_{i})_{i=1}^{4}$ be the integers defined
in (4.7). Let us set

$\displaystyle\alpha^{12}_{34}$
$\displaystyle=\alpha_{1}^{\delta_{\gamma_{1},-1}}\alpha_{2}^{-\delta_{\gamma_{2},1}}\alpha_{3}^{\delta_{\gamma_{3},-1}}\alpha_{4}^{-\delta_{\gamma_{4},1}},$
$\displaystyle\alpha^{14}_{23}$
$\displaystyle=\alpha_{1}^{-\delta_{\gamma_{1},1}}\alpha_{2}^{\delta_{\gamma_{2},-1}}\alpha_{3}^{-\delta_{\gamma_{3},1}}\alpha_{4}^{\delta_{\gamma_{4},-1}}.$

Then we have the following exchange relations in
$\zeta({\underline{\Delta}}_{\bullet},{\underline{\Delta}}_{\bullet}^{\prime})$:

$\displaystyle Z_{1,3}$ $\displaystyle=\hskip
1.99997pt\boldsymbol{:}\alpha^{12}_{34}Z_{2,4}^{-1}Z_{1,2}Z_{3,4}\boldsymbol{:}\hskip
1.99997pt+\hskip
1.99997pt\boldsymbol{:}\alpha^{14}_{23}Z_{2,4}^{-1}Z_{2,3}Z_{1,4}\boldsymbol{:}\hskip
1.99997pt,$ (4.8) $\displaystyle Z_{2,4}$ $\displaystyle=\hskip
1.99997pt\boldsymbol{:}\alpha^{12}_{34}Z_{1,3}^{-1}Z_{1,2}Z_{3,4}\boldsymbol{:}\hskip
1.99997pt+\hskip
1.99997pt\boldsymbol{:}\alpha^{14}_{23}Z_{1,3}^{-1}Z_{2,3}Z_{1,4}\boldsymbol{:}\hskip
1.99997pt,$ (4.9)

where the Weyl ordering is taken with respect to the set of generators
$\\{{Z}_{e},\alpha_{i}\\}$.

###### Proof..

We must translate the identity (4.6), which holds in the larger algebra

$\widetilde{\zeta}({\underline{\Delta}},{\underline{\Delta}}^{\prime})\simeq\widetilde{\zeta}({{\underline{\Delta}}})\otimes_{\zeta(\mathbb{D}_{2})^{\otimes
4}}\widetilde{\zeta}_{\mathbb{D}_{4}}({\underline{\Delta}},{\underline{\Delta}}^{\prime}),$

into an identity between elements of the subalgebra
${\zeta}({\underline{\Delta}}_{\bullet},{\underline{\Delta}}_{\bullet}^{\prime})$
of $\mathbf{T}^{\prime}_{\underline{\Delta}}$–invariants in
$\widetilde{\zeta}({\underline{\Delta}},{\underline{\Delta}}^{\prime})$. To do
this, for each $T$–region $i$ of $\mathbb{D}_{4}$, let $w_{i}$ be the first
vertex of the fencing graph for $\mathbb{D}_{4}$ with respect to the clockwise
order within $i$. Moreover, let $v_{i}$ be the distinguished vertex in the
$T$-region of $\mathbb{S}$ that contains the $T$–region $i$ of the subsurface
$\mathbb{D}_{4}$. Now write $\pi(i)$ for the set of short edges that make up
the minimal path in the fencing graph
$\Gamma_{{\underline{\Delta}},{\underline{\Delta}}^{\prime}}$ starting at
$v_{i}$ and travelling clockwise to vertex $w_{i}$. Then multiplying both
sides of the relation (4.6) by the product of short edges

$\hskip
1.99997pt\boldsymbol{:}\prod_{e\in\pi(1)}\prod_{f\in\pi(3)}a_{e}a_{f}\boldsymbol{:}\hskip
1.99997pt,$

we obtain the relation (4.8). ∎

The following is an easy direct computation.

######  Corollary 4.26.

The isomorphisms $\iota_{{\underline{\Delta}}_{\bullet}}$ between
$\mathcal{X}^{q}_{\underline{\Delta}}$ and
$\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}})$ intertwine the cluster
mutation and the restricted flips functors
$\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}},{\underline{\Delta}}^{\prime})$
obtained from
$\zeta_{\operatorname{SL}_{2}}({\underline{\Delta}}_{\bullet},{\underline{\Delta}}_{\bullet}^{\prime})$.

###  4.9. Topological remarks

In this final section we collect a number of remarks concerning the
topological invariance built into our construction.

Let $\operatorname{Aut}(\mathbb{S})$ denote the 2-group of automorphisms of
$\mathbb{S}$ in $\mathbb{S}\mathrm{urf}$, i.e. the category with a single
object, whose 1-morphisms are self-diffeomorphisms of $\mathbb{S}$ which
respect the stratification and the labeling by $G$ and $T$, and whose
2-morphisms are stratified isotopies of such (equivalently
$\operatorname{Aut}(\mathbb{S})$ may be regarded as a monoidal category in
which all objects and morphisms are invertible). We note that the
$1$-truncation $\pi_{\leq 1}\operatorname{Aut}(\mathbb{S})$ is the precisely
the marked mapping class group, since by definition this means that we take
isotopy equivalence classes. Given a set $\mathcal{G}$ of $G$\- and $T$-gates,
let us also denote by $\operatorname{Aut}_{\mathcal{G}}(\mathbb{S})$ the sub
$2$-group of $\operatorname{Aut}(\mathbb{S})$ of diffeomorphisms and isotopies
which fix $\mathcal{G}$ componentwise.

Simply because $\mathcal{Z}$ is a functor, the category
$\mathcal{Z}(\mathbb{S})$ inherits a canonical action of the 2-group
$\operatorname{Aut}(\mathbb{S})$. This data can be understood as an action of
the mapping class group, in which functors compose by the group law only up to
a coherent isomorphism, encoded by a 2-cocycle with values in the center of
the category. We note that the distinguished object, being defined by the
empty embedding is preserved canonically (i.e. up to a canonical isomorphism)
by any diffeomorphism, so that it obtains a canonical structure of a
$\operatorname{Aut}(\mathbb{S})$-fixed point of the category. In particular
the mapping class group acts strictly on the vector space
$\operatorname{End}(\mathrm{Dist}_{\mathbb{S}})$, and in such a way that it
intertwines the cluster charts coming from different triangulations.

Suppose we are given two triangulations ${\underline{\Delta}}$ and
${\underline{\Delta}}^{\prime}$ related by an isotopy $\gamma$. We may assume
$\gamma$ at time $t=0$ to be the identity automorphism of $\mathbb{S}$, so
that as $t$ runs to $t=1$, it induces isomorphisms between objects of
$\mathcal{Z}({\underline{\Delta}})$ and
$\mathcal{Z}({\underline{\Delta}}^{\prime})$. Since both categories are closed
under isomorphisms, it means two subcategories related by an isotopy are
simply equal.

An interesting extra feature comes from the
$\mathrm{Rep}_{q}(\mathcal{G})$-action at the gates. In the presence of gates
$\mathcal{G}$, any functor arising coming from
$\operatorname{Aut}_{\mathcal{G}}(\mathbb{S})$ is canonically promoted to a
$\mathcal{G}$-module functor, and any isotopy to a $\mathcal{G}$-module
natural isomorphism. This additional naturality has a consequence for the
charts $\zeta({\underline{\Delta}}_{\bullet})$ and their mutations. The
algebras $\zeta({\underline{\Delta}}_{\bullet})$ are defined with respect to a
notched triangulation, yet their presentation, as encoded by the fencing
graph, depended only on the associated notched ideal triangulation. We note
that two isotopy classes of notched triangulations give rise to the same
notched ideal triangulation if and only if they differ by iterated application
of the Dehn twist implementing $2\pi$ rotation at some puncture. This Dehn
twist preserves the notching, hence it gives a $\mathrm{Rep}_{q}(T)$-module
auto-equivalence of $\mathcal{Z}(\mathbb{S})$. However, the natural isotopy
undoing the Dehn twist _does not_ preserve the notch.

This means that the Dehn twist _is isomorphic_ to the identity functor as a
plain functor of categories, but is _not isomorphic_ as a
$\mathrm{Rep}_{q}(T)$-module category. Hence the Dehn twist does not act
trivially on internal endomorphism algebras such as
$\zeta({\underline{\Delta}}_{\bullet})$. Indeed, a direct computation implies
that the Dehn twist induces a homomorphism
$\widetilde{\zeta}({\underline{\Delta}}_{\bullet})\to\widetilde{\zeta}({\underline{\Delta}}_{\bullet}^{\prime})$
given by $Z_{e}\mapsto qZ_{e}\alpha_{p}$, for all edges $e$ incident to the
puncture.

And additional interesting subtlety comes when we consider the appearance of
topological framings of $\mathbb{S}$. In Section 4.3, we have implicitly fixed
a 2-framing of the surface $\mathbb{S}$ when computing the charts
$\widetilde{\zeta}$. The data of the 2-framing can be prescribed by a
_jellyfish_ diagram as in Fig. 9: the triangles are drawn in some order in a
line, and the attaching digons are stretched out to long legs connecting the
triangles. Such a prescription topologically fixes a 2-framing on the surface
– the blackboard framing – and also a total ordering on edge-ends of the
triangulation. This data entered into our computation of the generators and
relations for $\zeta({\underline{\Delta}}_{\bullet})$. However, the reader
will note that the algebra $\widetilde{\zeta}({\underline{\Delta}})$ has been
defined in such a way that it _does not depend_ on this data of framing.

Hence, while neither the _algebras_ $\widetilde{\zeta}({\underline{\Delta}})$,
$\zeta({\underline{\Delta}}_{\bullet})$, nor the _flips_
$\widetilde{\zeta}({\underline{\Delta}},{\underline{\Delta}}^{\prime})$,
$\zeta({\underline{\Delta}}_{\bullet},{\underline{\Delta}}_{\bullet}^{\prime})$
depend on the choice of 2-framing, their _presentation_ by generators and
relations (that is, their distinguished choice of generators), do indeed
depend on such a choice (in fact, less data, the choice of a 3-framing at the
quantum level, a spin structure classically). Algebraically, this is because
we have chosen the elements $D_{ij}$ as generators of a 1-dimensional space of
invariants in $\mathcal{F}_{q}(N\backslash
G)\otimes\mathcal{F}_{q}(N\backslash G),$ but this space doesn’t naturally
come with a distinguished vector.

We wish to stress that in traditional approach to quantization via quantum
cluster algebras, one _must fix_ these choices a priori in order to _define_
the quantum cluster charts and their flips in the first place. By contrast, we
_may make_ such choices in order to _compute_ our quantum cluster charts and
their flips a posteriori.

This distinction also resolves a potential point of confusion about
$\mathcal{A}$-varieties in our set-up. Traditionally, these are defined as
moduli spaces of _twisted_ local systems on the surface with $B$-reductions.
Any choice of spin structure on the surface (in particular any 3-framing, and
in particular any 2-framing) gives an isomorphism between the moduli spaces of
twisted and ordinary local systems. If defining charts and flips by formulas,
one must work with twisted local systems and non-canonically identify them
with ordinary local systems by a choice of spin structure. In our approach,
this is reversed, and ordinary local systems are what naturally appear.

##  5\. Examples

In this section we consider the examples when $S$ is a cylinder with one
marked point on each boundary circle, a punctured torus, and a three punctured
sphere. On Fig. 11, we show a pair
${\underline{\Delta}},{\underline{\Delta}}^{\prime}$ of superimposed
triangulations of each surface. The triangulation ${\underline{\Delta}}$
consists of edges labelled $\left\\{1,2,3\right\\}$ or
$\left\\{1,2,3,4\right\\}$, and ${\underline{\Delta}}^{\prime}$ is obtained
from ${\underline{\Delta}}$ by flipping the edge 3 to the one labelled
$3^{\prime}$. We denote the tori
$\zeta_{\operatorname{SL}_{2}}({\underline{\Delta}})$ and
$\zeta_{\operatorname{SL}_{2}}({\underline{\Delta}}^{\prime})$ by $\zeta$ and
$\zeta^{\prime}$ respectively. Similarly, we let $\chi$ and $\chi^{\prime}$
denote the tori $\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}})$ and
$\chi_{\operatorname{PGL}_{2}}({\underline{\Delta}}^{\prime})$. We let
$Z_{j},\alpha_{p}$ be the generators of $\zeta$, while setting
$Z^{\prime}_{3}:=Z_{3^{\prime}}\in\zeta^{\prime}$. The generators of $\chi$
and $\chi^{\prime}$ are denoted respectively by $\left\\{X_{i}\right\\}$ and
$\left\\{X^{\prime}_{i}\right\\}$. Finally, we let
$\mu:=\mu_{{\underline{\Delta}},{\underline{\Delta}}^{\prime}}$ be the flip
between the triangulaitons ${\underline{\Delta}}$ and
${\underline{\Delta}}^{\prime}$.

$2$$1$$2$$4$$3^{\prime}$$3$$2$$1$$2$$1$$3$$3^{\prime}$$\alpha$$1$$1$$2$$2$$3^{\prime}$$3$$\alpha_{1}$$\alpha_{2}$$\alpha_{3}$
Figure 11. At left: a cylinder with a marked point on each boundary circle. In
the middle: a punctured torus. At right: a 3-punctured sphere.

###  5.1. The cylinder

We have

$\displaystyle\zeta_{1}$
$\displaystyle=\mathbb{C}[q^{\pm\frac{1}{2}}]\left\langle Z_{1}^{\pm
1},Z_{2}^{\pm 1},Z_{3}^{\pm 1},Z_{4}^{\pm 1}\right\rangle,$
$\displaystyle\zeta_{2}$
$\displaystyle=\mathbb{C}[q^{\pm\frac{1}{2}}]\big{\langle}Z_{1}^{\pm
1},Z_{2}^{\pm 1},{Z^{\prime}_{3}}^{\pm 1},Z_{4}^{\pm 1}\big{\rangle},$

where the elements $Z_{1}$ and $Z_{4}$ are central in both tori, the other
relations read

$Z_{2}Z_{3}=qZ_{3}Z_{2}\qquad\text{and}\qquad
Z^{\prime}_{3}Z_{2}=qZ_{2}Z^{\prime}_{3},$

and the flip $\mu$ takes form

$Z^{\prime}_{3}Z_{3}=Z_{1}Z_{4}+qZ_{2}^{2}.$

The $X$-variables of the torus $\chi_{1}$ then read

$X_{1}=\hskip 1.99997pt\boldsymbol{:}Z_{1}Z_{2}Z_{3}^{-1}\boldsymbol{:}\hskip
1.99997pt,\qquad X_{2}=\hskip
1.99997pt\boldsymbol{:}Z_{3}^{2}Z_{1}^{-1}Z_{4}^{-1}\boldsymbol{:}\hskip
1.99997pt,\qquad X_{3}=\hskip
1.99997pt\boldsymbol{:}Z_{1}Z_{4}Z_{2}^{-2}\boldsymbol{:}\hskip
1.99997pt,\qquad X_{4}=\hskip
1.99997pt\boldsymbol{:}Z_{4}Z_{2}Z_{3}^{-1}\boldsymbol{:}\hskip 1.99997pt,$

and satisfy the relations

$\displaystyle X_{1}X_{2}$ $\displaystyle=q^{2}X_{2}X_{1},$ $\displaystyle
X_{1}X_{3}$ $\displaystyle=q^{-2}X_{3}X_{1},$ $\displaystyle X_{1}X_{4}$
$\displaystyle=X_{4}X_{1},$ $\displaystyle X_{4}X_{2}$
$\displaystyle=q^{2}X_{2}X_{4},$ $\displaystyle X_{4}X_{3}$
$\displaystyle=q^{-2}X_{3}X_{4},$ $\displaystyle X_{2}X_{3}$
$\displaystyle=q^{4}X_{3}X_{2}.$

Similarly, the $X$-variables of the torus $\chi_{2}$ are given by

$X^{\prime}_{1}=\hskip
1.99997pt\boldsymbol{:}Z_{1}Z^{\prime}_{3}Z_{2}^{-1}\boldsymbol{:}\hskip
1.99997pt,\qquad X^{\prime}_{2}=\hskip
1.99997pt\boldsymbol{:}Z_{1}Z_{4}{Z^{\prime}_{3}}^{-2}\boldsymbol{:}\hskip
1.99997pt,\qquad X^{\prime}_{3}=\hskip
1.99997pt\boldsymbol{:}Z_{2}^{2}Z_{1}^{-1}Z_{4}^{-1}\boldsymbol{:}\hskip
1.99997pt,\qquad X^{\prime}_{4}=\hskip
1.99997pt\boldsymbol{:}Z_{4}Z^{\prime}_{3}Z_{2}^{-1}\boldsymbol{:}\hskip
1.99997pt,$

and satisfy the relations

$\displaystyle X^{\prime}_{1}X^{\prime}_{2}$
$\displaystyle=q^{-2}X^{\prime}_{2}X^{\prime}_{1},$ $\displaystyle
X^{\prime}_{1}X^{\prime}_{3}$
$\displaystyle=q^{2}X^{\prime}_{3}X^{\prime}_{1},$ $\displaystyle
X^{\prime}_{1}X^{\prime}_{4}$ $\displaystyle=X^{\prime}_{4}X^{\prime}_{1},$
$\displaystyle X^{\prime}_{4}X^{\prime}_{2}$
$\displaystyle=q^{-2}X^{\prime}_{2}X^{\prime}_{4},$ $\displaystyle
X^{\prime}_{4}X^{\prime}_{3}$
$\displaystyle=q^{2}X^{\prime}_{3}X^{\prime}_{4},$ $\displaystyle
X^{\prime}_{2}X^{\prime}_{3}$
$\displaystyle=q^{-4}X^{\prime}_{3}X^{\prime}_{2}.$

Finally the flip $\mu$ reads

$X^{\prime}_{3}=X_{3}^{-1},\qquad X^{\prime}_{1}=X_{1}(1+qX_{3}),\qquad
X^{\prime}_{4}=X_{4}(1+qX_{1}),$

and

$X^{\prime}_{2}=X_{2}(1+qX_{3}^{-1})^{-1}(1+q^{3}X_{3}^{-1})^{-1}.$

###  5.2. The punctured torus

In this case, we get

$\displaystyle\zeta_{1}$
$\displaystyle=\mathbb{C}[q^{\pm\frac{1}{2}}]\left\langle Z_{1}^{\pm
1},Z_{2}^{\pm 1},Z_{3}^{\pm 1},\alpha^{\pm 1}\right\rangle,$
$\displaystyle\zeta_{2}$
$\displaystyle=\mathbb{C}[q^{\pm\frac{1}{2}}]\big{\langle}Z_{1}^{\pm
1},Z_{2}^{\pm 1},{Z^{\prime}_{3}}^{\pm 1},\alpha^{\pm 1}\big{\rangle}.$

The generators satisfy relations

$\displaystyle Z_{2}Z_{1}$ $\displaystyle=qZ_{1}Z_{2},$ $\displaystyle
Z_{2}Z_{3}$ $\displaystyle=qZ_{3}Z_{2},$ $\displaystyle Z_{3}Z_{1}$
$\displaystyle=qZ_{1}Z_{3},$ $\displaystyle Z_{2}Z^{\prime}_{3}$
$\displaystyle=qZ^{\prime}_{3}Z_{2},$ $\displaystyle Z_{1}Z^{\prime}_{3}$
$\displaystyle=qZ^{\prime}_{3}Z_{1},$

as well as

$\alpha Z_{x}=q^{2}Z_{x}\alpha$

for any edge $x\in\left\\{1,2,3,3^{\prime}\right\\}$. The flip $\mu$ reads

$Z^{\prime}_{3}Z_{3}=q^{2}Z_{1}^{2}\alpha+qZ_{2}^{2}.$

The $X$-generators of the torus $\chi_{1}$ read

$X_{1}=\hskip
1.99997pt\boldsymbol{:}Z_{2}^{2}Z_{3}^{-2}\alpha\boldsymbol{:}\hskip
1.99997pt,\qquad X_{2}=\hskip
1.99997pt\boldsymbol{:}Z_{3}^{2}Z_{1}^{-2}\alpha\boldsymbol{:}\hskip
1.99997pt,\qquad X_{3}=\hskip
1.99997pt\boldsymbol{:}Z_{1}^{2}Z_{2}^{-2}\alpha^{-1}\boldsymbol{:}\hskip
1.99997pt,$

and satisfy the relations

$X_{i}X_{i+1}=q^{4}X_{i+1}X_{i},$

for $i\in\mathbb{Z}/3\mathbb{Z}$. Similarly, the $X$-generators of the torus
$\chi_{2}$ take form

$X^{\prime}_{1}=\hskip
1.99997pt\boldsymbol{:}{Z^{\prime}_{3}}^{2}Z_{2}^{-2}\alpha\boldsymbol{:}\hskip
1.99997pt,\qquad X^{\prime}_{2}=\hskip
1.99997pt\boldsymbol{:}Z_{1}^{2}{Z^{\prime}_{3}}^{-2}\alpha^{-1}\boldsymbol{:}\hskip
1.99997pt,\qquad X^{\prime}_{3}=\hskip
1.99997pt\boldsymbol{:}Z_{2}^{2}Z_{1}^{-2}\alpha\boldsymbol{:}\hskip
1.99997pt,$

and satisfy the relations

$X^{\prime}_{i}X^{\prime}_{i+1}=q^{-4}X^{\prime}_{i+1}X^{\prime}_{i},$

for $i\in\mathbb{Z}/3\mathbb{Z}$. Finally, the flip of $X$-variables is

$X^{\prime}_{1}=X_{1}(1+qX_{3})(1+q^{3}X_{3}),\qquad
X^{\prime}_{2}=X_{2}(1+qX_{3}^{-1})^{-1}(1+q^{3}X_{3}^{-1})^{-1},\qquad
X^{\prime}_{3}=X_{3}^{-1}.$

###  5.3. The 3-punctured sphere

We have

$\displaystyle\zeta_{1}$
$\displaystyle=\mathbb{C}[q^{\pm\frac{1}{2}}]\left\langle Z_{1}^{\pm
1},Z_{2}^{\pm 1},Z_{3}^{\pm 1},\alpha_{1}^{\pm 1},\alpha_{2}^{\pm
1},\alpha_{3}^{\pm 1}\right\rangle,$ $\displaystyle\zeta_{2}$
$\displaystyle=\mathbb{C}[q^{\pm\frac{1}{2}}]\big{\langle}Z_{1}^{\pm
1},Z_{2}^{\pm 1},{Z^{\prime}_{3}}^{\pm 1},\alpha_{1}^{\pm 1},\alpha_{2}^{\pm
1},\alpha_{3}^{\pm 1}\big{\rangle}.$

The generators satisfy relations

$\displaystyle Z_{2}Z_{1}$ $\displaystyle=q^{\frac{1}{2}}Z_{1}Z_{2},$
$\displaystyle Z_{3}Z_{2}$ $\displaystyle=q^{\frac{1}{2}}Z_{2}Z_{3},$
$\displaystyle Z_{3}Z_{1}$ $\displaystyle=q^{\frac{1}{2}}Z_{1}Z_{3},$
$\displaystyle Z^{\prime}_{3}Z_{2}$ $\displaystyle=Z_{2}Z^{\prime}_{3},$
$\displaystyle Z^{\prime}_{3}Z_{1}$ $\displaystyle=qZ_{1}Z^{\prime}_{3},$

as well as

$[\alpha_{i},\alpha_{j}]=0\qquad\text{and}\qquad\alpha_{i}Z_{j}=q^{1-\delta_{ij}}Z_{j}\alpha_{i},$

for any $1\leqslant i,j\leqslant 3$. The flip $\mu$ reads

$Z^{\prime}_{3}Z_{3}=q^{\frac{1}{2}}\\!\\!\hskip
1.99997pt\boldsymbol{:}\alpha_{1}\alpha_{2}Z_{1}Z_{2}\boldsymbol{:}\hskip
1.99997pt+q^{-\frac{1}{2}}\\!\\!\hskip
1.99997pt\boldsymbol{:}\alpha_{3}Z_{1}Z_{2}\boldsymbol{:}\hskip 1.99997pt.$

The torus $\chi_{1}$ is commutative and is generated by the elements

$X_{j}=\frac{\alpha_{1}\alpha_{2}\alpha_{3}}{\alpha_{j}^{2}},$

for $1\leqslant i,j\leqslant 3$. The torus $\chi_{2}$ is commutative as well,
and is generated by

$X^{\prime}_{1}=X_{1},\qquad X^{\prime}_{2}=X_{2},\qquad
X^{\prime}_{3}=X_{3}^{-1}.$

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