# Positivity for quantum cluster algebras from orbifolds

Min Huang  School of Mathematics (Zhuhai)
Sun Yat-sen University
<EMAIL_ADDRESS>
###### Abstract.

Let $(S,M,U)$ be a marked orbifold with or without punctures and let
$\mathcal{A}_{v}$ be a quantum cluster algebra from $(S,M,U)$ with arbitrary
coefficients and quantization. We provide combinatorial formulas for quantum
Laurent expansion of quantum cluster variables of $\mathcal{A}_{v}$ concerning
an arbitrary quantum seed. Consequently, the positivity for the quantum
cluster algebra $\mathcal{A}_{v}$ is proved.

###### Contents

  1. 1 Introduction
  2. 2 Preliminary and some notation
    1. 2.1 Poset and partition bijection
      1. 2.1.1 Poset
      2. 2.1.2 Partition bijection
    2. 2.2 Cluster algebra
    3. 2.3 Quantum cluster algebra
    4. 2.4 (Quantum) Cluster algebras from orbifolds
    5. 2.5 Orbifold morphism and canonical polygon
    6. 2.6 Snake graph and perfect matching
    7. 2.7 Complete $(T^{o},\gamma)$-path
  3. 3 An isomorphism of quantum cluster algebras
  4. 4 Three lattices associated with tagged arcs
    1. 4.1 $\mathcal{L}(T^{o},\widetilde{\beta})$
    2. 4.2 $\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$
      1. 4.2.1 $\widetilde{\beta}\notin T^{o}$
      2. 4.2.2 $\widetilde{\beta}\in T^{o}$.
    3. 4.3 $\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$
      1. 4.3.1 $\widetilde{\beta}\notin T^{o}$.
      2. 4.3.2 $\widetilde{\beta}\in T^{o}$
  5. 5 Valuation maps
    1. 5.1 Valuation map on $\mathcal{L}(T^{o},\widetilde{\beta})$
    2. 5.2 Valuation map on $\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$
    3. 5.3 Valuation map on $\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$
  6. 6 Expansion formulas and the positivity
    1. 6.1 Expansion formula for ordinary arcs.
    2. 6.2 Expansion formula for one end tagged notched arcs.
    3. 6.3 Expansion formula for two ends tagged notched arcs.
  7. 7 Partition bijection between perfect matchings
    1. 7.1 Polygon case
      1. 7.1.1 Partition bijection $\phi^{T^{o}}_{\alpha}$
      2. 7.1.2 Partition bijection $\phi^{T^{o}}_{\alpha_{1},\cdots,\alpha_{k}}$
      3. 7.1.3 Partition bijection $\varphi^{T^{o}}_{\alpha}$
      4. 7.1.4 Partition bijection $\varphi^{T^{o}}_{\alpha_{1},\cdots,\alpha_{k}}$
    2. 7.2 General case
      1. 7.2.1 Partition bijection $\phi^{T^{o}}_{\alpha}$
      2. 7.2.2 Partition bijection $\varphi^{T^{o}}_{\alpha}$
  8. 8 Partition bijection between triangles
    1. 8.1 Once-puncture polygon case
    2. 8.2 General case
  9. 9 Partition bijection between lattices $\mathcal{L}$ and $\mathcal{L}^{\prime}$
  10. 10 Compatibility of the partition bijections and the lattice structures
  11. 11 Two equivalent relations in $\mathcal{L}$.
  12. 12 Proof for the expansion formulas
    1. 12.1 Proofs of Theorems 6.5, 6.9
    2. 12.2 Proof of Theorem 6.14 in case $p\neq q$
    3. 12.3 Proof of Theorem 6.14 in case $p=q$
    4. 12.4 Proof of Proposition 12.18.

##  1\. Introduction

Cluster algebras are commutative algebras introduced by Fomin-Zelevinsky [18]
around the year 2000. Quantum cluster algebras were later introduced by
Berenstein-Zelevinsky, [3]. The theory of cluster algebras is related to
numerous other fields, including Lie theory, representation theory of
algebras, dynamical systems, Teichmüller theory, combinatorics, number theory,
topology, and mathematical physics.

A cluster algebra is a subalgebra of a rational function field with a
distinguished set of generators, called cluster variables. Different cluster
variables are related by an iterated procedure, called mutation. Cluster
variables are rational functions by construction. In [18], Fomin and
Zelevinsky proved that they are Laurent polynomials of initial cluster
variables, known as the Laurent phenomenon. Laurent polynomials were proven to
have non-negative coefficients, known as positivity, see [29, 21].

Berenstein-Zelevinsky [3] proved that the Laurent phenomenon has a quantum
version in the quantum setting. The cluster variables are quantum Laurent
polynomials of initial ones. The coefficients, were conjectured to be in
$\mathbb{N}[q^{\pm 1/2}]$, known as positivity conjecture for quantum cluster
algebras, where $q$ is the quantum parameter. Kimura and Qin [28] proved the
positivity conjecture for the acyclic skew-symmetric quantum cluster algebras,
and Davison [11] proved this conjecture for the skew-symmetric case. The
author [23] proved the case for quantum cluster algebras from unpunctured
orbifolds. There is no noteworthy work performed for skew-symmetrizable cases
in general.

The original motivation of Fomin and Zelevinsky was to provide a combinatorial
characterization of the (dual) canonical bases in quantum groups (see [31,
27]) and the total positivity in algebraic groups. They conjectured that the
cluster structure should serve as an algebraic framework for the study of the
“dual canonical bases” in various coordinate rings and their $q$-deformations.
In Particular, they conjectured all cluster monomials belong to the dual
canonical bases. This was recently proved in [26, 39, 40]. Generally, it can
be very difficult to write the dual canonical bases explicitly. From this
point of view, finding an explicit (quantum) Laurent expansion formula for
(quantum) cluster monomials represents a noteworthy step in research in
cluster theory.

Finding an explicit formula for cluster variables attracted a lot of
attention. The Caldero-Chapoton map or cluster character is a way to provide
the expansion formula in terms of Euler-Poincaré characteristics, see [4, 5,
6, 20, 36, 37, 38, 12, 25, 7]. Rupel [42] generalized the Caldero-Chapoton map
to the quantum case. However, since Euler-Poincaré can be negative, the
positivity of the coefficients is not immediately implied. Extensive research
has been conducted in the other direction on (quantum) cluster algebras from
surfaces, introduced by Fomin-Shapiro-Thurston [16]. The expansion formula is
well-studied by Musiker, Schiffler, Thomas, Williams, et al. in a series of
studies, see [44, 43, 33, 34, 45]. Rupel [41] gave the explicit quantum
Laurent expansion formula for the quantum cluster algebras of type $A$.
Canakci-Lampe [8] gave an explicit quantum Laurent expansion formula for the
quantum cluster algebras of type $A$ and of the Kronecker type. The author
[22, 23] provided an expansion formula for quantum cluster algebras from
unpunctured surfaces (orbifolds). Berenstein-Retakh [2] deals with a fully
noncommutative version of cluster algebras arising from surfaces.

(Quantum) cluster algebras from orbifolds are an essential class of (quantum)
cluster algebras. Almost all skew-symmetrizable (quantum) cluster algebras of
finite mutation type are in this class, see [14]. Using the unfolding method,
the positivity for cluster algebras from orbifolds can be deduced from the
positivity for the cluster algebras from surfaces, see [14]. However, the
unfolding method does not provide any information on the $q$-coefficients.
Thus, we do not know the quantum Laurent expansion formula for quantum cluster
algebras from orbifolds, even that is known for quantum cluster algebras from
surfaces. The study solves the positivity conjecture for such a class of
quantum cluster algebras by giving an expansion formula of any quantum cluster
variable for any quantum cluster. We would generalize the methods in [24] from
the commutative case to the quantum case.

The study is organized as follows. Sections 2 are devoted to preliminaries. In
Subsection 2.1, we recall the key tool partition bijection which is defined in
[22] and give a criterion for two partition bijections are inverse to each
other, Proposition 2.6. Subsections 2.2, 2.3 and 2.4 are preliminaries on
(quantum) cluster algebras and (quantum) cluster algebras from orbifolds. In
Subsection 2.5, we recall orbifold morphism and canonical polygon defined in
[2]. Subsections 2.6, 2.7 are preliminaries on some heart combinatoric
notations, snake graphs, perfect matchings, and complete
$(T^{o},\gamma)$-path.

In Section 3, for an orbifold $\Sigma$, we generalize the result in [34] and
show that changing the tags at a puncture $p$ gives an isomorphism between two
quantum cluster algebras from $\Sigma$, see Proposition 3.2. To give a Laurent
expansion formula for any quantum cluster variables concerning any cluster, by
Proposition 3.2, it suffices to restrict to three cases, see Remark 3.3.

In Sections 4 and 5, given an ideal triangulation $T^{o}$, we introduce three
classes of lattices for ordinary arcs, one end tagged arcs and two ends tagged
arcs, and construct valuation maps on them. These lattices are the index sets
for the Laurent expansion formulas and the valuation maps provide the powers
of the quantum parameter.

The main results in this paper are given in Section 6. To be precise, we
provide Laurent expansion formulas in Theorems 6.5, 6.9, 6.14 and the
positivity for this class of quantum cluster algebras, Theorem 6.15.

We prove Theorems 6.5, 6.9, 6.14 in Section 12. Our strategy is the following:
for a given tagged arc $\beta$, $\beta^{(q)}$ or $\beta^{(p,q)}$, we first
show that the expansion formulas given in Theorems 6.5, 6.9, 6.14 do not
depend on the choice of ideal triangulations. Since any two ideal
triangulations are connected by a sequence of flips, it suffices to show the
expansion formulas given in Theorems 6.5, 6.9, 6.14 are invariant concerning
two ideal triangulations related by a flip, see Theorem 12.10. We then show
that the expansion formulas hold for a specially chosen ideal triangulation,
see Propositions 12.13, 12.14, 12.17, 12.18 and 12.24.

Sections 7, 8, 9, 10 and 11 are devoted to the proof of Theorem 12.10. The
naive idea is the following: for any ideal triangulation $T^{o}$ and a non-
self-folded arc $\alpha$, denote $T^{\prime o}=\mu_{\alpha}(T^{o})$ and by
$LP(T^{o})$ the Laurent expansion formula concerning $T^{o}$ for a given
tagged arc $\beta$, $\beta^{(q)}$ or $\beta^{(p,q)}$. For each term $X$ in
$LP(T^{o})$, assume that the index of the quantum cluster variable
$X_{\alpha}$ in $X$ is $n(X)$. By the exchange formula of quantum cluster
variables, we have $X$ is the sum of $2^{n(x)}$ terms in $LP(T^{\prime o})$ if
$n(X)\geq 0$ and the sum of $X$ with $2^{-n(x)}-1$ terms in $LP(T^{o})$ is one
term in $LP(T^{\prime o})$ if $n(X)<0$. This allows us to give a partition
bijection between the terms of $LP(T^{o})$ and $LP(T^{\prime o})$,
accordingly, we have a partition bijection between the index sets. We realize
this in Section 9. Sections 7 and 8 are preliminaries to give the partition
bijection in Section 9. We prove in 10 that has a nice compatible property
with the lattice structures in Section 4. In Section 11, we deal with the
powers of the quantum parameter and coefficients under the partition
bijection.

Acknowledgements: The author would like to thank Prof. Arkady Berenstein,
Prof. Vladimir Retakh and Dr. Eugen Rogozinnikov for inspirit discussion. This
project is supported by the National Natural Science Foundation of China
(No.12101617).

##  2\. Preliminary and some notation

###  2.1. Poset and partition bijection

####  2.1.1. Poset

Herein, we recall some notation in poset.

######  Definition 2.1.

A binary relation $\leq$ is a _partial order_ on a set $\mathcal{P}$ if it
satisfies the following properties.

  1. (1)

(reflexivity) $P\leq P$ for all $P\in\mathcal{P}$;

  2. (2)

(antisymmetry) $P\leq Q$ and $Q\leq P$ imply $P=Q$;

  3. (3)

(transitivity) $P\leq Q$ and $Q\leq R$ imply $P\leq R$.

Denote $P\geq Q$ if $Q\leq P$. A _poset_ is a set with a partial order.

We say that a poset $\mathcal{P}$ is _connected_ if for any
$P,Q\in\mathcal{P}$ there exists a sequence of elements
$P_{0}=P,P_{1},\cdots,P_{s}=Q$ such that $P_{i}\leq P_{i+1}$ if $i$ is even
and $P_{i}\geq P_{i+1}$ if $i$ is odd.

######  Definition 2.2.

Let $\mathcal{P}$ be a poset. For any $P,Q\in\mathcal{P}$, we say that $Q$
_covers_ $P$ if $P<Q$ and there is no other element $R$ such that $P<R<Q$.

####  2.1.2. Partition bijection

######  Definition 2.3.

Let $\mathcal{P}$ be a finite set.

  1. $(1)$

A _partition_ of $\mathcal{P}$ is a finite collection of subsets
$\mathcal{P}_{i},i\in I$ such that $\cup_{i\in I}\mathcal{P}_{i}=\mathcal{P}$
and $\mathcal{P}_{i}\cap\mathcal{P}_{j}=\emptyset$ for any different $i,j\in
I$.

  2. $(2)$

Let $\mathcal{P},\mathcal{P}^{\prime}$ be finite subsets. A _partition
bijection_ from $\mathcal{P}$ to $\mathcal{P}^{\prime}$ is a bijection from
some partition of $\mathcal{P}$ to some partition of $\mathcal{P}^{\prime}$,
denoted by $\varphi:\mathcal{P}\overset{par}{\to}\mathcal{P}^{\prime}$.

  3. $(3)$

Let $\varphi:\mathcal{P}\overset{par}{\to}\mathcal{P}^{\prime}$ and
$\varphi^{\prime}:\mathcal{P}^{\prime}\overset{par}{\to}\mathcal{P}$ be two
partition bijections. We say that $\varphi$ and $\varphi^{\prime}$ are
_inverse_ to each other if they are inverse to each other as bijective maps.

######  Example 2.4.

Let $\mathcal{P}=\\{x,y,x^{-2}y^{2},x^{-2}yz,x^{-2}zy,x^{-2}z^{2}\\}$ and
$\mathcal{P}^{\prime}=\\{x^{\prime-1}y,x^{\prime-1}z,y,x^{\prime 2}\\}$. Then
$\\{\\{x\\},\\{y\\},\\{x^{-2}y^{2},x^{-2}yz,x^{-2}zy,x^{-2}z^{2}\\}\\}$ is a
partition of $\mathcal{P}$ and
$\\{\\{x^{\prime-1}y,x^{\prime-1}z\\},\\{y\\},\\{x^{\prime 2}\\}\\}$ is a
partition of $\mathcal{P}^{\prime}$.

One see that $\\{x\\}\to\\{x^{\prime-1}y,x^{\prime-1}z\\}$,
$\\{y\\}\to\\{y\\}$ and
$\\{x^{-2}y^{2},x^{-2}yz,x^{-2}zy,x^{-2}z^{2}\\}\to\\{x^{\prime 2}\\}$ give a
partition bijection from $\mathcal{P}$ to $\mathcal{P}^{\prime}$.

######  Remark 2.5.

Let $\mathcal{P},\mathcal{P}^{\prime}$ be finite sets. From Definition 2.3, to
give a partition bijection from $\mathcal{P}$ to $\mathcal{P}^{\prime}$ is
equivalent to associating each $P\in\mathcal{P}^{\prime}$ with a non-empty
subset $\varphi(P)\in\mathcal{P}^{\prime}$ such that the following conditions
hold.

  1. (i)

For any $P,Q\in\mathcal{P}$, we have either
$\varphi(P)\cap\varphi(Q)=\emptyset$ or $\varphi(P)=\varphi(Q)$;

  2. (ii)

$\bigcup_{P\in\mathcal{P}}\varphi(P)=\mathcal{P}^{\prime}$.

We will frequently use this equivalence definition throughout this paper. The
following result is useful to judge if two given partition bijections are
inverse to each other.

######  Proposition 2.6.

Let $\mathcal{P},\mathcal{P}^{\prime}$ be finite sets. Let
$\varphi:\mathcal{P}\overset{par}{\to}\mathcal{P}^{\prime}$ and
$\varphi^{\prime}:\mathcal{P}^{\prime}\overset{par}{\to}\mathcal{P}$ be two
partition bijections. Then $\varphi$ and $\varphi^{\prime}$ are inverse to
each other is equivalent to the following condition: for any
$P\in\mathcal{P},P^{\prime}\in\mathcal{P}^{\prime}$ we have
$P^{\prime}\in\varphi(P)$ if and only if $P\in\varphi^{\prime}(P^{\prime})$.

###### Proof.

The necessity is clear. Now we prove the sufficiency. As
$\varphi:\mathcal{P}\overset{par}{\to}\mathcal{P}^{\prime}$ is a partition
bijection, $\varphi$ gives a bijection between the partition
$\bigsqcup_{P\in\mathcal{P}}\\{Q\mid\varphi(Q)=\varphi(P)\\}$ of $\mathcal{P}$
and the partition bijection $\bigsqcup_{P\in\mathcal{P}}\varphi(P)$ of
$\mathcal{P}^{\prime}$. Similarly, $\varphi^{\prime}$ gives a bijection
between the partition
$\bigsqcup_{P^{\prime}\in\mathcal{P}^{\prime}}\\{Q^{\prime}\mid\varphi^{\prime}(Q^{\prime})=\varphi^{\prime}(P^{\prime})\\}$
of $\mathcal{P}^{\prime}$ and the partition bijection
$\bigsqcup_{P^{\prime}\in\mathcal{P}^{\prime}}\varphi^{\prime}(P^{\prime})$ of
$\mathcal{P}$. According to the condition, for any $P\in\mathcal{P}$ and
$P^{\prime}\in\varphi(P)$ we have
$\\{Q\mid\varphi(Q)=\varphi(P)\\}=\varphi^{\prime}(P^{\prime})$. Thus the
partitions $\bigsqcup_{P\in\mathcal{P}}\\{Q\mid\varphi(Q)=\varphi(P)\\}$ and
$\bigsqcup_{P^{\prime}\in\mathcal{P}^{\prime}}\varphi^{\prime}(P^{\prime})$ of
$\mathcal{P}$ coincide. Similarly, the partitions
$\bigsqcup_{P^{\prime}\in\mathcal{P}^{\prime}}\\{Q^{\prime}\mid\varphi^{\prime}(Q^{\prime})=\varphi^{\prime}(P^{\prime})\\}$
and $\bigsqcup_{P\in\mathcal{P}}\varphi(P)$ of $\mathcal{P}^{\prime}$
coincide. The result follows. $\square$

###  2.2. Cluster algebra

In this subsection, we recall the definitions of cluster algebra and geometric
type cluster algebra in [18]. Throughout, for any $a\in\mathbb{Z}$, denote by
$[a]_{+}=max\\{0,a\\}$. Given a vector ${\bf
a}=(a_{1},\cdots,a_{k})\in\mathbb{Z}^{k}$ for some $k\in\mathbb{N}$, denote by
${\bf a}_{+}=([a_{1}]_{+},\cdots,[a_{k}]_{+})$ and denote by ${\bf
a}_{-}=(-{\bf a})_{+}$.

A triple $(\mathbb{P},\oplus,\cdot)$ is called a _semifield_ if
$(\mathbb{P},\cdot)$ is an abelian multiplicative group and
$(\mathbb{P},\oplus)$ is a commutative semigroup such that $``\oplus"$ is
distributive with respect to $``\cdot"$. A _tropical semifield_ ${\rm
trop}(u_{1},\cdots,u_{l})$ is a semifield freely generated by
$u_{1},\cdots,u_{l}$ as abelian groups with $\oplus$ defined by
$\prod_{j}u_{j}^{a_{j}}\oplus\prod_{j}u_{j}^{b_{j}}=\prod_{j}u_{j}^{min(a_{j},b_{j})}$.
Let $(\mathbb{P},\oplus,\cdot)$ be a semifield. The group ring $\mathbb{ZP}$
will be used as _ground ring_. Give an integer $n$, let $\mathcal{F}$ be the
rational functions field in $n$ independent variables, with coefficients in
$\mathbb{QP}$.

######  Definition 2.7.

A seed $t$ in $\mathcal{F}$ consists a triple $(x(t),y(t),B(t))$, where

  * •

$x(t)=\\{x_{1}(t),\cdots,x_{n}(t)\\}$ such that $\mathcal{F}$ is freely
generated by $x(t)$ over $\mathbb{QP}$.

  * •

$y(t)=\\{y_{1}(t),\cdots,y_{n}(t)\\}\subseteq\mathbb{P}$.

  * •

$B(t)=(b_{ij})$ is an $n\times n$ skew-symmetrizable integer matrix.

######  Definition 2.8.

Given a seed $t$ in $\mathcal{F}$, for any $k\in[1,n]$, we define the
_mutation_ of $t$ at the $k$-th direction to be the new seed
$t^{\prime}=\mu_{k}(t)=(x(t^{\prime}),y(t^{\prime}),B(t^{\prime}))$, where

  * •

$\begin{array}[]{ccl}x_{i}(t^{\prime})&=&\left\\{\begin{array}[]{ll}x_{i}(t),&\mbox{if
$i\neq k$},\\\ \frac{y_{k}(t)\prod x_{i}(t)^{[b_{ik}]_{+}}+\prod
x_{i}(t)^{[-b_{ik}]_{+}}}{(y_{k}(t)\oplus
1)x_{k}(t)},&\mbox{otherwise}.\end{array}\right.\end{array}$

  * •

(1)
$\begin{array}[]{ccl}y_{i}(t^{\prime})&=&\left\\{\begin{array}[]{ll}y^{-1}_{k}(t),&\mbox{if
$i=k$},\\\ y_{i}(t)y_{k}(t)^{[b_{ki}]_{+}}(1\oplus
y_{k}(t))^{-b_{ki}},&\mbox{otherwise}.\end{array}\right.\end{array}$

  * •

$B(t^{\prime})=(b^{\prime}_{ij})$ is determined by $B(t)=(b_{ij})$:

$\begin{array}[]{ccl}b^{\prime}_{ij}&=&\left\\{\begin{array}[]{ll}-b_{ij},&\mbox{if
$i=k$ or $j=k$},\\\
b_{ij}+[b_{ik}]_{+}[b_{kj}]_{+}-[-b_{ik}]_{+}[-b_{kj}]_{+},&\mbox{otherwise}.\end{array}\right.\end{array}$

######  Definition 2.9.

A _cluster algebra_ $\mathcal{A}$ (of rank $n$) over $\mathbb{P}$ is defined
by the following steps.

  1. $(1)$

Choose an initial seed $t_{0}=(x(t_{0}),y(t_{0}),B(t_{0}))$.

  2. $(2)$

Get all the seeds $t$ which are obtained from $t_{0}$ by iterated mutations at
directions $k\in[1,n]$.

  3. $(3)$

Define $\mathcal{A}=\mathbb{ZP}[x_{i}(t)]_{t,i\in[1,n]}$.

  4. $(4)$

$x(t)$ is called a _cluster_ for any $t$.

  5. $(5)$

$x_{i}(t)$ is called a _cluster variable_ for any $i\in[1,n]$ and $t$.

  6. $(6)$

A monomial in $x(t)$ is called a _cluster monomial_ for any $t$.

  7. $(7)$

$y(t)$ is called a _coefficient tuple_ for any $t$.

  8. $(8)$

$B(t)$ is called an _exchange matrix_ for any $t$.

In particular, when $\mathbb{P}={\rm Trop}(u_{1},\cdots,u_{l})$, we say that
$\mathcal{A}$ is of _geometric type_.

In case $\mathbb{P}={\rm Trop}(u_{1},\cdots,u_{l})$, denote by $m=n+l$. For a
seed $t$ in $\mathcal{F}$, $y_{j}(t)=\prod u_{i}^{c_{ij}}$ for some integers
$c_{ij}$. Thus we can write $t$ as $\widetilde{x}(t),\widetilde{B}(t)$, where

  1. $(1)$

$\widetilde{x}(t)=\\{x_{1}(t),\cdots,x_{n}(t),x_{n+1}(t)=u_{1},\cdots,x_{m}(t)=u_{l}\\}$.

  2. $(2)$

$\widetilde{B}(t)=(b_{ij})$ is an $m\times n$ with $b_{ij}=c_{i-n,j}$ for
$i\in[n+1,m]$.

Under this convention, the mutation of $t$ at direction $k$ is
$t^{\prime}=\mu_{k}(t)=(\widetilde{x}(t^{\prime}),\widetilde{B}(t^{\prime}))$,
where

  1. $(1)$

$\begin{array}[]{ccl}x_{i}(t^{\prime})&=&\left\\{\begin{array}[]{ll}x_{i}(t),&\mbox{if
$i\neq k$},\\\ \frac{\prod x_{i}(t)^{[b_{ik}]_{+}}+\prod
x_{i}(t)^{[-b_{ik}]_{+}}}{x_{k}(t)},&\mbox{otherwise}.\end{array}\right.\end{array}$

  2. $(2)$

$\widetilde{B}(t^{\prime})=(b^{\prime}_{ij})$ is determined by
$\widetilde{B}(t)=(b_{ij})$:

$\begin{array}[]{ccl}b^{\prime}_{ij}&=&\left\\{\begin{array}[]{ll}-b_{ij},&\mbox{if
$i=k$ or $j=k$},\\\
b_{ij}+[b_{ik}]_{+}[b_{kj}]_{+}-[-b_{ik}]_{+}[-b_{kj}]_{+},&\mbox{otherwise}.\end{array}\right.\end{array}$

###  2.3. Quantum cluster algebra

In this subsection, we recall the definition of quantum cluster algebra in
[3].

Fix two integers $n\leq m$. Let $\widetilde{B}$ be an $m\times n$ integer
matrix. Let $\Lambda$ be an $m\times m$ skew-symmetric integer matrix. We call
$(\widetilde{B},\Lambda)$ _compatible_ if $\widetilde{B}^{T}\Lambda=(D\;\;0)$
for some diagonal matrix $D$ with positive entries, where $\widetilde{B}^{T}$
is the transpose of $\widetilde{B}$. Note that in this case, the upper
$n\times n$ submatrix of $\widetilde{B}$ is skew-symmetrizable and
$\widetilde{B}$ is full rank.

######  Definition 2.10.

Let $v$ be the quantum parameter.

  1. $(1)$

A _quantum seed_ $t$ consists a compatible pair
$(\widetilde{B}(t),\Lambda(t))$ and a collection of indeterminate
$X_{i}(t),i\in[1,m]$, called _quantum cluster variables_ , $\widetilde{B}(t)$
is called _extended exchange matrix_ , $\Lambda(t)$ is called _quantum
commutative matrix_.

  2. $(2)$

Let $\\{{\bf e}_{i}\\}$ be the standard basis of $\mathbb{Z}^{m}$ and
$X(t)^{{\bf e}_{i}}=X_{i}(t)$. Define the corresponding _quantum torus_
$\mathcal{T}(t)$ to be the algebra which is freely generated by $X(t)^{{\bf
a}},{\bf a}\in\mathbb{Z}^{m}$ as $\mathbb{Z}[v^{\pm 1}]$-module, with
multiplication on these elements defined by

$X(t)^{{\bf a}}X(t)^{{\bf b}}=v^{\Lambda(t)({\bf a},{\bf b})}X(t)^{{\bf
a}+{\bf b}},$

where $\Lambda(t)(,)$ means the bilinear form on $\mathbb{Z}^{m}$ such that

$\Lambda(t)({\bf e}_{i},{\bf e}_{j})=\Lambda(t)_{ij}.$

######  Definition 2.11.

For any $k\in[1,n]$, we define the _mutation_ of $t$ at the $k$-th direction
to be the new seed
$t^{\prime}=\mu_{k}(t)=((X_{i}(t^{\prime})_{i\in[1,m]}),\widetilde{B}(t^{\prime}),\Lambda(t^{\prime}))$,
where

  1. $(1)$

$X_{i}(t^{\prime})=X_{i}(t)$ for $i\neq t$,

  2. $(2)$

$X_{k}(t^{\prime})=X(t)^{-{\bf e}_{k}+\sum_{i}[b_{ik}]_{+}{\bf
e}_{i}}+X(t)^{-{\bf e}_{k}+\sum_{i}[-b_{ik}]_{+}{\bf e}_{i}}$.

  3. $(3)$

$\widetilde{B}(t^{\prime})=(b^{\prime}_{ij})$ is determined by
$\widetilde{B}(t)=(b_{ij})$ such that

$\begin{array}[]{ccl}b^{\prime}_{ij}&=&\left\\{\begin{array}[]{ll}-b_{ij},&\mbox{if
$i=k$ or $j=k$},\\\
b_{ij}+[b_{ik}]_{+}[b_{kj}]_{+}-[-b_{ik}]_{+}[-b_{kj}]_{+},&\mbox{otherwise}.\end{array}\right.\end{array}$

  4. $(4)$

$\Lambda(t^{\prime})$ is skew-symmetric and satisfies:

$\begin{array}[]{ccl}\Lambda(t^{\prime})_{ij}&=&\left\\{\begin{array}[]{ll}\Lambda(t)_{ij},&\mbox{if
$i,j\neq k$},\\\ \Lambda(t)({\bf e}_{i},-{\bf e}_{k}+\sum_{l}[b_{lk}]_{+}{\bf
e}_{l}),&\mbox{if $i\neq k=j$},\end{array}\right.\end{array}$

One see that $(\widetilde{B}(t^{\prime}),\Lambda(t^{\prime}))$ is compatible
since
$\widetilde{B}^{T}(t)\Lambda(t)=\widetilde{B}^{T}(t^{\prime})\Lambda(t^{\prime})$.
The quantum torus $\mathcal{T}(t^{\prime})$ for the new seed $t^{\prime}$ can
be defined similarly.

######  Definition 2.12.

A _quantum cluster algebra_ $\mathcal{A}_{v}$ is defined by the following
steps.

  1. $(1)$

Choose an initial seed $t_{0}=((X_{1},\cdots,X_{m}),\widetilde{B},\Lambda)$.

  2. $(2)$

Get all the seeds $t$ are obtained from $t_{0}$ by iterated mutations at
directions $k\in[1,n]$.

  3. $(3)$

Define $\mathcal{A}_{v}=\mathbb{Z}[v^{\pm 1}]\langle
X_{i}(t)\rangle_{t,i\in[1,m]}$.

  4. $(4)$

$X_{n+1},\cdots,X_{m}$ are called _frozen variables_ or _coefficients_.

  5. $(5)$

A quantum cluster variable in $t$ is called a _quantum cluster variable_ of
$\mathcal{A}_{v}$.

  6. $(6)$

$X(t)^{{\bf a}}$ for some $t$ and ${\bf a}\in\mathbb{N}^{m}$ is called a
_quantum cluster monomial_.

Note that by specializing $v$ to 1, we get a commutative cluster algebra
$\mathcal{A}_{v}|_{v=1}$.

######  Theorem 2.13.

(Quantum Laurent Phenomenon, [3]) Let $\mathcal{A}_{v}$ be a quantum cluster
algebra and $t$ be a seed. For any quantum cluster variable $X$, we have
$X\in\mathcal{T}(t)$.

######  Conjecture 2.14.

(Positivity Conjecture, [3]) Let $\mathcal{A}_{v}$ be a quantum cluster
algebra and $t$ be a seed. For any quantum cluster variable $X$ of
$\mathcal{A}_{v}$,

$X\in\mathbb{N}[v^{\pm 1}]\langle X(t)^{{\bf a}}\mid{\bf
a}\in\mathbb{Z}^{m}\rangle.$

######  Remark 2.15.

The positivity conjecture was proved by Davison [11] for the skew-symmetric
case and by the author for the unpunctured orbifold case [23].

###  2.4. (Quantum) Cluster algebras from orbifolds

In this subsection, we recall some combinatorial notation from orbifolds,
please refer to [16, 14, 15] for more details.

Let $S$ be a connected oriented Riemann orbifold with boundary. Fix a non-
empty set $M$ of marked points in the closure of $S$ with at least one marked
point on each boundary component. Fix a finite set $U$ in the interior of $S$
such that $U\cap M=\emptyset$. We call the triple $(S,M,U)$ an _orbifold_.
Marked points in the interior of $S$ are called _punctures_. The points in $U$
are called _orbifold points._ Each orbifold point in $U$ comes with a weight
$\frac{1}{2}$ or $2$. Throughout this paper, we assume that $U$ contains no
orbifold point with weight $2$.

######  Definition 2.16.

[14, 15] An arc $\beta$ in $(S,M,U)$ is a curve in $S$ considered up to
relative isotopy (of $S\setminus(M\cup U)$) modulo endpoints such that

  1. $\bullet$

one of the following holds:

    1. $-$

either both endpoints of $\beta$ belong to $M$ (and then $\beta$ is called an
_ordinary arc_)

    2. $-$

or one endpoint belongs to $M$ and another belongs to $U$ (then $\beta$ is
called a _pending arc_);

  2. $\bullet$

$\beta$ has no self-intersections, except that its endpoints may coincide;

  3. $\bullet$

except for the endpoints, $\beta$ and $M\cup U\cup\partial S$ are disjoint;

  4. $\bullet$

$\beta$ does not cut out a monogon not containing points of $M$;

  5. $\bullet$

$\beta$ is not homotopic to a boundary segment.

If $\gamma$ is an arc incident to a puncture $p$, denote by $l_{p}(\gamma)$
the loop based on another endpoint of $\gamma$ and encloses $\gamma$.

An _oriented arc_ $\overrightarrow{\beta}$ is an arc $\beta$ with an
orientation. Denote by $s(\overrightarrow{\beta})$ and
$t(\overrightarrow{\beta})$ the starting point and ending point, respectively,
of $\overrightarrow{\beta}$.

Note that we do not allow both endpoints of $\beta$ to be in $U$.

######  Definition 2.17.

[14, 15] Two arcs $\beta$ and $\beta^{\prime}$ are _compatible_ if the
following hold:

  1. $\bullet$

they do not intersect in the interior of $S$;

  2. $\bullet$

if both $\beta$ and $\beta^{\prime}$ are pending arcs, then the endpoints of
$\beta$ and $\beta^{\prime}$ that are orbifold points do not coincide (i.e.,
two pending arcs may share a marked point, but not an orbifold point).

######  Definition 2.18.

[16, 14, 15] An _ideal triangulation_ of $(S,M,U)$ is a maximal collection of
distinct pairwise compatible arcs.

The arcs of an ideal triangulation cut $S$ into triangles. See Fig. 1 for a
list of possible ideal triangles.

Figure 1. List of possible ideal triangles

A triangle with two arcs folded is called a _self-folded triangle_. The folded
arc is called the _radius_. Let $T^{o}$ be an ideal triangulation of
$(S,M,U)$. If $\alpha$ is not the radius of some self-folded triangles, then
there is a unique arc $\alpha^{\prime}\neq\alpha$ such that
$(T^{o}\setminus\\{\alpha\\})\cup\\{\alpha^{\prime}\\}$ is an ideal
triangulation. Denote
$\mu_{\alpha}(T^{o})=(T^{o}\setminus\\{\alpha\\})\cup\\{\alpha^{\prime}\\}$
and we call $\mu_{\alpha}(T^{o})$ is the _flip_ of $T^{o}$ at $\alpha$ and
$\alpha^{\prime}$ is obtained from $T^{o}$ by flip at $\alpha$.

######  Lemma 2.19.

For each puncture $p$, let $T_{1}$ and $T_{2}$ be ideal triangulations without
self-folded triangles enclose $p$. Then there exists a sequence of ideal
triangulations $T_{1}=T^{0},T^{1},\cdots,T^{n}=T_{2}$ without self-folded
triangles enclose $p$ such that $T^{i}$ and $T^{i+1}$ are related by a flip
for all $i=0,\cdots,n-1$.

###### Proof.

By [16, Proposition 3.8], $T_{1}$ and $T_{2}$ are connected by a sequence of
flips. We may assume that there exists a sequence of ideal triangulations
$T_{1}=\hat{T}^{0},\hat{T}^{1},\cdots,\hat{T}^{m}=T_{2}$ without self-folded
triangles enclose $p$ such that $\hat{T}^{i}$ and $\hat{T}^{i+1}$ are related
by a flip for all $i=0,\cdots,m-1$. We prove the result by induction on the
number $N$ of the loops in $\bigcup_{i=0}^{m}\hat{T}^{i}$ enclosed only $p$.
If $N=0$ then we are done. Assume $N>1$ and
$l_{p}\in\hat{T}_{\ell}\cap\hat{T}_{\ell^{\prime}}$ is a loop enclose only $p$
such that $\ell<\ell^{\prime}$, $l_{p}\notin\hat{T}^{i}$ for $0\leq i<\ell$,
$l_{p}\in\hat{T}^{i}$ for $\ell\leq i\leq\ell^{\prime}$ and
$l_{p}\notin\hat{T}_{\ell^{\prime}+1}$. Then $l_{p}$ is in some self-folded
triangle in $\hat{T}^{i}$ for $\ell\leq i\leq\ell^{\prime}$. Denote by
$\gamma$ the radius. Thus
$\gamma\in\hat{T}_{\ell-1}\cap\hat{T}_{\ell^{\prime}+1}$. Cutting along
$\gamma$, we obtain a new orbifold $\widetilde{S}$. $\gamma$ becomes to two
boundary arcs $\gamma^{\pm}$ in $\hat{S}$. Since
$\gamma\in\hat{T}_{\ell-1}\cap\hat{T}_{ell^{\prime}+1}$, we have
$(\hat{T}_{\ell-1}\setminus\\{\gamma\\})\cup\\{\gamma^{\pm}\\}$ and
$(\hat{T}_{\ell+1}\setminus\\{\gamma\\})\cup\\{\gamma^{\pm}\\}$ are two
triangulations of $\widetilde{S}$. By [16, Proposition 3.8],
$(\hat{T}_{\ell-1}\setminus\\{\gamma\\})\cup\\{\gamma^{\pm}\\}$ and
$(\hat{T}_{\ell+1}\setminus\\{\gamma\\})\cup\\{\gamma^{\pm}\\}$ are connected
by a sequence of flips. Assume that
$\widetilde{T}_{1}=(\hat{T}_{\ell-1}\setminus\\{\gamma\\})\cup\\{\gamma^{\pm}\\},\widetilde{T}_{2},\cdots,\widetilde{T}_{m^{\prime}}=(\hat{T}_{\ell+1}\setminus\\{\gamma\\})\cup\\{\gamma^{\pm}\\}$
is a sequence of ideal triangulations of $\widetilde{S}$ such that
$\widetilde{T}_{i+1}$ and $\widetilde{T}_{i}$ are related by a flip for $1\leq
i\leq m^{\prime}-1$. For each $1\leq i\leq m^{\prime}-1$, let
$\bar{T}_{i}=(\widetilde{T}_{i}\setminus\\{\gamma^{\pm}\\})\cup\\{\gamma\\}$.
Then $\bar{T}_{i}$ is an ideal triangulation of $S$ and
$\bar{T}_{i},\bar{T}_{i+1}$ are related by a flip, and
$\bar{T}_{1}=\hat{T}_{\ell-1},\bar{T}_{m^{\prime}}=\hat{T}_{\ell+1}$. Consider
the flip sequence
$\hat{T}^{0}=T_{1},\cdots,\hat{T}_{\ell-1}=\bar{T}_{1},\bar{T}_{2},\cdots\bar{T}_{m^{\prime}}=\hat{T}_{\ell+1},\cdots,\hat{T}_{m}=T_{2}$.
We have the number of the loops in
$\bigcup_{i=0}^{\ell-1}\hat{T}^{i}\cup\bigcup_{i=\ell+1}^{m}\hat{T}^{i}\cup\bigcup_{i=0}^{m^{\prime}}\bar{T}^{i}$
enclosed only $p$ is strictly than $N$. Then the result is followed by
induction. $\square$

The following lemma is similar to Lemma 2.19.

######  Lemma 2.20.

For punctures $p$ and $q$, let $T_{1}$ and $T_{2}$ be ideal triangulations
without self-folded triangles enclose $p$ or $q$. Then there exists a sequence
of ideal triangulations $T_{1}=T^{0},T^{1},\cdots,T^{n}=T_{2}$ without self-
folded triangles enclose $p$ or $q$ such that $T^{i}$ and $T^{i+1}$ are
related by a flip for all $i=0,\cdots,n-1$.

######  Definition 2.21.

[16, Definition 7.1](Tagged arc) Each arc $\beta$ in $(S,M,U)$ has two ends
obtained by arbitrarily cutting $\beta$ into three pieces, then throwing out
the middle one. We think of the two ends as locations near the endpoints to be
used for labeling (“tagging”) an arc. A tagged arc is an ordinary arc in which
each end has been tagged in one of two ways, plain or notched, so that the
following conditions are satisfied:

  1. $\bullet$

the arc does not cut out a once-punctured monogon;

  2. $\bullet$

an endpoint lying on the boundary is tagged plain;

  3. $\bullet$

ends of a pending arc being orbifold points are always tagged plain; and

  4. $\bullet$

both ends of a loop are tagged in the same way.

######  Definition 2.22.

[16, Definition 7.4] (Compatibility of tagged arcs) Two tagged arcs $\beta$
and $\beta^{\prime}$ are called _compatible_ if the following conditions are
satisfied:

  1. $\bullet$

The untagged versions of $\beta$ and $\beta^{\prime}$ are compatible;

  2. $\bullet$

If the untagged versions of $\beta$ and $\beta^{\prime}$ are different, and
$\beta$ and $\beta^{\prime}$ share an endpoint $a$, then the ends of $\beta$
and $\beta^{\prime}$ connecting to $a$ must be tagged in the same way;

  3. $\bullet$

If the untagged versions of $\beta$ and $\beta^{\prime}$ coincide, then at
least one end of $\beta$ must be tagged in the same way as the corresponding
end of $\beta^{\prime}$.

######  Definition 2.23.

[16, 14, 15] A _tagged triangulation_ of $(S,M,U)$ is a maximal collection of
distinct pairwise compatible tagged arcs.

######  Definition 2.24.

Let $o$ be an orbifold point. For a tagged pending arc $\beta$ connecting a
marked point $i$ and $o$, denote by $sl(\beta)$ the tagged loop at $i$ around
$o$ which tagged the same with $\beta$ at $i$. We call $sl(\beta)$ the
_special loop_ associated with $\beta$. See Fig. 2.

Figure 2. Special loop

Note that special loop is not an arc in $\Sigma=(S,M,U)$.

Throughout this paper, the notation $\gamma$, $\gamma^{(q)}$ and
$\gamma^{(p,q)}$ respectively mean the following three cases,

  1. (i)

$\gamma$ is an ordinary arc or a pending arc starting from $p$ and ending at
$q$;

  2. (ii)

$\gamma^{(p)}$ is the tagged arc with underlying arc $\gamma$ and tagged
notched at $q$;

  3. (iii)

$\gamma^{(p,q)}$ is the tagged arc with underlying arc $\gamma$ and tagged
notched at $p,q$.

In particular, if $p=q$ then $\gamma^{(p)}=\gamma^{(p,q)}$.

For any ideal triangulation $T^{o}$, replace all the loops $l_{p}(\gamma)$ in
self-folded triangles enclose puncture $p$ with radius $\gamma$ by
$\gamma^{(p)}$, we obtain a tagged triangulation $T$, we call $T$ the tagged
triangulation corresponding to $T^{o}$.

Conversely, for any tagged triangulation $T$, first replace all pairs of
tagged arcs of the form $\gamma,\gamma^{(p)}$ by $\gamma,l_{p}(\gamma)$ and
then replace $\gamma^{(p)}$ and $\gamma^{(p,q)}$ by $\gamma$, we obtain an
ideal triangulation, denoted by $l(T)$.

If $\gamma$ is a pending arc which incident an orbifold point $o$, denote by
$l(\gamma)$ the loop cutting out a monogon enclosing $o$ and radius $\gamma$,
then the _crossing number_ $N(\gamma,\gamma^{\prime})$ of $\gamma$ with
$\gamma^{\prime}$ is the minimum of the numbers of crossings of arcs $\alpha$
and $\alpha^{\prime}$, where $\alpha$ is isotopic to $l(\gamma)$ and
$\alpha^{\prime}$ is isotopic to $\gamma^{\prime}$. Note that
$N(\gamma,\gamma^{\prime})\neq N(\gamma,\gamma^{\prime})$ generally if one is
an ordinary arc and another is a pending arc.

For $\tau\in T$, denote by $w(\tau)$ the weight of $\tau$.

For two non-boundary non-self-folded arcs $\tau$, $\tau^{\prime}$ in an ideal
triangulation $T^{o}$ and a non-self-folded triangle $\Delta$ in $T^{o}$,
define

$\begin{array}[]{ccl}b^{T^{o},\Delta}_{\tau\tau^{\prime}}=\left\\{\begin{array}[]{ll}w(\tau^{\prime}),&\mbox{if
$\tau,\tau^{\prime}$ are sides of $\Delta$ and $\tau^{\prime}$ following
$\tau$ in the clockwise order},\\\ -w(\tau^{\prime}),&\mbox{if
$\tau,\tau^{\prime}$ are sides of $\Delta$ and $\tau$ following
$\tau^{\prime}$ in the clockwise order},\\\
0,&\mbox{otherwise.}\end{array}\right.\end{array}$

and
$b^{T^{o}}_{\tau\tau^{\prime}}=\sum_{\Delta}b^{T^{o},\Delta}_{\tau\tau^{\prime}}$,
where $\Delta$ runs over all non-self-folded triangle $\Delta$ in $T^{o}$.
Denote by $l(\tau)$ the loop encloses $\tau$ if $\tau$ is a self-fold arc and
$\tau$ otherwise. For any two non-boundary arcs $\tau,\tau^{\prime}\in T^{o}$,
define
$b^{T^{o},\Delta}_{\tau\tau^{\prime}}=b^{T^{o},\Delta}_{l(\tau)l(\tau^{\prime})}$.

We say the matrix $B^{T^{o}}=(b^{T^{o}}_{\tau,\tau^{\prime}})$ the _signed
adjacency matrix_ of $T$, see [14, 15]. Then $B^{T^{o}}$ is a skew-
symmetrizable matrix. In fact, let $D^{T^{o}}=diag(w(\tau))_{\tau\in T^{o}}$,
then $D^{T^{o}}B^{T^{o}}$ is skew-symmetric.

For any non-self-folded $\tau\in T^{o}$, we have
$B^{\mu_{\tau}(T^{o})}=\mu_{\tau}(B^{T^{o}})$.

######  Definition 2.25.

Let $(S,M,U)$ be an orbifold.

  1. $(1)$

We say that a cluster algebra $\mathcal{A}$ is _coming from $(S,M,U)$_ if
there exists a tagged triangulation $T$ such that $B^{T}$ is an exchange
matrix of $\mathcal{A}$.

  2. $(2)$

We say that a quantum cluster algebra $\mathcal{A}_{v}$ is _coming from
$(S,M,U)$_ if the specialized cluster algebra $\mathcal{A}_{v}|_{v=1}$ is
coming from $(S,M,U)$.

######  Proposition 2.26.

[16, 14, 3] Let $\mathcal{A}_{v}$ be a quantum cluster algebra from
$\Sigma=(S,M,U)$.

  1. $(1)$

If $\Sigma$ is not a closed surface with one puncture, then there are
bijections

$\\{\text{Tagged arcs in }\Sigma\\}\rightarrow\\{\text{Quantum cluster
variables of }\mathcal{A}_{v}\\},\gamma\mapsto X_{\gamma}.$

$\\{\text{Tagged triangulation of }\Sigma\\}\rightarrow\\{\text{Quantum
clusters of }\mathcal{A}\\},T\mapsto X_{T}.$

  2. $(2)$

If $\Sigma$ is a closed surface with exactly one puncture, then there are
bijections

$\\{\text{Ordinary arcs in }\Sigma\\}\rightarrow\\{\text{Quantum cluster
variables of }\mathcal{A}\\},\gamma\mapsto X_{\gamma}.$

$\\{\text{Ideal triangulation of }\Sigma\\}\rightarrow\\{\text{Quantum
clusters of }\mathcal{A}\\},T\mapsto X_{T}.$

###  2.5. Orbifold morphism and canonical polygon

######  Definition 2.27.

[2, Definition 3.5] For two orbifolds $(S,M,U)$ and
$(S^{\prime},M^{\prime},U^{\prime})$, we say that a continuous map $f:S\to
S^{\prime}$ is a _morphism_ if

  1. $\bullet$

$f^{-1}(M^{\prime}\cup U^{\prime})\subseteq M\cup U$ and $f(U)\subseteq
U^{\prime}$;

  2. $\bullet$

For each point $p\in S\setminus I^{f}$, there exists a neighborhood $S_{p}$ of
$p$ in $S$ such that the restriction of $f$ to $S_{p}$ is injective, where
$I^{f}:=f^{-1}(U^{\prime})\setminus U$;

  3. $\bullet$

For each point $p\in I^{f}$, there exists a neighborhood $S_{p}$ of $p$ in $S$
such that the restriction of $f$ to $S_{p}$ is a two-fold cover of $f(S_{p})$
ramified at $p$.

######  Theorem 2.28.

[2, Theorem 3.21] Let $T^{o}$ be a triangulation of $(S,M,U)$. Then for each
$r$-gon $Q=(\gamma^{\prime}_{1},\cdots,\gamma^{\prime}_{r}$ in $(S,M,U)$,
there exists an $n$-gon
$P=(\gamma_{1},\cdots,\gamma_{n})\in(T_{0}(Q,T^{o}))^{n}$ for some $n\geq r$,
a triangulation $\Delta$ of $P_{n}$ (the $n$-gon), and an order-preserving
embedding $\iota:[r]\hookrightarrow[n]$ such that:

  1. $(a)$

$\gamma_{ij}\in T_{0}(Q,T^{o})$ if and only if $(i,j)\in\Delta$;

  2. $(b)$

$\gamma^{\prime}_{k}=\gamma_{\iota(k),\iota(k^{+})}$ for all $k\in[r]$ (i.e.,
$Q$ is a “sub-polygon” of $P$).

In particular, if $Q=(\gamma,\bar{\gamma})$ is the digon formed by an arc
$\gamma$ and its inverse, then we call the polygon $P_{T^{o}}(\gamma)$ the
_canonical polygon_ of $\gamma$ with respect to $T^{o}$.

######  Example 2.29.

In Figure 3, the left picture is an annulus with one marked point on each
boundary and triangulation $\\{1,2\\}$. The arc $\gamma$ crosses through
triangles $(1,3,2),(2,4,1),(1,3,2)$ and $(2,4,1)$ consecutively. Thus the
canonical polygon for $\gamma$ is as shown in the right picture.

Figure 3. Canonical polygon

###  2.6. Snake graph and perfect matching

We first recall the definition of an abstract snake graph [9]. A tile is
considered as a graph with four vertices and four edges in the obvious way.
Throughout this paper, we denote by $N(G)$ (resp. $S(G),W(G),E(G)$) the north
(resp. south, west, east) edge of a tile $G$.

######  Definition 2.30.

[9] A _snake graph_ $G$ is a connected graph consisting of a finite sequence
of tiles $G_{1},G_{2},\cdots,G_{c}$ with $c\geq 1$, such that for each
$i=1,2,\cdots,c-1$

  1. (i)

$G_{i}$ and $G_{i+1}$ share exactly one edge which is either the north edge of
$G_{i}$ and the south edge of $G_{i+1}$ or the east edge of $G_{i}$ and the
west edge of $G_{i+1}$.

  2. (ii)

$G_{i}$ and $G_{j}$ have no edge in common whenever $|i-j|\geq 2$.

  3. (iii)

$G_{i}$ and $G_{j}$ are disjoint whenever $|i-j|\geq 3$.

Let $H$ be a graph. In this paper, we denote by $edge(H)$ the edge set of $H$.

######  Definition 2.31.

([34, Definition 4.6]) A _perfect matching_ of a graph $G$ is a subset $P$ of
the edges of $G$ such that each vertex of $G$ is incident to exactly one edge
of $P$. Denote by $\mathcal{P}(G)$ the set of all perfect matchings of $G$.

######  Definition 2.32.

([35, 22, 23]) Let $P$ be a perfect matching of a snake graph $G$. We say that
$P$ can _twist_ on a tile $G_{i}$ if there are two edges of $G_{i}$ in $P$.
The perfect matching obtained by replacing the two edges with the remaining
two edges of $G_{i}$ is called the _twist_ of $P$ at $G_{i}$, denoted by
$\mu_{G_{i}}(P)$.

Twist of perfect matching

######  Definition 2.33.

For a perfect matching $P$, we call an edge in $P$ _$\tau$ -mutable_ if it is
an edge of some tile with diagonal labeled $\tau$ that $P$ can twist on.

Next, we recall the construction of the snake graph $G_{T,\gamma}$ and its
perfect matching. For more details, see [34, Section 4], [9].

Let $T^{o}$ be an ideal triangulation and $\gamma$ be a curve connecting two
marked points. Let $p_{0}$ be the starting point of $\gamma$, and let
$p_{c+1}$ be its endpoint. Assume that $\gamma$ crosses $T^{o}$ at
$p_{1},\cdots,p_{c}$ sequentially.

Let $\tau_{i_{j}}$ be the arc in $T^{o}$ containing $p_{j}$. Let
$\Delta_{j-1}$ and $\Delta_{j}$ be the two ideal triangles in $T$ on either
side of $\tau_{i_{j}}$.

For each $p_{j}$, we associate a _tile_ $G_{j}$ as follows. Define
$\Delta_{1}^{j}$ and $\Delta_{2}^{j}$ to be two triangles with edges labeled
as in $\Delta_{j-1}$ and $\Delta_{j}$, further, the orientations of
$\Delta_{1}^{j}$ and $\Delta_{2}^{j}$ both agree with those of $\Delta_{j-1}$
and $\Delta_{j}$ if $j$ is odd; the orientations of $\Delta_{1}^{j}$ and
$\Delta_{2}^{j}$ both disagree with those of $\Delta_{j-1}$ and $\Delta_{j}$
otherwise. We glue $\Delta_{1}^{j}$ and $\Delta_{2}^{j}$ at the edge labeled
$\tau_{i_{j}}$, so that the orientations of $\Delta_{1}^{j}$ and
$\Delta_{2}^{j}$ both either agree or disagree with those of $\Delta_{j-1}$
and $\Delta_{j}$. We say the edge labeled $\tau_{i_{j}}$ the _diagonal_ of
$G_{j}$.

The two arcs $\tau_{i_{j}}$ and $\tau_{i_{j+1}}$ form two edges of the
triangle $\Delta_{j}$. Denote the third edge of $\Delta_{j}$ by
$\tau_{[\gamma_{j}]}$. After gluing the tiles $G(p_{j})$ and $G(p_{j+1})$ at
the edge labeled $\tau_{[\gamma_{j}]}$ for $1\leq j<d-1$ step by step, we
obtain a graph, denote as $\overline{G_{T^{o},\gamma}}$. Let
$G_{T^{o},\gamma}$ be the graph obtained from $\overline{G_{T^{o},\gamma}}$ by
removing the diagonal of each tile.

In particular, when $\gamma\in T$, let $G_{T^{o},\gamma}$ be the graph with
one only edge labeled $\gamma$.

Denote

$\begin{array}[]{ccl}rel(G_{i},T^{o})=\left\\{\begin{array}[]{ll}1,&\mbox{if
the orientations of $G_{i}$ and $T^{o}$ coincide},\\\ -1,&\mbox{if the
orientations of $G_{i}$ and $T^{o}$ are
different.}\end{array}\right.\end{array}$

######  Definition 2.34.

([34, Definition 4.7]) Let $a_{1}$ and $a_{2}$ be the two edges of
$G_{T^{o},\gamma}$ which lie in the counterclockwise direction from the
diagonal of $G_{1}$. Then the _minimal matching_ $P_{-}$ is defined as the
unique matching which contains only boundary edges and does not contain edges
$a_{1}$ or $a_{2}$. The _maximal matching_ $P_{+}$ is the other matching with
only boundary edges.

######  Lemma 2.35.

[22, Lemma 2.4] Let $a$ be an edge of the tile $G_{j}$. If $a$ is in the
maximal/minimal perfect matching of $G_{T,\gamma}$, then $a$ lies in the
counterclockwise/clockwise direction from the diagonal of $G_{j}$ when $j$ is
odd and lies in the clockwise/counterclockwise direction from the diagonal of
$G_{j}$ when $j$ is even.

###  2.7. Complete $(T^{o},\gamma)$-path

Let $T^{o}$ be an ideal triangulation and $\gamma$ be an arc. Choose an
orientation of $\gamma$, assume that $\overrightarrow{\gamma}$ crosses $T^{o}$
at $p_{1},\cdots,p_{c}$ sequentially. Denote
$p_{0}=s(\overrightarrow{\gamma})$ and $p_{c+1}=t(\overrightarrow{\gamma})$.
Suppose that $p_{j}\in\tau_{i_{j}}$ for $j=1,\cdots,c$.

######  Definition 2.36.

[33, 43] A sequences of oriented arcs
$\overrightarrow{\xi}=(\overrightarrow{\xi}_{1},\cdots,\overrightarrow{\xi}_{2c+1})$
is called a _complete $(T^{o},\gamma)$-path_ if the following axioms hold:

  1. $(T1)$

$\xi_{i}\in T^{o}$ for all $i\in\\{1,\cdots,2c+1\\}$;

  2. $(T2)$

$s(\overrightarrow{\xi}_{1})=s(\overrightarrow{\gamma})$,
$t(\overrightarrow{\xi}_{2c+1})=t(\overrightarrow{\gamma})$ and
$s(\overrightarrow{\xi}_{i+1})=t(\overrightarrow{\xi}_{i})$ for all
$i\in\\{1,\cdots,2c\\}$;

  3. $(T3)$

The even arcs are precisely the arcs crossed by $\gamma$ in order, that is,
$\xi_{2k}=\tau_{i_{k}}$ for all $k\in\\{1,\cdots,c\\}$;

  4. $(T4)$

For all $k=0,1,2,\cdots,c$, the segment $\overrightarrow{\gamma}_{k}$ of
$\overrightarrow{\gamma}$ starting from $p_{k}$ and ending at $p_{k+1}$ is
homotopic to the segment of the path $\overrightarrow{\xi}$ starting at the
point $p_{k}$ following
$\overrightarrow{\xi}_{2k},\overrightarrow{\xi}_{2k+1}$ and
$\overrightarrow{\xi}_{2k+2}$ until the point $p_{k+1}$.

Denote by $\mathcal{CP}(T^{o},\gamma)$ the set of all complete
$(T^{o},\gamma)$-paths.

For any complete $(T^{o},\gamma)$-path
$\overrightarrow{\xi}=(\overrightarrow{\xi}_{1},\cdots,\overrightarrow{\xi}_{2c+1})$
and arc $\zeta\in T^{o}$, denote by

(2)
$m(\overrightarrow{\xi},\zeta)=\sum_{i=1}^{2c+1}(-1)^{i-1}\delta_{\xi_{i},\zeta},$

where

(3) $\delta_{\xi_{i},\zeta}=\begin{cases}1,&\mbox{if $\xi_{i}=\zeta$},\\\
0,&\mbox{if $\xi_{i}\neq\zeta$}.\end{cases}$

By [33, Theorem 4.4], there is a natural bijective map from
$\mathcal{P}(G_{T^{o},\gamma})$ to $\mathcal{CP}(T^{o},\gamma)$. Roughly
speaking, for any perfect matching $P$, by taking the diagonals of all the
tiles, we get a complete path, see the following figure for an illustration.

######  Definition 2.37.

We say that two complete $(T^{o},\gamma)$-paths $\overrightarrow{\xi}$ and
$\overrightarrow{\zeta}$ are related by a _twist at $\alpha\in T^{o}$_ if
there is an even $j$ such that
$\overrightarrow{\xi}_{i}=\overrightarrow{\zeta}_{i}$ for $i\neq j-1,j,j+1$
and
$(\overrightarrow{\xi}_{j-1},\overrightarrow{\zeta}_{{j+1}},\overleftarrow{\xi}_{j+1},\overleftarrow{\zeta}_{j-1})$
is a quadrilateral in $T^{o}$ with diagonal $\alpha$. In this case, we see
that $\xi_{j-1},\xi_{j+1}$ are $\alpha$-twist-able edges in
$\overrightarrow{\xi}$.

##  3\. An isomorphism of quantum cluster algebras

Let $\Sigma$ be an orbifold and $\mathcal{A}_{v}(\Sigma)$ be a quantum cluster
algebra from $\Sigma$. For any puncture $q$ and tagged arc $\beta$, denote by
$\beta^{(q)}$ the tagged arc obtained from $\beta$ by changing the tags at
$q$. Let $T$ be a tagged triangulation of $\Sigma$. Denote by
$T^{(q)}=\\{\beta^{(q)}\mid\beta\in T\\}$. Then $T^{(q)}$ is a tagged
triangulation and $B(T)=B(T^{(q)})$. Suppose that the extended exchange matrix
and the quantum commutative matrix of $\mathcal{A}_{v}(\Sigma)$ at $T$ are
$\widetilde{B}(T)=\left(\begin{array}[]{c}B(T)\\\ C(T)\end{array}\right)$ and
$\Lambda(T)$, respectively. Assume that $\Sigma$ is not a closed surface with
exactly one puncture, let $\mathcal{A}^{(q)}_{v}$ be the quantum cluster
algebra from $\Sigma$ such that the extended exchange matrix and quantum
commutative matrix at $T^{(q)}$ are $\widetilde{B}(T)$ and $\Lambda(T)$,
respectively.

By Proposition 2.26, for any tagged arc $\beta$, denote by $X_{\beta}$ and
$X^{(q)}_{\beta}$ the quantum cluster variables of $\mathcal{A}_{v}(\Sigma)$
and $\mathcal{A}^{(q)}_{v}(\Sigma)$, respectively corresponding to $\beta$.
Then we have the following proposition. Denote by $X_{n+1},\cdots,X_{m}$ the
frozen quantum cluster variables of $\mathcal{A}_{v}(\Sigma)$, denote by
$X^{(q)}_{n+1},\cdots,X^{(q)}_{m}$ the frozen quantum cluster variables of
$\mathcal{A}^{(q)}_{v}(\Sigma)$.

######  Lemma 3.1.

Let $T$ be a tagged triangulation and $q$ be a puncture. For any tagged arc
$\beta\in T$, we have $\mu_{\beta^{(q)}}(T^{(q)})=(\mu_{\beta}(T))^{(q)}.$

###### Proof.

Suppose that $\mu_{\beta}(T)=(T\setminus\\{\beta\\})\cup\\{\beta^{\prime}\\}$
for some tagged arc $\beta^{\prime}$. Then for any $\alpha\in
T\setminus\\{\beta\\}$ we have $\beta^{\prime}$ is compatible with $\alpha$,
by [17, Remark 5.13], $\beta^{\prime(q)}$ is compatible with $\alpha^{(q)}$.
It follows that $\mu_{\beta^{(q)}}(T^{(q)})=(\mu_{\beta}(T))^{(q)}.$ $\square$

######  Proposition 3.2.

Assume that $\Sigma$ is not a closed surface with exactly one puncture. Then
there is a $\mathbb{Z}[v^{\pm 1}]$-algebra isomorphism
$\sigma:\mathcal{A}_{v}(\Sigma)\to\mathcal{A}^{(q)}_{v}(\Sigma)$ which
satisfies

  1. $(1)$

$\sigma(X_{\beta})=X^{(q)}_{\beta^{(q)}}$ for all tagged arc $\beta$.

  2. $(2)$

$\sigma$ preserves mutations.

###### Proof.

For any tagged triangulation $T^{\prime}$, denote by
$\widetilde{B}^{(q)}(T^{\prime})$ and $\Lambda^{(q)}(T^{\prime})$ the extended
exchange matrix and quantum commutative matrix, respectively of
$\mathcal{A}^{(q)}_{v}(\Sigma)$ at $T^{\prime}$, denote by
$\mathcal{T}(T^{\prime})$ and $\mathcal{T}^{(q)}(T^{\prime})$ the quantum
torus at $T^{\prime}$ for $\mathcal{A}_{v}(\Sigma)$ and
$\mathcal{A}^{(q)}_{v}(\Sigma)$, respectively. Then we have
$\widetilde{B}^{(q)}(T^{(q)})=\widetilde{B}(T)$ and
$\Lambda^{(q)}(T^{(q)})=\Lambda(T)$, moreover, for any $\beta\in T$, the
column (resp. row) of $\widetilde{B}(T)$ which is indexed by $X_{\beta}$
equals the column (resp. row) of $\widetilde{B}^{(q)}(T^{(q)})$ which is
indexed by $X_{\beta^{(q)}}$; for any $i\in\\{n+1,\cdots,m\\}$ the row of
$\widetilde{B}(T)$ which is indexed by $X_{\beta}$ equals the row of
$\widetilde{B}^{(q)}(T^{(q)})$ which is indexed by $X_{\beta^{(q)}}$. It is
also true for the quantum commutative matrices $\Lambda(T)$ and
$\Lambda^{(q)}(T^{(q)})$.

Clearly, we have a $\mathbb{Z}[v^{\pm 1}]$-algebra isomorphism of quantum
torus

$\sigma:\mathcal{T}(T)\to\mathcal{T}^{(q)}(T^{(q)}),X_{\alpha}\mapsto
X^{(q)}_{\alpha^{(q)}}.$

For any $\beta\in T$, suppose that
$\mu_{\beta}(T)=(T\setminus\\{\beta\\})\cup\\{\beta^{\prime}\\}$ for some
tagged arc $\beta^{\prime}$. As
$\widetilde{B}^{(q)}(T^{(q)})=\widetilde{B}(T)$, by Lemma 3.1 we have
$\sigma(X_{\beta^{\prime}})=X_{\beta^{\prime(q)}}$, moreover, we have

$\widetilde{B}^{(q)}(\mu_{\beta^{(q)}}T^{(q)})=\mu_{X^{(q)}_{\beta^{(q)}}}(B^{(q)}(T^{(q)}))=\mu_{X_{\beta}}(B(T))=B(\mu_{\beta}T).$

Similarly, we have
$\Lambda^{(q)}(\mu_{\beta^{(q)}}T^{(q)})=\Lambda(\mu_{\beta}T).$

Therefore, $\sigma$ induces a $\mathbb{Z}[v^{\pm 1}]$-algebra isomorphism of
quantum torus

$\sigma:\mathcal{T}(\mu_{\tau}T)\to\mathcal{T}^{(q)}(\mu_{\tau^{(q)}}T^{(q)}),X_{\alpha}\mapsto
X^{(q)}_{\alpha^{(q)}}.$

As $\Sigma$ is not a closed surface with exactly one puncture, any two tagged
triangulations of $\Sigma$ are connected by a sequence of flips. By induction
we see that for any tagged triangulation $T^{\prime}$, $\sigma$ induces a
$\mathbb{Z}[v^{\pm 1}]$-algebra isomorphism of quantum torus

$\sigma:\mathcal{T}(T^{\prime})\to\mathcal{T}^{(q)}(T^{\prime(q)}),X_{\alpha}\mapsto
X^{(q)}_{\alpha^{(q)}}.$

The result follows. $\square$

######  Remark 3.3.

Let $T^{o}$ be an ideal triangulation of $\Sigma$ and $\beta$ be any arc.
Assume that $\widetilde{\beta}$ starts from $p$ and ends at $q$. Let $T$ be
the tagged triangulation corresponding to $T^{o}$. By Proposition 3.2, to give
an expansion formula of $X_{\gamma}$ concerning a quantum seed $X_{\Delta}$
for any tagged arc $\gamma$ and tagged triangulation $\Delta$, it suffices to
restrict to the following three cases:

  1. $(i)$

$\gamma=\widetilde{\beta}$ and $\Delta=T$;

  2. $(ii)$

$q$ is a puncture with $q\neq p$, $\gamma=\widetilde{\beta}^{(q)}$, $\Delta=T$
contains no arcs tagged notched at $q$;

  3. $(iii)$

$p,q$ are punctures, $\gamma=\widetilde{\beta}^{(p,q)}$, $\Delta=T$ contains
no arcs tagged notched at $p$ or $q$.

##  4\. Three lattices associated with tagged arcs

Fix an ideal triangulation $T^{o}$ of $\Sigma$ and an arc $\beta$. Let $T$ be
the corresponding tagged triangulation of $T^{o}$. Let

(4) $\widetilde{\beta}=\begin{cases}\beta,&\mbox{if $\beta$ is not a pending
arc of weight $1/2$},\vspace{1mm}\\\ sl(\beta),&\mbox{if $\beta$ is a pending
arc of weight $1/2$ }.\end{cases}$

Denote by $G_{1},\cdots,G_{c}$ the tiles of $G_{T^{o},\widetilde{\beta}}$ in
order. Assume that $\widetilde{\beta}$ starts from $p$ and ends at $q$. In
this section, we recall the three lattices
$\mathcal{L}(T^{o},\widetilde{\beta})$,
$\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$ and
$\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$ constructed in [24], which will
be the index sets for our expansion formulas in the quantum case. These three
lattices also appear in [1, 22] for providing expansion formulas for tagged
curves in (non-commutative) cluster algebras from $\Sigma$.

###  4.1. $\mathcal{L}(T^{o},\widetilde{\beta})$

Let
$\mathcal{L}(T^{o},\widetilde{\beta})=\mathcal{P}(G_{T^{o},\widetilde{\beta}})$.
For any $P\in\mathcal{L}(T^{o},\widetilde{\beta})$ can twist on a tile
$G_{i}$, let $P<\mu_{G_{i}}(P)$ if $W(G_{i}),E(G_{i})\in P,rel(G_{i},T^{o})=1$
or $N(G_{i}),S(G_{i})\in P,rel(G_{i},T^{o})=-1$.

######  Proposition 4.1.

[10, 35] $\mathcal{L}(T^{o},\widetilde{\beta})$ with above order forms a
lattice.

The following proposition is an immediate consequence.

######  Proposition 4.2.

  1. $(1)$

The maximum element $P_{+}$ in $\mathcal{P}(G_{T^{o},\widetilde{\beta}})$ is
the perfect matching contains only boundary edges such that for any $e\in
P_{+}\cap edge(G_{i})$ we have $e\in\\{N(G_{i}),S(G_{i})\\}$ if
$rel(G_{i},T^{o})=1$ and $e\in\\{W(G_{i}),E(G_{i})\\}$ if
$rel(G_{i},T^{o})=-1$;

  2. $(2)$

The minimum element $P_{-}$ in $\mathcal{P}(G_{T^{o},\widetilde{\beta}})$ is
the perfect matching contains only boundary edges such that for any $e\in
P_{+}\cap edge(G_{i})$ we have $e\in\\{W(G_{i}),E(G_{i})\\}$ if
$rel(G_{i},T^{o})=1$ and $e\in\\{N(G_{i}),S(G_{i})\\}$ if
$rel(G_{i},T^{o})=-1$.

It follows that any two perfect matchings are related by a sequence of twists.

######  Lemma 4.3.

The Hasse graph of $\mathcal{L}(T^{o},\widetilde{\beta})$ is connected.

###  4.2. $\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$

Herein we assume that $q$ is a puncture in $\Sigma$ and $q\neq p$. We label
clockwise the triangles in $T^{o}$ incident to $q$ by
$\Delta_{1}(q),\Delta_{2}(q),\cdots,\Delta_{t}(q)$ such that either
$\widetilde{\beta}$ crosses $\Delta_{1}(q)$ or $\widetilde{\beta}$ is the
common side of $\Delta_{1}(q)$ and $\Delta_{t}(q)$, see Figure 4. Denote

$\Delta(T^{o},q)=\\{\Delta_{1}(q),\Delta_{2}(q),\cdots,\Delta_{t}(q)\\}.$

Figure 4. Triangles incident to $q$

Let

(5)
$\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})=\mathcal{P}(G_{T^{o},\widetilde{\beta}})\times\\{\Delta_{1}(q),\cdots,\Delta_{t}(q)\\}.$

Set $\Delta_{t+1}(q)=\Delta_{1}(q)$. Denote by $\tau_{j}(q)$ the common side
of $\Delta_{j}(q)$ and $\Delta_{j+1}(q)$ for $j=1,\cdots,t$. Let
$\tau_{0}(q)=\tau_{t}(q)$. Thus $\tau_{j-1}(q)$ and $\tau_{j}(q)$ are two
sides of $\Delta_{j}(q)$. Denote by $\tau_{[j]}(q)$ the third side of
$\Delta_{j}(q)$. We have $\tau_{t}(q)=\widetilde{\beta}$ when
$\widetilde{\beta}\in T^{o}$.

####  4.2.1. $\widetilde{\beta}\notin T^{o}$

Herein we consider the case that $\widetilde{\beta}\notin T^{o}$. Define

(6) $E_{1}(q)=\begin{cases}E(G_{c}),&\mbox{if
$rel(G_{c},T^{o})=1$},\vspace{1mm}\\\ N(G_{c}),&\mbox{if
$rel(G_{c},T^{o})=-1$,}\end{cases}\;\;\;E_{2}(q)=\begin{cases}N(G_{c}),&\mbox{if
$rel(G_{c},T^{o})=1$},\vspace{1mm}\\\ E(G_{c}),&\mbox{if
$rel(G_{c},T^{o})=-1$.}\end{cases}$

Thus $E_{1}(q)$ is labeled $\tau_{t}(q)$ and $E_{2}(q)$ is labeled
$\tau_{1}(q)$. By Proposition 4.2, we have $E_{1}(q)\in P_{-}$ and
$E_{2}(q)\in P_{+}$.

For any $P\in\mathcal{P}(G_{T^{o},\widetilde{\beta}})$, it is clear that
either $E_{1}(q)\in P$ or $E_{2}(q)\in P$.

It is proved in [24] that $\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$ forms a
lattice with minimum element $(P_{-},\Delta_{1}(q))$ and maximum element
$(P_{+},\Delta_{1}(q))$, under the partial order induced by the following.

  1. (i)

For any $P\in\mathcal{P}(G_{T^{o},\widetilde{\beta}})$,

    1. (a)

if $E_{1}(q)\in P$ then

$(P,\Delta_{1}(q))<(P,\Delta_{2}(q))<(P,\Delta_{3}(q))<\cdots<(P,\Delta_{t}(q));$

    2. (b)

if $E_{2}(q)\in P$ then

$(P,\Delta_{2}(q))<\cdots<(P,\Delta_{t-1}(q))<(P,\Delta_{t}(q))<(P,\Delta_{1}(q)).$

  2. (ii)

For any $j\in\\{1,\cdots,t\\}$, $(P,\Delta_{j}(q))<(Q,\Delta_{j}(q))$ if
$P<Q$.

The following lemmas are immediate.

######  Lemma 4.4.

Let $P$ be a perfect matching which can twist on $G_{l}$ such that
$P>\mu_{G_{l}}P$. Then for any $\Delta_{j}(q)$, we have $(P,\Delta_{j}(q))$
covers $(\mu_{G_{l}}P,\Delta_{j}(q))$ unless $j=1$ and $l=c$.

######  Lemma 4.5.

For any $(P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$,

  1. $(1)$

if $E_{1}(q)\in P$, then $(P,\Delta_{j}(q))$ covers $(P,\Delta_{j-1}(q))$ for
all $j\neq 1$;

  2. $(2)$

if $E_{2}(q)\in P$, then $(P,\Delta_{j}(q))$ covers $(P,\Delta_{j-1}(q))$ for
all $j\neq 2$.

######  Example 4.6.

In Figure 5, we show an example of the Hasse graph of
$\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$.

Figure 5. Hasse graph of $\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$

####  4.2.2. $\widetilde{\beta}\in T^{o}$.

Herein we consider the case that $\widetilde{\beta}\in T^{o}$. It is clear
that $\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$ forms a lattice under the
following partial order.

$(P_{\widetilde{\beta}},\Delta_{1}(q))<(P_{\widetilde{\beta}},\Delta_{2}(q))<\cdots<(P_{\widetilde{\beta}},\Delta_{t-1}(q))<(P_{\widetilde{\beta}},\Delta_{t}(q)).$

The following Lemma follows by Lemma 4.3.

######  Lemma 4.7.

The Hasse graph of $\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$ is connected.

######  Lemma 4.8.

Assume that ${\bf P}=(P,\Delta_{j}(q))$ covers ${\bf
Q}=(\mu_{G_{l}}P,\Delta_{j}(q))$ such that
$\tau_{i_{l}}=\alpha_{k}\neq\alpha_{k-1},\alpha_{k+1}$ for some
$k\in\\{1,2,3,4\\}$.

  1. $(1)$

In case $k\in\\{1,3\\}$, if ${\bf P}_{1}=(P,\Delta_{j+1}(q))$ covers ${\bf P}$
with $\tau_{j}(q)=\alpha$, then ${\bf Q}_{1}=(\mu_{G_{l}}P,\Delta_{j+1}(q))$
covers ${\bf Q}=(\mu_{G_{l}}P,\Delta_{j}(q))$ and ${\bf P}_{1}$ covers ${\bf
Q}_{1}$.

  2. $(2)$

In case $k\in\\{2,4\\}$, if ${\bf P}$ covers ${\bf P}_{1}=(P,\Delta_{j-1}(q))$
with $\tau_{j-1}(q)=\alpha$, then ${\bf Q}=(\mu_{G_{l}}P,\Delta_{j}(q))$
covers ${\bf Q}_{1}=(\mu_{G_{l}}P,\Delta_{j-1}(q))$ covers and ${\bf P}_{1}$
covers ${\bf Q}_{1}$.

###### Proof.

We shall only prove (1). We may assume that $k=1$. We first prove ${\bf
Q}_{1}$ covers ${\bf Q}$. Otherwise, by Lemma 4.5, we have either
$E_{1}(q)\in\mu_{G_{l}}P,\Delta_{j+1}(q)=\Delta_{1}(q)$ or
$E_{2}(q)\in\mu_{G_{l}}P,\Delta_{j}(q)=\Delta_{1}(q)$. In case
$E_{1}(q)\in\mu_{G_{l}}P,\Delta_{j+1}(q)=\Delta_{1}(q)$, as ${\bf P}_{1}$
covers ${\bf P}$ we have $E_{2}(q)\in P$ by Lemma 4.5. Thus $G_{l}=G_{c}$. As
$\tau_{i_{c}}=\alpha_{1}$, we have $\tau_{t}(q)=\alpha_{4}$. It contradicts to
$\tau_{j}(q)=\tau_{t}(q)=\alpha$. We then prove ${\bf P}_{1}$ covers ${\bf
Q}_{1}$. Otherwise, by Lemma 4.4, we have $G_{l}=G_{c}$ and
$\Delta_{j+1}(q)=\Delta_{1}(q)$, it is impossible from the above discussion.
$\square$

######  Lemma 4.9.

Assume that ${\bf P}=(P,\Delta_{j}(q))$ covers ${\bf
Q}=(\mu_{G_{l}}P,\Delta_{j}(q))$ such that
$\tau_{i_{l}}=\alpha_{k}\neq\alpha_{k-1},\alpha_{k+1}$ for some
$k\in\\{1,2,3,4\\}$.

  1. $(1)$

In case $k\in\\{1,3\\}$, if $\tau_{j}(q)=\alpha$ but $(P,\Delta_{j+1}(q))$
does not cover ${\bf P}$, then $G_{l}\neq G_{c}$ and $m_{\eta_{\alpha}}({\bf
P})=m_{\eta_{\alpha}}({\bf Q})=1$.

  2. $(2)$

In case $k\in\\{2,4\\}$, if $\tau_{j-1}(q)=\alpha$ but ${\bf P}$ does not
cover $(P,\Delta_{j-1}(q))$, then $G_{l}\neq G_{c}$ and
$m_{\eta_{\alpha}}({\bf P})=m_{\eta_{\alpha}}({\bf Q})=1$.

###### Proof.

We shall only prove one. As ${\bf P}_{1}=(P,\Delta_{j+1}(q))$ does not cover
${\bf P}$, by Lemma 4.5, we have either $E_{1}(q)\in
P,\Delta_{j+1}(q)=\Delta_{1}(q)$ or $E_{2}(q)\in
P,\Delta_{j}(q)=\Delta_{1}(q)$.

In case $E_{1}(q)\in P,\Delta_{j+1}(q)=\Delta_{1}(q)$, as
$\tau_{j}(q)=\alpha$, we have $\tau_{[1]}(q)=\alpha_{2}$ or $\alpha_{4}$. Thus
$\tau_{i_{c}}=\tau_{[1]}(q)=\alpha_{2}$ or $\alpha_{4}$. It follows that
$E_{1}(q)$ is labeled $\alpha$ and thus $m_{\eta_{\alpha}}({\bf P})=1$.
$E_{1}(q)\in P$ implies $G_{l}\neq G_{c}$. Thus, $m_{\eta_{\alpha}}({\bf
Q})=1$.

In case $E_{2}(q)\in P,\Delta_{j}(q)=\Delta_{1}(q)$, as $\tau_{j}(q)=\alpha$,
we have $\tau_{[1]}(q)=\alpha_{1}$ or $\alpha_{3}$. Thus
$\tau_{i_{c}}=\tau_{[1]}(q)=\alpha_{1}$ or $\alpha_{3}$. It follows that
$E_{2}(q)$ is labeled $\alpha$ and thus $m_{\eta_{\alpha}}({\bf P})=1$. As
${\bf P}=(P,\Delta_{j}(q))$ covers ${\bf Q}=(\mu_{G_{l}}P,\Delta_{j}(q))$,
$G_{l}\neq G_{c}$. Thus, $m_{\eta_{\alpha}}({\bf Q})=1$.

From $\tau_{j}(q)=\alpha$, we see that $m_{\eta_{\alpha}+1}({\bf P})=-1$.
Therefore, in both cases we have $(\eta_{\alpha},\eta_{\alpha}+1)$ is an ${\bf
m}({\bf P};\alpha)$ and ${\bf m}({\bf Q};\alpha)$-pair. $\square$

######  Lemma 4.10.

Assume that ${\bf P}=(P,\Delta_{j+1}(q))$ covers ${\bf Q}=(P,\Delta_{j}(q))$
such that $\tau_{j}(q)=\alpha_{k}\neq\alpha_{k-1},\alpha_{k+1}$ for some
$k\in\\{1,2,3,4\\}$. Assume that $P$ can twist on some $G_{l}$ with
$\tau_{i_{l}}=\alpha$.

  1. $(1)$

In case $k\in\\{1,3\\}$, if $P<\mu_{G_{l}}P$, then
$(\mu_{G_{l}}P,\Delta_{j+1}(q))$ covers ${\bf P}$ and
$(\mu_{G_{l}}P,\Delta_{j}(q))$, $(\mu_{G_{l}}P,\Delta_{j}(q))$ covers ${\bf
Q}$.

  2. $(2)$

In case $k\in\\{2,4\\}$, if $P>\mu_{G_{l}}P$, then ${\bf P}$ covers
$(\mu_{G_{l}}P,\Delta_{j+1}(q))$, and $(\mu_{G_{l}}P,\Delta_{j}(q))$ is
covered by ${\bf Q}$ and $(\mu_{G_{l}}P,\Delta_{j+1}(q))$.

###### Proof.

We shall only prove (1).

We first prove $(\mu_{G_{l}}P,\Delta_{j+1}(q))$ covers ${\bf P}$. Otherwise,
by Lemma 4.4 we have $G_{l}=G_{c}$ and $\Delta_{j+1}(q)=\Delta_{1}(q)$. As
$P<\mu_{G_{l}}P$, we see that $E_{1}(q)\in P$. Thus ${\bf
P}=(P,\Delta_{1}(q))$ does not cover ${\bf Q}=(P,\Delta_{t}(q))$, a
contradiction.

We then prove $(\mu_{G_{l}}P,\Delta_{j+1}(q))$ covers
$(\mu_{G_{l}}P,\Delta_{j}(q))$. Otherwise, by Lemma 4.5 we have either
$E_{1}(q)\in\mu_{G_{l}}P,\Delta_{j+1}(q)=\Delta_{1}(q)$ or
$E_{2}(q)\in\mu_{G_{l}}P,\Delta_{j}(q)=\Delta_{1}(q)$. If
$E_{1}(q)\in\mu_{G_{l}}P,\Delta_{j+1}(q)=\Delta_{1}(q)$, as $P<\mu_{G_{l}}P$,
we have $E_{1}(q)\in P,\Delta_{j+1}(q)=\Delta_{1}(q)$, it is contradictions to
${\bf P}$ covers to ${\bf Q}$. If
$E_{2}(q)\in\mu_{G_{l}}P,\Delta_{j}(q)=\Delta_{1}(q)$, as ${\bf P}$ covers to
${\bf Q}$, we see that $G_{l}=G_{c}$. Thus $\tau_{[j]}(q)=\alpha$ and hence
$\tau_{j}(q)\in\\{\alpha_{2},\alpha_{4}\\}$, contradicts to
$\tau_{j}(q)\in\\{\alpha_{1},\alpha_{3}\\}$.

Last, we prove $(\mu_{G_{l}}P,\Delta_{j}(q))$ covers ${\bf Q}$. Otherwise, by
Lemma 4.4 we have $G_{l}=G_{c}$ and $\Delta_{j}(q)=\Delta_{1}(q)$. Thus
$\tau_{[j]}(q)=\alpha$ and hence $\tau_{j}(q)\in\\{\alpha_{2},\alpha_{4}\\}$,
contradicts to $\tau_{j}(q)\in\\{\alpha_{1},\alpha_{3}\\}$.

The proof is complete. $\square$

###  4.3. $\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$

In this section, we assume that $p,q$ are punctures. As the once notched case,
we label clockwise the triangles in $T^{o}$ incident to $p$ by
$\Delta_{1}(p),\Delta_{2}(p),\cdots,\Delta_{s}(p)$ such that either
$\widetilde{\beta}$ crosses $\Delta_{1}(p)$ or $\widetilde{\beta}$ is the
common side of $\Delta_{1}(p)$ and $\Delta_{s}(p)$. See Figure 6. Denote

$\Delta(T^{o},p)=\\{\Delta_{1}(p),\Delta_{2}(p),\cdots,\Delta_{s}(p)\\}.$

Set $\Delta_{s+1}(p)=\Delta_{1}(p)$. Denote by $\tau_{i}(p)$ the common side
of the triangles $\Delta_{i}(p)$ and $\Delta_{i+1}(p)$ for $i=1,\cdots,s$. Set
$\tau_{0}(p)=\tau_{s}(p)$. Thus $\tau_{i-1}(p)$ and $\tau_{i}(p)$ are two
sides of $\Delta_{i}(p)$. Denote by $\tau_{[i]}(p)$ the third edge of
$\Delta_{i}(p)$.

In particular, if $\widetilde{\beta}\in T^{o}$ then
$\tau_{s}(p)=\tau_{t}(q)=\widetilde{\beta}$,
$\tau_{[s]}(p)=\tau_{1}(q),\tau_{[t]}(q)=\tau_{1}(p)$.

Figure 6. Triangles incident to $p$ and $q$

Let

(7)
$\begin{array}[]{rcl}\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})&=&\\{\Delta_{1}(p),\cdots,\Delta_{s}(p)\\}\times\mathcal{P}(G_{T^{o},\widetilde{\beta}})\times\\{\Delta_{1}(q),\cdots,\Delta_{t}(q)\\}\vspace{2mm}\\\
&=&\\{\Delta_{1}(p),\cdots,\Delta_{s}(p)\\}\times\mathcal{L}(T^{o},\widetilde{\beta}^{(q)}).\end{array}$

####  4.3.1. $\widetilde{\beta}\notin T^{o}$.

Herein we suppose that $\widetilde{\beta}\notin T^{o}$. Similar to (6) in
Section 4.2, define

(8) $E_{1}(p)=\begin{cases}W(G_{1}),&\mbox{if
$rel(G_{1},T^{o})=1$},\vspace{1mm}\\\ S(G_{1}),&\mbox{if
$rel(G_{1},T^{o})=-1$,}\end{cases}\;\;\;E_{2}(p)=\begin{cases}S(G_{1}),&\mbox{if
$rel(G_{1},T^{o})=1$},\vspace{1mm}\\\ W(G_{1}),&\mbox{if
$rel(G_{1},T^{o})=-1$.}\end{cases}$

Then $E_{1}(p)$ is labeled $\tau_{s}(p)$ and $E_{2}(p)$ is labeled
$\tau_{1}(p)$. By Proposition 4.2, we have $E_{1}(p)\in P_{-}$ and
$E_{2}(p)\in P_{+}$.

For any $P\in\mathcal{P}(G_{T^{o},\beta})$, it is clear that either
$E_{1}(p)\in P$ or $E_{2}(p)\in P$.

It is proved in [24] that $\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$ forms
a lattice with minimum element $(\Delta_{1}(p),P_{-},\Delta_{1}(q))$ and
maximum element $(\Delta_{1}(p),P_{+},\Delta_{1}(q))$, under the partial order
induced by the following relations.

  1. (i)

For any
$(P,\Delta_{j}(q))\in\mathcal{P}(G_{T^{o},\widetilde{\beta}})\times\Delta(T^{o},q)$,

    1. (a)

if $E_{1}(p)\in P$ then

$(\Delta_{1}(q),P,\Delta_{j}(q))<(\Delta_{2}(q),P,\Delta_{j}(q))<\cdots<(\Delta_{s}(q),P,\Delta_{j}(q));$

    2. (b)

if $E_{2}(q)\in P$ then

$(\Delta_{2}(q),P,\Delta_{j}(q))<(\Delta_{3}(q),P,\Delta_{j}(q))<\cdots<(\Delta_{s}(q),P,\Delta_{j}(q))<(\Delta_{1}(q),P,\Delta_{j}(q));$

  2. (ii)

For any $i\in\\{1,\cdots,t\\}$,
$(\Delta_{i}(p),P,\Delta_{a}(q))<(\Delta_{i}(p),Q,\Delta_{b}(q))$ if
$(P,\Delta_{a}(q))<(Q,\Delta_{b}(q))$ in
$\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$.

The following two lemmas follow immediately by the partial order defined
above, Lemma 4.4 and Lemma 4.5.

######  Lemma 4.11.

For any $P$ can twist on $G_{l}$ with $\mu_{G_{l}}P>P$, then for any
$\Delta_{i}(p)$ and $\Delta_{j}(q)$, we have
$(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q))$ covers
$(\Delta_{i}(p),P,\Delta_{j}(q))$ unless $i=1,l=1$ or $j=1,l=c$.

######  Lemma 4.12.

For any
$(\Delta_{i}(p),P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$,

  1. $(1)$

if $E_{1}(q)\in P$, then $(\Delta_{i}(p),P,\Delta_{j}(q))$ covers
$(\Delta_{i}(p),P,\Delta_{j-1}(q))$ for all $j\neq 1$;

  2. $(2)$

if $E_{2}(q)\in P$, then $(\Delta_{i}(p),P,\Delta_{j}(q))$ covers
$(\Delta_{i}(p),P,\Delta_{j-1}(q))$ for all $j\neq 2$;

  3. $(3)$

if $(\Delta_{i}(p),P,\Delta_{j}(q))>(\Delta_{i}(p),P,\Delta_{j-1}(q))$ then
$(\Delta_{i}(p),P,\Delta_{j}(q))$ covers $(\Delta_{i}(p),P,\Delta_{j-1}(q))$;

  4. $(4)$

if $E_{1}(p)\in P$, then $(\Delta_{i}(p),P,\Delta_{j}(q))$ covers
$(\Delta_{i-1}(p),P,\Delta_{j}(q))$ for all $i\neq 1$;

  5. $(5)$

if $E_{2}(p)\in P$, then $(\Delta_{i}(p),P,\Delta_{j}(q))$ covers
$(\Delta_{i-1}(p),P,\Delta_{j}(q))$ for all $i\neq 2$;

  6. $(6)$

if $(\Delta_{i}(p),P,\Delta_{j}(q))>(\Delta_{i-1}(p),P,\Delta_{j}(q))$ then
$(\Delta_{i}(p),P,\Delta_{j}(q))$ covers $(\Delta_{i-1}(p),P,\Delta_{j}(q))$.

See the following example of the Hasse diagram for
$\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$, where $\Sigma$ is the twice
punctured digon and $T^{o}$ is shown in the left below Figure.

Figure 7. Hasse diagram of $\mathcal{L}(T^{o},\widetilde{\beta}^{(0,1)})$

####  4.3.2. $\widetilde{\beta}\in T^{o}$

Herein we suppose that $\widetilde{\beta}\in T^{o}$.

$\bullet$ If $s,t\geq 2$, then $\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$
forms a lattice with the following order.

  1. (i)

$(\Delta_{1}(p),P_{\widetilde{\beta}},\Delta_{t}(q))<(\Delta_{1}(p),P_{\widetilde{\beta}},\Delta_{1}(q))<\cdots<(\Delta_{1}(p),P_{\widetilde{\beta}},\Delta_{t-2}(q))<(\Delta_{1}(p),P_{\widetilde{\beta}},\Delta_{t-1}(q)).$

  2. (ii)

For $i=2,\cdots,s-1$,

$(\Delta_{i}(p),P_{\widetilde{\beta}},\Delta_{1}(q))<(\Delta_{i}(p),P_{\widetilde{\beta}},\Delta_{2}(q))<\cdots<(\Delta_{i}(p),P_{\widetilde{\beta}},\Delta_{t-1}(q))<(\Delta_{i}(p),P_{\widetilde{\beta}},\Delta_{t}(q)).$

  3. (iii)

$(\Delta_{s}(p),P_{\widetilde{\beta}},\Delta_{2}(q))<(\Delta_{s}(p),P_{\widetilde{\beta}},\Delta_{3}(q))<\cdots<(\Delta_{s}(p),P_{\widetilde{\beta}},\Delta_{t}(q))<(\Delta_{s}(p),P_{\widetilde{\beta}},\Delta_{1}(q)).$

  4. (iv)

For $j=1,\cdots,t$,

$(\Delta_{1}(p),P_{\widetilde{\beta}},\Delta_{j}(q))<(\Delta_{2}(p),P_{\widetilde{\beta}},\Delta_{j}(q))<\cdots<(\Delta_{s-1}(p),P_{\widetilde{\beta}},\Delta_{j}(q))<(\Delta_{s}(p),P_{\widetilde{\beta}},\Delta_{j}(q)).$

See the Hasse diagram of $\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$ in
Figure 8.

Figure 8. Hasse diagram of $\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$ in
case $\widetilde{\beta}\in T^{o}$ and $s\geq 2$

$\bullet$ If $s=1$ or $t=1$, we may assume that $s=1$, then
$\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$ forms a lattice with the
following order.

$(\Delta_{1}(p),P_{\widetilde{\beta}},\Delta_{t}(q))<(\Delta_{1}(p),P_{\widetilde{\beta}},\Delta_{1}(q))<\cdots<(\Delta_{1}(p),P_{\widetilde{\beta}},\Delta_{t-2}(q))<(\Delta_{1}(p),P_{\widetilde{\beta}},\Delta_{t-1}(q)).$

The following lemma follows by Lemma 4.7.

######  Lemma 4.13.

The Hasse graph of $\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$ is
connected.

The following two lemmas are similar to Lemma 4.9.

######  Lemma 4.14.

Assume that ${\bf P}=(\Delta_{i}(p),P,\Delta_{j}(q))$ covers ${\bf
Q}=(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q))$ such that
$\tau_{i_{l}}=\alpha_{k}\neq\alpha_{k-1},\alpha_{k+1}$ for some
$k\in\\{1,2,3,4\\}$.

  1. $(1)$

In case $k\in\\{1,3\\}$, if $\tau_{i}(p)=\alpha$ but
$(\Delta_{i+1}(p),P,\Delta_{j}(q))$ does not cover ${\bf P}$, then $G_{l}\neq
G_{1}$ and $m_{1}({\bf P})=m_{1}({\bf Q})=1$.

  2. $(2)$

In case $k\in\\{1,3\\}$, if $\tau_{j}(q)=\alpha$ but
$(\Delta_{i}(p),P,\Delta_{j+1}(q))$ does not cover ${\bf P}$, then $G_{l}\neq
G_{c}$ and $m_{\eta_{\alpha}}({\bf P})=m_{\eta_{\alpha}}({\bf Q})=1$.

  3. $(3)$

In case $k\in\\{2,4\\}$, if $\tau_{i-1}(p)=\alpha$ but ${\bf P}$ does not
cover $(\Delta_{i-1}(p),P,\Delta_{j}(q))$, then $G_{l}\neq G_{1}$ and
$m_{1}({\bf P})=m_{1}({\bf Q})=1$.

  4. $(4)$

In case $k\in\\{2,4\\}$, if $\tau_{j-1}(q)=\alpha$ but ${\bf P}$ does not
cover $(\Delta_{i}(p),P,\Delta_{j-1}(q))$, then $G_{l}\neq G_{c}$ and
$m_{\eta_{\alpha}}({\bf P})=m_{\eta_{\alpha}}({\bf Q})=1$.

######  Lemma 4.15.

Assume that ${\bf P}=(\Delta_{i+1}(p),P,\Delta_{j}(q))$ covers ${\bf
Q}=(\Delta_{i}(p),P,\Delta_{j}(q))$ such that
$\tau_{i}(p)=\alpha_{k}\neq\alpha_{k-1},\alpha_{k+1}$ for some
$k\in\\{1,2,3,4\\}$.

  1. $(1)$

In case $k\in\\{1,3\\}$, if $\tau_{j}(q)=\alpha$ but
$(\Delta_{i+1}(p),P,\Delta_{j+1}(q))$ does not cover ${\bf P}$, then
$m_{\eta_{\alpha}}({\bf P})=m_{\eta_{\alpha}}({\bf Q})=1$.

  2. $(2)$

In case $k\in\\{2,4\\}$, if $\tau_{j-1}(q)=\alpha$ but ${\bf P}$ does not
cover $(\Delta_{i+1}(p),P,\Delta_{j-1}(q))$, then $m_{\eta_{\alpha}}({\bf
P})=m_{\eta_{\alpha}}({\bf Q})=1$.

######  Lemma 4.16.

Assume that ${\bf P}=(\Delta_{i}(p),P,\Delta_{j+1}(q))$ covers ${\bf
Q}=(\Delta_{i}(p),P,\Delta_{j}(q))$ such that
$\tau_{j}(q)=\alpha_{k}\neq\alpha_{k-1},\alpha_{k+1}$ for some
$k\in\\{1,2,3,4\\}$.

  1. $(1)$

In case $k\in\\{1,3\\}$, if $\tau_{i}(p)=\alpha$ but
$(\Delta_{i+1}(p),P,\Delta_{j+1}(q))$ does not cover ${\bf P}$, then
$m_{1}({\bf P})=m_{1}({\bf Q})=1$.

  2. $(2)$

In case $k\in\\{2,4\\}$, if $\tau_{i-1}(p)=\alpha$ but ${\bf P}$ does not
cover $(\Delta_{i-1}(p),P,\Delta_{j+1}(q))$, then $m_{1}({\bf P})=m_{1}({\bf
Q})=1$.

The following two lemmas can be proved similar to Lemma 4.8.

######  Lemma 4.17.

Assume that ${\bf P}=(\Delta_{i+1}(p),P,\Delta_{j}(q))$ covers ${\bf
Q}=(\Delta_{i}(p),P,\Delta_{j}(q))$ such that
$\tau_{i}(p)=\alpha_{k}\neq\alpha_{k-1},\alpha_{k+1}$ for some
$k\in\\{1,2,3,4\\}$.

  1. $(1)$

In case $k\in\\{1,3\\}$, if $(\Delta_{i+1}(p),P,\Delta_{j+1}(q))$ covers ${\bf
P}$ with $\tau_{j}(q)=\alpha$, then $(\Delta_{i+1}(p),P,\Delta_{j+1}(q))$
covers $(\Delta_{i}(p),P,\Delta_{j+1}(q))$ and
$(\Delta_{i}(p),P,\Delta_{j+1}(q))$ covers ${\bf Q}$.

  2. $(2)$

In case $k\in\\{2,4\\}$, if ${\bf P}$ covers
$(\Delta_{i+1}(p),P,\Delta_{j-1}(q))$ with $\tau_{j-1}(q)=\alpha$, then
$(\Delta_{i}(p),P,\Delta_{j-1}(q))$ is covered by ${\bf Q}$ and
$(\Delta_{i+1}(p),P,\Delta_{j-1}(q))$.

######  Lemma 4.18.

Assume that ${\bf P}=(\Delta_{i}(p),P,\Delta_{j+1}(q))$ covers ${\bf
Q}=(\Delta_{i}(p),P,\Delta_{j}(q))$ such that
$\tau_{j}(q)=\alpha_{k}\neq\alpha_{k-1},\alpha_{k+1}$ for some
$k\in\\{1,2,3,4\\}$.

  1. $(1)$

In case $k\in\\{1,3\\}$, if $(\Delta_{i+1}(p),P,\Delta_{j+1}(q))$ covers ${\bf
P}$ with $\tau_{i}(p)=\alpha$, then $(\Delta_{i+1}(p),P,\Delta_{j+1}(q))$
covers $(\Delta_{i+1}(p),P,\Delta_{j}(q))$ and
$(\Delta_{i+1}(p),P,\Delta_{j}(q))$ covers ${\bf Q}$.

  2. $(2)$

In case $k\in\\{2,4\\}$, if ${\bf P}$ covers
$(\Delta_{i-1}(p),P,\Delta_{j+1}(q))$ with $\tau_{i-1}(p)=\alpha$, then
$(\Delta_{i-1}(p),P,\Delta_{j}(q))$ is covered by ${\bf Q}$ and
$(\Delta_{i-1}(p),P,\Delta_{j+1}(q))$.

The following two lemmas can be proved similar to Lemma 4.10.

######  Lemma 4.19.

Assume that ${\bf P}=(\Delta_{i}(p),P,\Delta_{j+1}(q))$ covers ${\bf
Q}=(\Delta_{i}(p),P,\Delta_{j}(q))$ such that
$\tau_{j}(q)=\alpha_{k}\neq\alpha_{k-1},\alpha_{k+1}$ for some
$k\in\\{1,2,3,4\\}$. Assume that $P$ can twist on some $G_{l}$ with
$\tau_{i_{l}}=\alpha$.

  1. $(1)$

In case $k\in\\{1,3\\}$, if $P<\mu_{G_{l}}P$, then
$(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j+1}(q))$ covers ${\bf P}$ and
$(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q))$,
$(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q))$ covers ${\bf Q}$.

  2. $(2)$

In case $k\in\\{2,4\\}$, if $P>\mu_{G_{l}}P$, then ${\bf P}$ covers
$(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j+1}(q))$, and
$(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q))$ is covered by ${\bf Q}$ and
$(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j+1}(q))$.

######  Lemma 4.20.

Assume that ${\bf P}=(\Delta_{i+1}(p),P,\Delta_{j}(q))$ covers ${\bf
Q}=(\Delta_{i}(p),P,\Delta_{j}(q))$ such that
$\tau_{i}(p)=\alpha_{k}\neq\alpha_{k-1},\alpha_{k+1}$ for some
$k\in\\{1,2,3,4\\}$. Assume that $P$ can twist on some $G_{l}$ with
$\tau_{i_{l}}=\alpha$.

  1. $(1)$

In case $k\in\\{1,3\\}$, if $P<\mu_{G_{l}}P$, then
$(\Delta_{i+1}(p),\mu_{G_{l}}P,\Delta_{j}(q))$ covers ${\bf P}$ and
$(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q))$,
$(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q))$ covers ${\bf Q}$.

  2. $(2)$

In case $k\in\\{2,4\\}$, if $P>\mu_{G_{l}}P$, then ${\bf P}$ covers
$(\Delta_{i+1}(p),\mu_{G_{l}}P,\Delta_{j}(q))$, and
$(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q))$ is covered by ${\bf Q}$ and
$(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j+1}(q))$.

##  5\. Valuation maps

Let $T^{o}$ be an ideal triangulation and $\beta$ be an arc. Assume that
$s(\widetilde{\beta})=p,t(\widetilde{\beta}=q)$. In this section, for
$\mathcal{L}=\mathcal{L}(T^{o},\widetilde{\beta}),\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$
or $\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$ we construct a map
$w:\mathcal{L}\to\mathbb{Z}$, which will act as the quantum coefficients in
the expansion formula for quantum cluster variables. To this end, we should
introduce some integers.

Let $\alpha$ be an arc in $T^{o}$. For any perfect matching
$P\in\mathcal{P}(G_{T^{o},\widetilde{\beta}})$, denote

(9) $m(P;\alpha)=\text{number of edges labeled }\alpha\text{ in }P.$

For any snake subgraph $G$ of $G_{T^{o},\widetilde{\beta}}$, denote

(10) $n(G;\alpha)=\text{number of diagonals labeled }\alpha\text{ of }G.$

In the case that $\widetilde{\beta}\in T^{o}$, $G_{T^{o},\widetilde{\beta}}$
is the graph with two vertices and one edge labeled $\widetilde{\beta}$
connecting them, it has a unique perfect matching $P_{\widetilde{\beta}}$. Set
$m(P_{\widetilde{\beta}};\alpha)=1$ and
$n(G_{T^{o},\widetilde{\beta}};\alpha)=0$.

With the notation in Subsection 4.2, for any $j=1,\cdots,t$, denote

(11) $\begin{array}[]{rcl}m(\Delta_{j}(q);\alpha)&=&\text{number of edges
labeled }\alpha\text{ in }\\{\tau_{[j]}(q)\\}\vspace{2mm}\\\ &-&\text{number
of edges labeled }\alpha\text{ in
}\\{\tau_{j-1}(q),\tau_{j}(q)\\}.\end{array}$

Similarly, with the notation in Subsection 4.3, for any $i=1,\cdots,s$, denote

(12) $\begin{array}[]{rcl}m(\Delta_{i}(p);\alpha)&=&\text{number of edges
labeled }\alpha\text{ in }\\{\tau_{[i]}(p)\\}\vspace{2mm}\\\ &-&\text{number
of edges labeled }\alpha\text{ in
}\\{\tau_{i-1}(p),\tau_{i}(p)\\}.\end{array}$

For any $l=1,\cdots,c$, denote by $G^{+}_{l}$ (resp. $G^{-}_{l}$) the snake
subgraph of $G_{T^{o},\beta}$ formed by the tiles $G_{l+1},\cdots,G_{c}$
(resp. $G_{1},\cdots,G_{l-1}$).

Moreover, if $P$ can twist on a tile $G_{l}$ for some $l=1,\cdots,c$, denote

$m^{\pm}(P,G_{l};\alpha)=\text{number of edges labeled }\alpha\text{ in
}P\cap(edge(G^{\pm}_{l})\setminus edge(G_{l})).$

###  5.1. Valuation map on $\mathcal{L}(T^{o},\widetilde{\beta})$

In this section, we construct a map

$w:\mathcal{L}(T^{o},\widetilde{\beta})\to\mathbb{Z}.$

######  Definition 5.1.

Let $P$ be a perfect matching of $G_{T^{o},\widetilde{\beta}}$. Suppose that
$P$ can twist on a tile $G_{l}$ for some $l=1,\cdots,c$. Assume that the
diagonal of $G_{l}$ is labeled $\tau_{i_{l}}$. The _gradient number
$\Omega(P;G_{l})$ of $P$ at $G_{l}$_ is defined to be

$\Omega(P;G_{l})=d(\tau_{i_{l}})[\left(m^{+}(P,G_{l};\tau_{i_{l}})-m^{-}(P,G_{l};\tau_{i_{l}})\right)-\left(n(G^{+}_{l};\tau_{i_{l}})-n(G^{-}_{l};\tau_{i_{l}})\right)].$

######  Lemma 5.2.

Suppose that $P$ can twist on two tiles $G_{l}$ and $G_{r}$ with $r-l>1$ such
that $\mu_{G_{l}}P,\mu_{G_{r}}P>P$. Then

$\Omega(P;G_{l})+\Omega(\mu_{G_{l}}P;G_{r})=\Omega(P;G_{r})+\Omega(\mu_{G_{r}}P;G_{l}).$

###### Proof.

We first consider the case that $b^{T^{o}}_{\tau_{i_{l}}\tau_{i_{r}}}=0$. Then
we have

$m^{\pm}(P,G_{l};\tau_{i_{l}})=m^{\pm}(\mu_{G_{r}}P,G_{l};\tau_{i_{l}}),\hskip
14.22636ptm^{\pm}(P,G_{r};\tau_{i_{r}})=m^{\pm}(\mu_{G_{l}}P,G_{r};\tau_{i_{r}}).$

Thus the result follows in this case.

We then consider the case that $b^{T^{o}}_{\tau_{i_{l}}\tau_{i_{r}}}\neq 0$.
We may assume that $b^{T^{o}}_{\tau_{i_{l}}\tau_{i_{r}}}>0$. As
$\mu_{G_{l}}P,\mu_{G_{r}}P>P$, the edge labeled $\tau_{i_{r}}$ of $G_{i_{l}}$
is in $P$ and the edge labeled $\tau_{i_{l}}$ of $G_{i_{r}}$ is not in $P$.
Therefore, we have

$m^{+}(\mu_{G_{r}}P,G_{l};\tau_{i_{l}})=m^{+}(P,G_{l};\tau_{i_{l}})+b^{T^{o}}_{\tau_{i_{l}}\tau_{i_{r}}},\hskip
14.22636ptm^{-}(\mu_{G_{r}}P,G_{l};\tau_{i_{l}})=m^{-}(P,G_{l};\tau_{i_{l}}),$
$m^{-}(\mu_{G_{l}}P,G_{r};\tau_{i_{l}})=m^{-}(P,G_{r};\tau_{i_{l}})+b^{T^{o}}_{\tau_{i_{r}}\tau_{i_{l}}},\hskip
14.22636ptm^{+}(\mu_{G_{l}}P,G_{r};\tau_{i_{l}})=m^{+}(P,G_{r};\tau_{i_{l}}).$

Then the result follows by
$d(\tau_{i_{l}})b^{T^{o}}_{\tau_{i_{l}}\tau_{i_{r}}}=-d(\tau_{i_{r}})b^{T^{o}}_{\tau_{i_{r}}\tau_{i_{l}}}$,
as $B(T^{o})$ is skew-symmetrizable. $\square$

######  Proposition 5.3.

There is a unique map

$w:\mathcal{L}(T^{o},\widetilde{\beta})\to\mathbb{Z}$

satisfies the following conditions:

  1. $(1)$

(Initial condition) $w(P_{-})=0$, where $P_{-}$ is the minimum element in
$\mathcal{L}(T^{o},\widetilde{\beta})$,

  2. $(2)$

(Recurrence condition) $w(\mu_{G_{l}}(P))-w(P)=\Omega(P;G_{l})$ for any
$P\in\mathcal{L}(T^{o},\beta)$ such that $P$ can twist on $G_{l}$ with
$P<\mu_{G_{l}}(P)$.

###### Proof.

We shall only consider the case that $\widetilde{\beta}\notin T^{o}$. The
uniqueness of $w$ is clear since the Hasse graph of
$\mathcal{P}(G_{T^{o},\widetilde{\beta}})$ is connected (by Lemma 4.3).

We now show the existence. For any $P\in\mathcal{P}(G_{T^{o},\beta})$ and two
chains $P_{-}=P_{0}<P_{1}<\cdots<P_{a}=P$,
$P_{-}=P^{\prime}_{0}<P^{\prime}_{1}<\cdots<P^{\prime}_{a^{\prime}}=P$ such
that $P_{i+1}$ covers $P_{i}$ and $P^{\prime}_{i+1}$ covers $P^{\prime}_{i}$
for any $i$, it suffices to prove that $w_{1}(P)=w_{2}(P)$, where $w_{1}(P)$
(resp. $w_{2}(P)$) is obtained via the chain
$P_{-}=P_{0}<P_{1}<\cdots<P_{a}=P$ (resp.
$P_{-}=P^{\prime}_{0}<P^{\prime}_{1}<\cdots<P^{\prime}_{a^{\prime}}=P$).
Suppose that $P\ominus P_{-}$ encloses the union of tiles $\bigcup_{j\in
J}G_{j}$ for some $J\subseteq\\{1,\cdots,c\\}$. Denote $\Psi(P)=\prod_{j\in
J}\chi_{j}$. It is clear that $\frac{\Psi(\mu_{G_{l}}Q)}{\Psi(Q)}=\chi_{l}$
for any $Q$ can twist on $G_{l}$ with $Q<\mu_{G_{l}}Q$. Therefore
$a=a^{\prime}=|J|$. We say that $a=|J|$ is the distance between $P$ and
$P_{-}$.

Next, we show that $w_{1}(P)=w_{2}(P)$ by induction on the distance $a$
between $P$ and $P_{-}$. If $a=0$ then $P=P_{-}$ and thus
$w_{1}(P)=w_{2}(P)=0$. Suppose that $P_{a-1}=\mu_{G_{l}}P$ and
$P^{\prime}_{a-1}=\mu_{G_{r}}P$. Assume that $w_{1}(P)=w_{2}(P)$ holds for all
the cases that the distance is less than $a$. In particular, we have
$w_{1}(P_{a-1})=w_{2}(P^{\prime}_{a-1})$.

If $P_{a-1}=P^{\prime}_{a-1}$ then we have

$w_{1}(P)=w_{1}(P_{a-1})+\Omega(P_{a-1};G_{l})=w_{2}(P_{a-1})+\Omega(P_{a-1};G_{l})=w_{2}(P).$

If $P_{a-1}\neq P^{\prime}_{a-1}$, we have $|l-r|>1$ as
$P_{a-1},P^{\prime}_{a-1}<P$. Thus
$\mu_{G_{l}}\mu_{G_{r}}P=\mu_{G_{r}}\mu_{G_{l}}P$. Therefore,

$\begin{array}[]{rcl}w_{1}(P)&=&w_{1}(P_{s-1})+\Omega(P_{a-1};G_{l})\vspace{2mm}\\\
&=&\left(w_{1}(\mu_{G_{r}}\mu_{G_{l}}P)+\Omega(\mu_{G_{r}}\mu_{G_{l}}P;G_{r})\right)+\Omega(\mu_{G_{l}}P;G_{l})\vspace{2mm}\\\
&=&\left(w_{2}(\mu_{G_{l}}\mu_{G_{r}}P)+\Omega(\mu_{G_{l}}\mu_{G_{r}}P;G_{l})\right)+\Omega(\mu_{G_{r}}P;G_{r})\vspace{2mm}\\\
&=&w_{2}(\mu_{G_{r}}P)+\Omega(\mu_{G_{r}}P;G_{r})\vspace{2mm}\\\
&=&w_{2}(P^{\prime}_{a-1})+\Omega(\mu_{G_{r}}P;G_{r})\vspace{2mm}\\\
&=&w_{2}(P),\end{array}$

where the third equality follows by induction hypothesis and Lemma 5.2. The
result follows. $\square$

###  5.2. Valuation map on $\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$

Suppose that $q$ is a puncture and $q\neq p$. Assume that $T$ contains no arc
tagged notched at $q$. In this section, we construct a map

$w:\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})\to\mathbb{Z}.$

######  Definition 5.4.

For $(P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$ and
$\zeta\in T^{o}$, the _gradient number
$\Omega(P,\underline{\Delta_{j}(q)};\zeta)$ of $(P,\Delta_{j}(q))$ at
$\Delta_{j}(q)$_ is defined to be

$\Omega(P,\underline{\Delta_{j}(q)};\zeta)=d(\zeta)\left(-m(P;\zeta)+n(G_{T^{o},\widetilde{\beta}};\zeta)\right)=-d(\zeta)\hat{m}(P;\zeta).$

In particular, if $\zeta=\tau_{j}(q)$, we write
$\Omega(P,\underline{\Delta_{j}(q)};\zeta)$ as
$\Omega(P,\underline{\Delta_{j}(q)})$.

######  Definition 5.5.

For $(P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$, assume
that $P$ can twist on a tile $G_{l}$ with diagonal labeled $\tau_{i_{l}}$, the
_gradient number $\Omega(P,\Delta_{j}(q);G_{l})$ of $(P,\Delta_{j}(q))$ at
$G_{l}$_ is defined to be

$\Omega(P,\Delta_{j}(q);G_{l})=\Omega(P;G_{l})+d(\tau_{i_{l}})m(\Delta_{j}(q);\tau_{i_{l}}).$

######  Lemma 5.6.

Suppose that $P$ can twist on two tiles $G_{l}$ and $G_{r}$ with $r-l>1$ such
that $\mu_{G_{l}}P,\mu_{G_{r}}P>P$. Then for any
$\Delta_{j}(q)\in\Delta(T^{o},q)$ we have

$\Omega(P,\Delta_{j}(q);G_{l})+\Omega(\mu_{G_{l}}P,\Delta_{j}(q);G_{r})=\Omega(P,\Delta_{j}(q);G_{r})+\Omega(\mu_{G_{r}}P,\Delta_{j}(q);G_{l}).$

###### Proof.

It follows by Lemma 5.2 and Definition 5.5. $\square$

######  Lemma 5.7.

Suppose that $P$ can twist on a tile $G_{l}$ with $\mu_{G_{l}}P>P$. Then for
any $\Delta_{j}(q)\in\Delta(T^{o},q)$ we have

$\Omega(P,\Delta_{j}(q);G_{l})+\Omega(\mu_{G_{l}}P,\underline{\Delta_{j}(q)})=\Omega(P,\underline{\Delta_{j}(q)})+\Omega(P,\Delta_{j+1}(q);G_{l}).$

###### Proof.

By definition, we have

$\begin{array}[]{rcl}\Omega(P,\Delta_{j}(q);G_{l})+\Omega(\mu_{G_{l}}P,\underline{\Delta_{j}(q)})&=&\Omega(P;G_{l})+d(\tau_{i_{l}})m(\Delta_{j}(q);\tau_{i_{l}})\vspace{2mm}\\\
&+&d(\tau_{j}(q))(-m(\mu_{G_{l}}(P);\tau_{j}(q))+n(G_{T^{o},\widetilde{\beta}};\tau_{j}(q))),\end{array}$
$\begin{array}[]{rcl}\Omega(P,\underline{\Delta_{j}(q)})+\Omega(P,\Delta_{j+1}(q);G_{l})&=&d(\tau_{j}(q))(-m(P;\tau_{j}(q))+n(G_{T^{o},\widetilde{\beta}};\tau_{j}(q)))\vspace{2mm}\\\
&+&\Omega(P;G_{l})+d(\tau_{i_{l}})m(\Delta_{j+1}(q);\tau_{i_{l}}),\end{array}$

We first consider the case that $b^{T^{o}}_{\tau_{i_{l}},\tau_{j}(q)}=0$. Then
we have

$m(P;\tau_{j}(q))=m(\mu_{G_{l}}P;\tau_{j}(q)),\hskip
14.22636ptm(\Delta_{j}(q);\tau_{i_{l}})=m(\Delta_{j+1}(q);\tau_{i_{l}}).$

Thus the result follows in this case.

We then consider the case that $b^{T^{o}}_{\tau_{i_{l}},\tau_{j}(q)}\neq 0$.
We may assume that $b^{T^{o}}_{\tau_{i_{l}},\tau_{j}(q)}>0$. As
$\mu_{G_{l}}P>P$, the edge labeled $\tau_{j}(q)$ of $G_{i_{l}}$ is in $P$.
Therefore, we have

$m(P;\tau_{j}(q))=m(\mu_{G_{l}}P;\tau_{j}(q))-b^{T^{o}}_{\tau_{j}(q),\tau_{i_{l}}},\hskip
14.22636ptm(\Delta_{j}(q);\tau_{i_{l}})=m(\Delta_{j+1}(q);\tau_{i_{l}})-b^{T^{o}}_{\tau_{i_{l}},\tau_{j}(q)}.$

Then the result follows by
$d(\tau_{i_{l}})b^{T^{o}}_{\tau_{i_{l}}\tau_{j}(q)}=-d(\tau_{j}(q))b^{T^{o}}_{\tau_{j}(q)\tau_{i_{l}}}$,
as $B(T^{o})$ is skew-symmetrizable.

The proof is complete. $\square$

######  Proposition 5.8.

Assume that $q$ is a puncture and $q\neq p$. Assume that $T$ contains no arc
tagged notched at $q$. Then there is a unique map

$w:\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})\to\mathbb{Z}$

satisfies the following conditions:

  1. $(1)$

(Initial condition) $w({\bf P}_{-})=0$, where ${\bf
P}_{-}=(P_{-},\Delta_{1}(q))$ is the minimum element in
$\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$,

  2. $(2)$

(Recurrence condition)

    1. (a)

For any $(P,\Delta_{j}(q))$ such that $(P,\Delta_{j+1}(q))$ covers
$(P,\Delta_{j}(q))$,

$w(P,\Delta_{j+1}(q))-w(P,\Delta_{j}(q))=\Omega(P,\underline{\Delta_{j}(q)}).$

    2. (b)

For any $(P,\Delta_{j}(q)),(Q,\Delta_{j}(q))\in\mathcal{L}(T^{o},\beta^{(q)})$
such that $(Q,\Delta_{j}(q))$ covers $(P,\Delta_{j}(q))$, in particular $P<Q$
are related by a twist on some tile $G_{l}$,

$w(Q,\Delta_{j}(q))-w(P,\Delta_{j}(q))=\Omega(P,\Delta_{j}(q);G_{l}).$

###### Proof.

The result holds for the case $\widetilde{\beta}\in T^{o}$ as the Hasse graph
of $\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$ does not have any un-oriented
cycles in this case.

We now consider the case that $\widetilde{\beta}\notin T^{o}$. The uniqueness
holds as the Hasse graph of $\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$ is
connected (By Lemma 4.7). We now show the existence. For any
$\textbf{P}=(P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$,
for any two chains
$\textbf{P}_{0}=\textbf{P}_{-}<\textbf{P}_{1}<\cdots<\textbf{P}_{a}=\textbf{P}$
and
$\textbf{P}^{\prime}_{0}=\textbf{P}_{-}<\textbf{P}^{\prime}_{1}<\cdots<\textbf{P}^{\prime}_{a^{\prime}}=\textbf{P}$
such that $\textbf{P}_{i+1}$ covers $\textbf{P}_{i}$ and
$\textbf{P}^{\prime}_{i+1}$ covers $\textbf{P}^{\prime}_{i}$ for all $i$, it
suffices to show that $w_{1}(\textbf{P})=w_{2}(\textbf{P})$, where
$w_{1}(\textbf{P})$ (resp. $w_{1}(\textbf{P})$) is determined by the chain
$\textbf{P}_{0}=\textbf{P}_{-}<\textbf{P}_{1}<\cdots<\textbf{P}_{a}=\textbf{P}$
(resp.
$\textbf{P}^{\prime}_{0}=\textbf{P}_{-}<\textbf{P}^{\prime}_{1}<\cdots<\textbf{P}^{\prime}_{a^{\prime}}=\textbf{P}$).

Suppose that $P\ominus P_{-}$ encloses the union of tiles $\bigcup_{i\in
J}G_{i}$ for some $J\subseteq\\{1,\cdots,c\\}$. Denote

$\Psi(\textbf{P})=\begin{cases}\prod_{i\in J}\chi_{i},&\mbox{if $E_{1}(q)\in
P$ and $j=1$},\\\ \prod_{i\in
J}\chi_{i}(\kappa_{1}\kappa_{2}\cdots\kappa_{t-1}\kappa_{t}),&\mbox{if
$E_{2}(q)\in P$ and $j=1$},\\\ \prod_{i\in
J}\chi_{i}(\kappa_{1}\kappa_{2}\cdots\kappa_{j-1}),&\mbox{otherwise},\end{cases}$

where $E_{1}(q)$ and $E_{2}(q)$ are the edges of $G_{c}$ determined by (6) in
Section 4. Thus $\Psi(\mu_{G_{l}}Q,\Delta)=\Psi(Q,\Delta)\chi_{l}$ if
$(\mu_{G_{l}}Q,\Delta)$ covers $(Q,\Delta)$ and
$\Psi(Q,\Delta_{l+1}(q))=\Psi(Q,\Delta_{l}(q))\kappa_{l}$ if
$(Q,\Delta_{l+1}(q))$ covers $(Q,\Delta_{l}(q))$. It follows that
$a=a^{\prime}$ equals the exponent of $\Psi(\textbf{P})$. We say that
$a=a^{\prime}$ is the distance between $\textbf{P}_{-}$ and P. We prove that
$w_{1}(\textbf{P})=w_{2}(\textbf{P})$ by induction on the distance $a$.

In case $a=0$ we have $\textbf{P}=\textbf{P}_{-}$, it follows that
$w_{1}(\textbf{P})=w_{2}(\textbf{P})=0$. Assume that
$w_{1}(\textbf{P})=w_{2}(\textbf{P})$ holds for any case that the distance is
less than $a$.

If $\textbf{P}_{a-1}=\textbf{P}^{\prime}_{a-1}$ then

$w_{1}(\textbf{P})-w_{1}(\textbf{P}_{a-1})=w_{2}(\textbf{P})-w_{2}(\textbf{P}_{a-1}).$

Thus by hypothesis, we have

$w_{1}(\textbf{P})=w_{2}(\textbf{P}).$

If $\textbf{P}_{a-1}\neq\textbf{P}^{\prime}_{a-1}$, we have either
$\textbf{P}_{a-1}=(\mu_{G_{l}}P,\Delta_{j}(q)),\textbf{P}^{\prime}_{a-1}=(\mu_{G_{r}}P,\Delta_{j}(q))$
for some $l,r$ with $P$ can twist on $G_{l}$ and $G_{r}$ or
$\textbf{P}_{a-1}=(\mu_{G_{l}}P,\Delta_{j}(q)),\textbf{P}^{\prime}_{a-1}=(P,\Delta_{j-1}(q))$
for some $l$.

(1) If
$\textbf{P}_{a-1}=(\mu_{G_{l}}P,\Delta_{j}(q)),\textbf{P}^{\prime}_{a-1}=(\mu_{G_{r}}P,\Delta_{j}(q))$,
as $\textbf{P}_{a-1},\textbf{P}^{\prime}_{a-1}<\textbf{P}$, we have that
$\mu_{G_{l}}P,\mu_{G_{r}}P<P$ and $|l-r|>1$. Consequently,
$\mu_{G_{l}}\mu_{G_{r}}P=\mu_{G_{r}}\mu_{G_{l}}P$, moreover, as P covers
$\textbf{P}_{s-1}$, $l=c$ and $j=1$ can not hold simultaneously by Lemma 4.4.
Similarly, $r=c$ and $j=1$ can not hold simultaneously. Thus by Lemma 4.4, we
have both $\textbf{P}_{a-1}$ and $\textbf{P}^{\prime}_{a-1}$ cover
$(\mu_{G_{r}}\mu_{G_{l}}P,\Delta_{j}(q))$. Therefore,

$\begin{array}[]{rcl}w_{1}(\textbf{P})&=&w_{1}(\textbf{P}_{a-1})+\Omega(\mu_{G_{l}}P,\Delta_{j}(q);G_{l})\vspace{2mm}\\\
&=&\left(w_{1}(\mu_{G_{r}}\mu_{G_{l}}P,\Delta_{j}(q))+\Omega(\mu_{G_{r}}\mu_{G_{l}}P,\Delta_{j}(q);G_{r})\right)+\Omega(\mu_{G_{l}}P,\Delta_{j}(q);G_{l})\vspace{2mm}\\\
&=&w_{2}(\mu_{G_{l}}\mu_{G_{r}}P,\Delta_{j}(q))+\left(\Omega(\mu_{G_{l}}\mu_{G_{r}}P,\Delta_{j}(q);G_{l})+\Omega(\mu_{G_{r}}P,\Delta_{j}(q);G_{r})\right)\vspace{2mm}\\\
&=&w_{2}(\mu_{G_{r}}P,\Delta_{j}(q))+\Omega(\mu_{G_{r}}P,\Delta_{j}(q);G_{r})\vspace{2mm}\\\
&=&w_{2}(\textbf{P}^{\prime}_{a-1})+\Omega(\mu_{G_{r}}P,\Delta_{j}(q);G_{r})\vspace{2mm}\\\
&=&w_{2}(\textbf{P}),\end{array}$

where the third equality follows by the induction hypothesis and Lemma 5.6.

(2) If
$\textbf{P}_{a-1}=(\mu_{G_{l}}P,\Delta_{j}(q)),\textbf{P}^{\prime}_{a-1}=(P,\Delta_{j-1}(q))$,
we see that $l=c$ and $j-1=1$ can not hold simultaneously, since otherwise
$\textbf{P}=(P,\Delta_{j}(q))<\textbf{P}^{\prime}_{a-1}=(P,\Delta_{j-1}(q))$.
Thus, $\textbf{P}^{\prime}_{a-1}=(P,\Delta_{j-1}(q))$ covers
$(\mu_{G_{l}}P,\Delta_{j-1}(q))$ by Lemma 4.4. We also see that $l=c$ and
$j=1$ can not hold simultaneously, since otherwise
$\textbf{P}=(P,\Delta_{j}(q))$ does not cover
$\textbf{P}_{a-1}=(\mu_{G_{l}}P,\Delta_{j}(q))$. As
$\textbf{P}=(P,\Delta_{j}(q))$ covers
$\textbf{P}^{\prime}_{a-1}=(P,\Delta_{j-1}(q))$, by Lemma 4.5, we thus have
$\textbf{P}_{a-1}=(\mu_{G_{l}}P,\Delta_{j}(q))$ covers
$(\mu_{G_{l}}P,\Delta_{j-1}(q))$.

Therefore,

$\begin{array}[]{rcl}w_{1}(\textbf{P})&=&w_{1}(\textbf{P}_{a-1})+\Omega(\mu_{G_{l}}P,\Delta_{j}(q);G_{l})\vspace{2mm}\\\
&=&\left(w_{1}(\mu_{G_{l}}P,\Delta_{j-1}(q))+\Omega(\mu_{G_{l}}P,\underline{\Delta_{j-1}(q)})\right)+\Omega(\mu_{G_{l}}P,\Delta_{j}(q);G_{l})\vspace{2mm}\\\
&=&w_{2}(\mu_{G_{l}}P,\Delta_{j-1}(q))+\Omega(\mu_{G_{l}}P,\Delta_{j-1}(q);G_{l})+\Omega(P,\underline{\Delta_{j-1}(q)})\vspace{2mm}\\\
&=&w_{2}(P,\Delta_{j-1}(q))+\Omega(P,\underline{\Delta_{j-1}(q)})\vspace{2mm}\\\
&=&w_{2}(\textbf{P}^{\prime}_{a-1})+\Omega(P,\underline{\Delta_{j-1}(q)})\vspace{2mm}\\\
&=&w_{2}(\textbf{P}),\end{array}$

where the third equality follows by induction hypothesis and Lemma 5.7.

The proof is complete. $\square$

###  5.3. Valuation map on $\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$

Suppose that $p$ and $q$ are punctures. Assume that $T$ contains no arcs
tagged notched at $p$ or $q$. In this section, we construct a map

$w:\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})\to\mathbb{Z}.$

######  Definition 5.9.

For any
$(\Delta_{i}(p),P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$
and $\zeta\in T^{o}$, the _gradient number
$\Omega(\underline{\Delta_{i}(p)},P,\Delta_{j}(q);\zeta)$ of
$(\Delta_{i}(p),P,\Delta_{j}(q))$ at $\Delta_{i}(p)$_ is defined to be

$\Omega(\underline{\Delta_{i}(p)},P,\Delta_{j}(q);\zeta)=d(\zeta)[m(P;\zeta)-n(G_{T^{o},\widetilde{\beta}};\zeta)+m(\Delta_{j}(q);\zeta)].$

In particular, if $\zeta=\tau_{i}(p)$, we write
$\Omega(\underline{\Delta_{i}(p)},P,\Delta_{j}(q);\zeta)$ as
$\Omega(\underline{\Delta_{i}(p)},P,\Delta_{j}(q))$.

######  Definition 5.10.

For any
$(\Delta_{i}(p),P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$
and $\zeta\in T^{o}$, the _gradient number
$\Omega(\Delta_{i}(p),P,\underline{\Delta_{j}(q)};\zeta)$ of
$(\Delta_{i}(p),P,\Delta_{j}(q))$ at $\Delta_{j}(q)$_ is defined to be

$\Omega(\Delta_{i}(p),P,\underline{\Delta_{j}(q)};\zeta)=-d(\zeta)[m(P;\zeta)-n(G_{T^{o},\widetilde{\beta}};\zeta)+m(\Delta_{i}(p);\zeta)].$

In particular, if $\zeta=\tau_{j}(q)$, we write
$\Omega(\Delta_{i}(p),P,\underline{\Delta_{j}(q)};\zeta)$ as
$\Omega(\Delta_{i}(p),P,\underline{\Delta_{j}(q)})$.

######  Definition 5.11.

Assume that
$(\Delta_{i}(p),P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$
and $P$ can twist on a tile $G_{l}$ for some $l\in\\{1,\cdots,c\\}$. Assume
that the diagonal of $G_{l}$ is labeled $\tau_{i_{l}}$. The _gradient number
$\Omega(\Delta_{i}(p),P,\Delta_{j}(q);G_{l})$ of
$(\Delta_{i}(p),P,\Delta_{j}(q))$ at $G_{l}$_ is defined to be

$\Omega(\Delta_{i}(p),P,\Delta_{j}(q);G_{l})=\Omega(P;G_{l})+d(\tau_{i_{l}})[m(\Delta_{j}(q);\tau_{i_{l}})-m(\Delta_{i}(p);\tau_{i_{l}})].$

######  Lemma 5.12.

Suppose that $P$ can twist on two tiles $G_{l}$ and $G_{r}$ with $r-l>1$ such
that $\mu_{G_{l}}P,\mu_{G_{r}}P>P$. Then for any
$\Delta_{i}(p)\in\Delta(T^{o},p)$, $\Delta_{j}(q)\in\Delta(T^{o},q)$ we have

$\begin{array}[]{rcl}&&\Omega(\Delta_{i}(p),P,\Delta_{j}(q);G_{l})+\Omega(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q);G_{r})\vspace{2mm}\\\
&=&\Omega(\Delta_{i}(p),P,\Delta_{j}(q);G_{r})+\Omega(\Delta_{i}(p),\mu_{G_{r}}P,\Delta_{j}(q);G_{l}).\end{array}$

###### Proof.

It follows by Lemma 5.2 and Definition 5.11. $\square$

######  Lemma 5.13.

Suppose that $P$ can twist on a tile $G_{l}$ such that $\mu_{G_{l}}P>P$. Then
for any $\Delta_{i}(p)\in\Delta(T^{o},p)$, $\Delta_{j}(q)\in\Delta(T^{o},q)$
we have

  1. $(1)$

$\begin{array}[]{rcl}&&\Omega(\Delta_{i}(p),P,\Delta_{j}(q);G_{l})+\Omega(\Delta_{i}(p),\mu_{G_{l}}P,\underline{\Delta_{j}(q)})\vspace{2mm}\\\
&=&\Omega(\Delta_{i}(p),P,\underline{\Delta_{j}(q)})+\Omega(\Delta_{i}(p),P,\Delta_{j+1}(q);G_{l}).\end{array}$

  2. $(2)$

$\begin{array}[]{rcl}&&\Omega(\Delta_{i}(p),P,\Delta_{j}(q);G_{l})+\Omega(\underline{\Delta_{i}(p)},\mu_{G_{l}}P,\Delta_{j}(q))\vspace{2mm}\\\
&=&\Omega(\underline{\Delta_{i}(p)},P,\Delta_{j}(q))+\Omega(\Delta_{i+1}(p),P,\Delta_{j}(q);G_{l}).\end{array}$

###### Proof.

We shall only prove (1) as the proof of (2) is similar.

By definition, we have

$\begin{array}[]{rcl}&&\Omega(\Delta_{i}(p),P,\Delta_{j}(q);G_{l})+\Omega(\Delta_{i}(p),\mu_{G_{l}}P,\underline{\Delta_{j}(q)})\vspace{2mm}\\\
&=&\Omega(P;G_{l})+d(\tau_{i_{l}})[m(\Delta_{j}(q);\tau_{i_{l}})-m(\Delta_{i}(p);\tau_{i_{l}})]\vspace{2mm}\\\
&-&d(\tau_{j}(q))[m(\mu_{G_{l}}P;\tau_{j}(q))-n(G_{T^{o},\widetilde{\beta}};\tau_{j}(q))+m(\Delta_{i}(p);\tau_{j}(q))]\end{array}$

and

$\begin{array}[]{rcl}&&\Omega(\Delta_{i}(p),P,\underline{\Delta_{j}(q)})+\Omega(\Delta_{i}(p),P,\Delta_{j+1}(q);G_{l})\vspace{2mm}\\\
&=&-d(\tau_{j}(q))[m(P;\tau_{j}(q))-n(G_{T^{o},\widetilde{\beta}};\tau_{j}(q))+m(\Delta_{i}(p);\tau_{j}(q))]\vspace{2mm}\\\
&+&\Omega(P;G_{l})+d(\tau_{i_{l}})[m(\Delta_{j+1}(q);\tau_{i_{l}})-m(\Delta_{i}(p);\tau_{i_{l}})].\end{array}$

We first consider the case that $b^{T^{o}}_{\tau_{i_{l}},\tau_{j}(q)}=0$. Then
we have

$m(P;\tau_{j}(q))=m(\mu_{G_{l}}P;\tau_{j}(q)),\hskip
14.22636ptm(\Delta_{j}(q);\tau_{i_{l}})=m(\Delta_{j+1}(q);\tau_{i_{l}}).$

Thus the result follows in this case.

We then consider the case that $b^{T^{o}}_{\tau_{i_{l}},\tau_{j}(q)}\neq 0$.
We may assume that $b^{T^{o}}_{\tau_{i_{l}},\tau_{j}(q)}>0$. As
$\mu_{G_{l}}P>P$, the edge labeled $\tau_{j}(q)$ of $G_{i_{l}}$ is in $P$.
Therefore, we have

$m(P;\tau_{j}(q))=m(\mu_{G_{l}}P;\tau_{j}(q))-b^{T^{o}}_{\tau_{j}(q),\tau_{i_{l}}},\hskip
14.22636ptm(\Delta_{j}(q);\tau_{i_{l}})=m(\Delta_{j+1}(q);\tau_{i_{l}})-b^{T^{o}}_{\tau_{i_{l}},\tau_{j}(q)}.$

Then the result follows by
$d(\tau_{i_{l}})b^{T^{o}}_{\tau_{i_{l}},\tau_{j}(q)}=-d(\tau_{j}(q))b^{T^{o}}_{\tau_{j}(q),\tau_{i_{l}}}$,
as $B(T^{o})$ is skew-symmetrizable.

The proof is complete. $\square$

######  Lemma 5.14.

For any
$(\Delta_{i}(p),P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$
we have

$\begin{array}[]{rcl}&&\Omega(\underline{\Delta_{i}(p)},P,\Delta_{j}(q))+\Omega(\Delta_{i+1}(p),P,\underline{\Delta_{j}(q)})\vspace{2mm}\\\
&=&\Omega(\Delta_{i}(p),P,\underline{\Delta_{j}(q)})+\Omega(\underline{\Delta_{i}(p)},P,\Delta_{j+1}(q)).\end{array}$

###### Proof.

By definition, we have

$\begin{array}[]{rcl}&&\Omega(\underline{\Delta_{i}(p)},P,\Delta_{j}(q))+\Omega(\Delta_{i+1}(p),P,\underline{\Delta_{j}(q)})\vspace{2mm}\\\
&=&d(\tau_{i}(p))[m(P;\tau_{i}(p))-n(G_{T^{o},\widetilde{\beta}};\tau_{i}(p))+m(\Delta_{j}(q);\tau_{i}(p))]\vspace{2mm}\\\
&-&d(\tau_{j}(q))[m(P;\tau_{j}(q))-n(G_{T^{o},\widetilde{\beta}};\tau_{j}(q))+m(\Delta_{i+1}(p);\tau_{j}(q))],\end{array}$

$\begin{array}[]{rcl}&&\Omega(\Delta_{i}(p),P,\underline{\Delta_{j}(q)})+\Omega(\underline{\Delta_{i}(p)},P,\Delta_{j+1}(q))\vspace{2mm}\\\
&=&-d(\tau_{j}(q))[m(P;\tau_{j}(p))-n(G_{T^{o},\widetilde{\beta}};\tau_{j}(q))+m(\Delta_{i}(p);\tau_{j}(q))]\vspace{2mm}\\\
&+&d(\tau_{i}(p))[m(P;\tau_{i}(p))-n(G_{T^{o},\widetilde{\beta}};\tau_{i}(p))+m(\Delta_{j+1}(q);\tau_{i}(p))],\end{array}$

We first consider the case that $b^{T^{o}}_{\tau_{i}(p),\tau_{j}(q)}=0$. Then
we have

$m(\Delta_{i}(p);\tau_{j}(q))=m(\Delta_{i+1}(p);\tau_{j}(q)),\hskip
14.22636ptm(\Delta_{j}(q);\tau_{i}(p))=m(\Delta_{j+1}(q);\tau_{i}(p)).$

Thus the result follows in this case.

We then consider the case that $b^{T^{o}}_{\tau_{i}(p),\tau_{j}(q)}\neq 0$. We
may assume that $b^{T^{o}}_{\tau_{i}(p),\tau_{j}(q)}>0$. Then we have

$m(\Delta_{i}(p);\tau_{j}(q))=m(\Delta_{i+1}(p);\tau_{j}(q))-b^{T^{o}}_{\tau_{j}(q),\tau_{i}(p)},\hskip
5.69054ptm(\Delta_{j}(q);\tau_{i}(p))=m(\Delta_{j+1}(q);\tau_{i}(p))-b^{T^{o}}_{\tau_{i}(p),\tau_{j}(q)}.$

Then the result follows by
$d(\tau_{i}(p))b^{T^{o}}_{\tau_{i}(p),\tau_{j}(q)}=-d(\tau_{j}(q))b^{T^{o}}_{\tau_{j}(q),\tau_{i}(p)}$,
as $B(T^{o})$ is skew-symmetrizable.

The proof is complete. $\square$

######  Proposition 5.15.

With the foregoing notation. Suppose that $p$ and $q$ are punctures and $T$
contains no arcs tagged notched at $p$ or $q$. There is a unique map

$w:\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})\to\mathbb{Z}$

satisfies the following conditions:

(1) (Initial condition) $w({\bf P}_{-})=0$, where ${\bf P}_{-}$ is the minimum
element in $\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$,

(2) (Recurrence condition)

(a) For any $(\Delta_{i}(p),P,\Delta_{j}(q))$ such that
$(\Delta_{i+1}(p),P,\Delta_{j}(q))$ covers $(\Delta_{i}(p),P,\Delta_{j}(q))$,

$w(\Delta_{i+1}(p),P,\Delta_{j}(q))-w(\Delta_{i}(p),P,\Delta_{j}(q))=\Omega(\underline{\Delta_{i}(p)},P,\Delta_{j}(q)).$

(b) For any $(\Delta_{i}(p),P,\Delta_{j}(q))$ such that
$(\Delta_{i}(p),P,\Delta_{j+1}(q))$ covers $(\Delta_{i}(p),P,\Delta_{j}(q))$,

$w(\Delta_{i}(p),P,\Delta_{j+1}(q))-w(\Delta_{i}(p),P,\Delta_{j}(q))=\Omega(\Delta_{i}(p),P,\underline{\Delta_{j}(q)}).$

(c) For any
$(\Delta_{i}(p),Q,\Delta_{j}(q)),(\Delta_{i}(p),P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$
such that $(\Delta_{i}(p),Q,\Delta_{j}(q))$ covers
$(\Delta_{i}(p),P,\Delta_{j}(q))$, in particular, $Q>P$ are related by a twist
on some tile $G_{l}$,

$w(\Delta_{i}(p),Q,\Delta_{j}(q))-w(\Delta_{i}(p),P,\Delta_{j}(q))=\Omega(\Delta_{i}(p),P,\Delta_{j}(q);G_{l}).$

###### Proof.

The uniqueness holds as the Hasse graph of
$\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$ is connected (by Lemma 4.13).
We now show the existence. For any
$\textbf{P}=(\Delta_{i}(p),P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$,
for any two chains
$\textbf{P}_{0}=\textbf{P}_{-}<\textbf{P}_{1}<\cdots<\textbf{P}_{a}=\textbf{P}$
and
$\textbf{P}^{\prime}_{0}=\textbf{P}_{-}<\textbf{P}^{\prime}_{1}<\cdots<\textbf{P}^{\prime}_{a^{\prime}}=\textbf{P}$
such that $\textbf{P}_{i+1}$ covers $\textbf{P}_{i}$ and
$\textbf{P}^{\prime}_{i+1}$ covers $\textbf{P}^{\prime}_{i}$ for all $i$, it
suffices to show that $w_{1}(\textbf{P})=w_{2}(\textbf{P})$, where
$w_{1}(\textbf{P})$ (resp. $w_{2}(\textbf{P})$) is determined by the chain
$\textbf{P}_{0}=\textbf{P}_{-}<\textbf{P}_{1}<\cdots<\textbf{P}_{a}=\textbf{P}$
(resp.
$\textbf{P}^{\prime}_{0}=\textbf{P}_{-}<\textbf{P}^{\prime}_{1}<\cdots<\textbf{P}^{\prime}_{a^{\prime}}=\textbf{P}$).

If $\widetilde{\beta}\notin T^{o}$, denote

$\Psi(\Delta_{i}(p),P,\Delta_{j}(q))=\begin{cases}\Psi(P,\Delta_{j}(q)),&\mbox{if
$E_{1}(p)\in P$ and $i=1$},\\\
\Psi(P,\Delta_{j}(q))\vartheta_{1}\vartheta_{2}\cdots\vartheta_{s-1}\vartheta_{s},&\mbox{if
$E_{2}(p)\in P$ and $i=1$},\\\
\Psi(P,\Delta_{j}(q))\vartheta_{1}\vartheta_{2}\cdots\vartheta_{i-1},&\mbox{otherwise},\end{cases}$

where $E_{1}(p)$ and $E_{2}(p)$ are the edges of $G_{1}$ determined by (8) in
Section 4 and $\Psi(P,\Delta_{j}(q))$ is given in the proof of Proposition
5.8.

If $\widetilde{\beta}\in T^{o}$, denote

(13)
$\Psi(\Delta_{i}(p),P_{\widetilde{\beta}},\Delta_{j}(q))=\begin{cases}1,&\mbox{if
$i=1$ and $j=t$},\\\
\vartheta_{1}\vartheta_{2}\cdots\vartheta_{s}\kappa_{1}\cdots\kappa_{t},&\mbox{if
$i=s$ and $j=1$},\\\
\vartheta_{1}\vartheta_{2}\cdots\vartheta_{i-1}\kappa_{1}\cdots\kappa_{j-1}\kappa_{t},&\mbox{otherwise}.\end{cases}$

Thus,
$\Psi(\Delta,\mu_{G_{l}}Q,\Delta^{\prime})=\Psi(\Delta,Q,\Delta^{\prime})\chi_{l}$
if $(\Delta,\mu_{G_{l}}Q,\Delta^{\prime})$ covers
$(\Delta,Q,\Delta^{\prime})$,
$\Psi(\Delta,Q,\Delta_{l+1}(q))=\Psi(\Delta,Q,\Delta_{l}(q))\kappa_{l}$ if
$(\Delta,Q,\Delta_{l+1}(q))$ covers $(\Delta,Q,\Delta_{l}(q))$, and
$\Psi(\Delta_{l+1}(p),Q,\Delta^{\prime})=\Psi(\Delta_{l}(p),Q,\Delta^{\prime})\vartheta_{l}$
if $(\Delta_{l+1}(p),Q,\Delta^{\prime})$ covers
$(\Delta_{l}(p),Q,\Delta^{\prime})$. It follows that $a=a^{\prime}$ equals the
exponent of $\Psi(\textbf{P})$. We say that $a=a^{\prime}$ is the distance
between $\textbf{P}_{-}$ and P. We prove that
$w_{1}(\textbf{P})=w_{2}(\textbf{P})$ by induction on the distance $a$.

In case $a=0$ we have $\textbf{P}=\textbf{P}_{-}$. It follows that
$w_{1}(\textbf{P})=w_{2}(\textbf{P})=0$ in this case. Assume that
$w_{1}(\textbf{P})=w_{2}(\textbf{P})$ holds for any case that the distance is
less than $a$.

If $\textbf{P}_{a-1}=\textbf{P}^{\prime}_{a-1}$ then
$w_{1}(\textbf{P})-w_{1}(\textbf{P}_{a-1})=w_{2}(\textbf{P})-w_{2}(\textbf{P}_{a-1}).$
Thus by the hypothesis we have $w_{1}(\textbf{P})=w_{2}(\textbf{P}).$

We now consider the case that $\textbf{P}_{a-1}\neq\textbf{P}^{\prime}_{a-1}$.
There are four possibilities:

  1. $(1)$

$\textbf{P}_{a-1}=(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q))$ and
$\textbf{P}^{\prime}_{a-1}=(\Delta_{i}(p),\mu_{G_{r}}P,\Delta_{j}(q))$ for
some $G_{l}$ and $G_{r}$ such that $P$ can twist on $G_{l}$ and $G_{r}$;

  2. $(2)$

$\textbf{P}_{a-1}=(\Delta_{i-1}(p),P,\Delta_{j}(q))$ and
$\textbf{P}^{\prime}_{a-1}=(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q))$ for
some $G_{l}$ such that $P$ can twist on $G_{l}$;

  3. $(3)$

$\textbf{P}_{a-1}=(\Delta_{i}(p),P,\Delta_{j-1}(q))$ and
$\textbf{P}^{\prime}_{a-1}=(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q))$ for
some $G_{l}$ such that $P$ can twist on $G_{l}$;

  4. $(4)$

$\textbf{P}_{a-1}=(\Delta_{i-1}(p),P,\Delta_{j}(q))$ and
$\textbf{P}^{\prime}_{a-1}=(\Delta_{i}(p),P,\Delta_{j-1}(q))$.

We shall only prove cases (1) (3) and (4) as cases (2) and (3) are dual.

For the case (1), we have $|l-r|>1$ and $\textbf{P}_{a-1}$ covers
$(\Delta_{i}(p),\mu_{G_{r}}\mu_{G_{l}}P,\Delta_{j}(q))$ and
$\textbf{P}^{\prime}_{a-1}$ covers
$(\Delta_{i}(p),\mu_{G_{l}}\mu_{G_{r}}P,\Delta_{j}(q))$. Thus we have

$\begin{array}[]{rcl}w_{1}(\textbf{P})&=&w_{1}(\textbf{P}_{a-1})+\Omega(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q);G_{l})\vspace{2mm}\\\
&=&\left(w_{1}(\Delta_{i}(p),\mu_{G_{r}}\mu_{G_{l}}P,\Delta_{j}(q))+\Omega(\Delta_{i}(p),\mu_{G_{r}}\mu_{G_{l}}P,\Delta_{j}(q);G_{r})\right)\vspace{2mm}\\\
&+&\Omega(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q);G_{l})\vspace{2mm}\\\
&=&w_{2}(\Delta_{i}(p),\mu_{G_{l}}\mu_{G_{r}}P,\Delta_{j}(q))+(\Omega(\Delta_{i}(p),\mu_{G_{l}}\mu_{G_{r}}P,\Delta_{j}(q);G_{l})\vspace{2mm}\\\
&+&\Omega(\Delta_{i}(p),\mu_{G_{r}}P,\Delta_{j}(q);G_{r}))\vspace{2mm}\\\
&=&w_{2}(\Delta_{i}(p),\mu_{G_{r}}P,\Delta_{j}(q))+\Omega(\Delta_{i}(p),\mu_{G_{r}}P,\Delta_{j}(q);G_{r}))\vspace{2mm}\\\
&=&w_{2}(\textbf{P}^{\prime}_{a-1})+\Omega(\Delta_{i}(p),\mu_{G_{r}}P,\Delta_{j}(q);G_{r}))\vspace{2mm}\\\
&=&w_{2}(\textbf{P}),\end{array}$

where the third equality follows by the induction hypothesis and Lemma 5.12.

For the case (3), as $\textbf{P}=(\Delta_{i}(p),P,\Delta_{j}(q))$ covers
$\textbf{P}_{a-1}=(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q))$, we have
$E_{2}(q)\in P$ and $j\neq 1$. Thus, $E_{1}(q)\in\mu_{G_{l}}P$ and
$\textbf{P}_{a-1}=(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q))$ covers
$(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j-1}(q))$ by Lemma 4.12. We also have
that $l=c$ and $j-1=1$ can not hold simultaneously, since otherwise
$\textbf{P}=(\Delta_{i}(p),P,\Delta_{j}(q))<\textbf{P}^{\prime}_{a-1}=(\Delta_{i}(p),P,\Delta_{j-1}(q))$.
Thus by Lemma 4.11,
$\textbf{P}^{\prime}_{a-1}=(\Delta_{i}(p),P,\Delta_{j-1}(q))$ covers
$(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j-1}(q))$.

Therefore,

$\begin{array}[]{rcl}w_{1}(\textbf{P})&=&w_{1}(\textbf{P}_{a-1})+\Omega(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q);G_{l})\vspace{2mm}\\\
&=&w_{1}(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j-1}(q))+\Omega(\Delta_{i}(p),\mu_{G_{l}}P,\underline{\Delta_{j-1}(q)})\vspace{2mm}\\\
&+&\Omega(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j}(q);G_{l})\vspace{2mm}\\\
&=&w_{2}(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j-1}(q))+\Omega(\Delta_{i}(p),\mu_{G_{l}}P,\Delta_{j-1}(q);G_{l})\vspace{2mm}\\\
&+&\Omega(\Delta_{i}(p),P,\underline{\Delta_{j-1}(q)})\vspace{2mm}\\\
&=&w_{2}(\Delta_{i}(p),P,\Delta_{j-1}(q))+\Omega(\Delta_{i}(p),P,\underline{\Delta_{j-1}(q)})\vspace{2mm}\\\
&=&w_{2}(\textbf{P}^{\prime}_{a-1})+\Omega(\Delta_{i}(p),P,\underline{\Delta_{j-1}(q)})\vspace{2mm}\\\
&=&w_{2}(\textbf{P}),\end{array}$

where the third equality follows by induction hypothesis and Lemma 5.13.

For the case (4), as P covers
$\textbf{P}_{a-1}=(\Delta_{i-1}(p),P,\Delta_{j}(q))$ and
$\textbf{P}^{\prime}_{a-1}=(\Delta_{i}(p),P,\Delta_{j-1}(q))$, we have
$\textbf{P}_{a-1}=(\Delta_{i-1}(p),P,\Delta_{j}(q))$ and
$\textbf{P}^{\prime}_{a-1}=(\Delta_{i}(p),P,\Delta_{j-1}(q))$ covers
$(\Delta_{i-1}(p),P,\Delta_{j-1}(q))$ by Lemma 4.12.

Therefore,

$\begin{array}[]{rcl}w_{1}(\textbf{P})&=&w_{1}(\textbf{P}_{a-1})+\Omega(\underline{\Delta_{i-1}(p)},\mu_{G_{l}}P,\Delta_{j}(q))\vspace{2mm}\\\
&=&w_{1}(\Delta_{i-1}(p),P,\Delta_{j-1}(q))+\Omega(\Delta_{i-1}(p),P,\underline{\Delta_{j-1}(q)})\vspace{2mm}\\\
&+&\Omega(\underline{\Delta_{i-1}(p)},\mu_{G_{l}}P,\Delta_{j}(q))\vspace{2mm}\\\
&=&w_{2}(\Delta_{i-1}(p),P,\Delta_{j-1}(q))+\Omega(\underline{\Delta_{i-1}(p)},P,\Delta_{j-1}(q))\vspace{2mm}\\\
&+&\Omega(\Delta_{i}(p),\mu_{G_{l}}P,\underline{\Delta_{j-1}(q)})\vspace{2mm}\\\
&=&w_{2}(\Delta_{i}(p),P,\Delta_{j-1}(q))+\Omega(\Delta_{i}(p),\mu_{G_{l}}P,\underline{\Delta_{j-1}(q)})\vspace{2mm}\\\
&=&w_{2}(\textbf{P}^{\prime}_{a-1})+\Omega(\Delta_{i}(p),\mu_{G_{l}}P,\underline{\Delta_{j-1}(q)})\vspace{2mm}\\\
&=&w_{2}(\textbf{P}),\end{array}$

where the third equality follows by induction hypothesis and Lemma 5.14.

The proof is complete. $\square$

##  6\. Expansion formulas and the positivity

Let $\Sigma$ be an orbifold and $\mathcal{A}_{v}(\Sigma)$ be a quantum cluster
algebra arising from $\Sigma$. Let $\beta$ be an arc in $\Sigma$ and $T^{o}$
be an ideal triangulation. Let $\widetilde{\beta}$ be the curve associated
with $\beta$ given by (4). Let $T$ be the corresponding tagged triangulation.
Assume that $\widetilde{\beta}$ starts from $p$ and ends at $q$. In this
section using the lattices $\mathcal{L}(T^{o},\widetilde{\beta})$,
$\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$ and
$\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$, respectively in Section 4, we
give expansion formulas for quantum cluster variables $\widetilde{\beta}$,
$\widetilde{\beta}^{(q)}$ and $\widetilde{\beta}^{(p,q)}$, respectively. As a
corollary, we obtain the positivity for quantum cluster algebras from
orbifolds.

###  6.1. Expansion formula for ordinary arcs.

######  Definition 6.1.

[34] Suppose that the diagonal of $G_{T^{o},\widetilde{\beta}}$ are labeled
$\tau_{i_{1}},\cdots,\tau_{i_{c}}$. Let $P$ be a perfect matching of
$G_{T^{o},\widetilde{\beta}}$ with edges labeled by
$\tau_{k_{1}},\cdots,\tau_{k_{r}}$. Suppose that $P\ominus P_{-}$ encloses the
union of tiles $\bigcup_{j\in J}G_{j}$ for some $J\subseteq\\{1,\cdots,c\\}$.

  1. $(1)$

The _weight_ $x^{T}(P)$ of $P$ is defined to be

$x^{T}(P):=\frac{x_{\tau_{k_{1}}}\cdots
x_{\tau_{k_{r}}}}{x_{\tau_{i_{1}}}\cdots x_{\tau_{i_{c}}}},$

  2. $(2)$

The _height monomial_ $h^{T}(P)$ of $P$ is defined to be

$h^{T}(P):=\prod_{j\in J}h^{T}_{i_{j}},$

  3. $(3)$

The _specialized height monomial_ $y^{T}(P)$ of $P$ is defined to be

$y^{T}(P):=\Phi(h^{T}(P)),$

where $\Phi$ is given by

(14) $\Phi(h^{T}_{\tau_{i}})=\begin{cases}y^{T}_{\tau_{i}},&\mbox{if
$\tau_{i}$ is not a side of a self-folded triangle},\vspace{1mm}\\\
\frac{y^{T}_{r}\vspace{2mm}}{\vspace{2mm}y^{T}_{r^{(p)}}},&\mbox{if $\tau_{i}$
is a radius $r$ to puncture $p$ in a self-folded triangle},\vspace{1mm}\\\
y^{T}_{r^{(p)}},&\mbox{if $\tau_{i}$ is a loop in a self-folded triangle with
radius $r$ to a puncture $p$.}\end{cases}$

  4. $(4)$

The _quantum weight_ $X^{T}(P)$ of $P$ is defined to be the element $X^{a(P)}$
such that

$X^{a(P)}|_{v=1}=\frac{x^{T}(P)y^{T}(P)}{\bigoplus_{R\in\mathcal{P}(G_{T^{o},\widetilde{\beta}})}y^{T}(R)}.$

In particular, if $\widetilde{\beta}\in T^{o}$ we have
$x^{T}(P_{\widetilde{\beta}})=x_{\widetilde{\beta}},y^{T}(P_{\widetilde{\beta}})=1$,
where $P_{\widetilde{\beta}}$ is the unique perfect matching of
$G_{T^{o},\widetilde{\beta}}$.

For an arc $\gamma$ incident to a puncture $p$, we let the weight
$x_{l_{q}(\gamma)}=x_{\gamma}x_{\gamma^{(q)}}$.

######  Remark 6.2.

  1. $(1)$

Note that the weight of a perfect matching slightly differs from the original
definition in [34], here we quotient the crossing monomial.

  2. $(2)$

For convenience, we denote

$y^{T^{o}}_{\tau_{i}}=\begin{cases}y^{T}_{\tau_{i}},&\mbox{if $\tau_{i}$ is
not a side of a self-folded triangle},\vspace{1mm}\\\
\frac{y^{T}_{r}\vspace{2mm}}{\vspace{2mm}y^{T}_{r^{(p)}}},&\mbox{if $\tau_{i}$
is a radius $r$ to puncture $p$ in a self-folded triangle},\vspace{1mm}\\\
y^{T}_{r^{(p)}},&\mbox{if $\tau_{i}$ is a loop in a self-folded triangle with
radius $r$ to a puncture $p$.}\end{cases}$

Then $\Phi(h^{T}_{\tau_{i}})=y^{T^{o}}_{\tau_{i}}.$

For $i\in\\{1,2,3,4\\}$, if $\alpha_{i}$ is a radius of some self-folded
triangle in $T^{o}$, then $\alpha$ is the loop.

######  Lemma 6.3.

Let $T^{o}$ be an ideal triangulation and $T$ be the corresponding tagged
triangulation. For any arc $\alpha\in T^{o}$ which can be flipped, denote
$T^{\prime o}=(T^{o}\setminus\\{\alpha\\})\cup\\{\alpha^{\prime}\\}$ be
flipped ideal triangulation and $T^{\prime}$ the corresponding tagged
triangulation. Suppose that $\alpha$ is a diagonal of the quadrilateral
$(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$ in $T^{o}$ and
$\alpha_{1},\alpha_{3}$ are in the clockwise direction of $\alpha$. Then

  1. $(1)$

if $\tau\neq\alpha,\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$, then
$y^{T^{\prime o}}_{\tau}=y^{T^{o}}_{\tau}$,

  2. $(2)$

$y^{T^{\prime o}}_{\alpha^{\prime}}=(y^{T^{o}}_{\alpha})^{-1}$,

  3. $(3)$

For $i\in\\{1,2,3,4\\}$,

    1. (a)

if $\alpha_{i}$ is not a radius of any self-folded triangles in $T^{o}$ or
$T^{\prime o}$, i.e., $\alpha_{i}\neq\alpha_{i-1},\alpha_{i+1}$, then we have
$y_{\alpha_{i}}^{T^{o}}=y_{\alpha_{i}}^{T^{\prime
o}}(y_{\alpha^{\prime}}^{T^{\prime
o}})^{[-b^{T^{o}}_{\alpha\alpha_{i}}]_{+}}(1\oplus
y_{\alpha^{\prime}}^{T^{\prime o}})^{b^{T^{o}}_{\alpha\alpha_{i}}},$

    2. (b)

if $\alpha_{i}$ is a radius of some self-folded triangle in $T^{\prime o}$ to
some puncture $p$, then we have

$y_{\alpha_{i}}^{T^{o}}=y_{\alpha_{i}}^{T^{\prime
o}}y_{l_{p}(\alpha_{i})}^{T^{\prime o}},$

  4. $(4)$

$x_{\alpha}=\frac{x_{\alpha_{2}}x_{\alpha_{4}}+x_{\alpha_{1}}x_{\alpha_{3}}y^{T^{\prime
o}}_{\alpha^{\prime}}}{x_{\alpha^{\prime}}\cdot(1\oplus y^{T^{\prime
o}}_{\alpha^{\prime}})}$.

###### Proof.

(1) It is clear.

(2) It is clear if $\alpha,\alpha^{\prime}$ are not sides of self-folded
triangles. We may assume one of $\alpha,\alpha^{\prime}$ is a side of a self-
folded triangle, without loss of generality, we may assume that $\alpha$ is a
loop of a self-folded triangle with the radius $\alpha_{3}=\alpha_{4}$ to a
puncture $p$. Then $y^{T^{\prime
o}}_{\alpha^{\prime}}=y^{T^{\prime}}_{\alpha^{\prime}}=(y^{T}_{\alpha_{3}^{(p)}})^{-1}=(y^{T^{o}}_{\alpha})^{-1}$.

(3) We may assume $i=1$. For the statement (a), if $\alpha_{1}$ and $\alpha$
are not a side of a self-folded triangle in $T^{o}$ or $T^{\prime o}$, then

$y^{T^{\prime
o}}_{\alpha_{1}}=y^{T^{\prime}}_{\alpha_{1}}=y^{T}_{\alpha_{1}}(y^{T}_{\alpha})^{b^{T^{o}}_{\alpha\alpha_{1}}}(1\oplus
y^{T^{o}}_{\alpha})^{-b^{T^{o}}_{\alpha\alpha_{1}}}=y^{T^{o}}_{\alpha_{1}}(1\oplus(y^{T}_{\alpha})^{-1})^{-b^{T^{o}}_{\alpha\alpha_{1}}}=y^{T^{o}}_{\alpha_{1}}(1\oplus
y^{T^{\prime o}}_{\alpha^{\prime}})^{-b^{T^{o}}_{\alpha\alpha_{1}}}.$

If $\alpha_{1}$ is not a side of a self-folded triangle in $T^{o}$ or
$T^{\prime o}$ but $\alpha$ is a side of a self-folded triangle, then
$\alpha_{2}=\alpha_{3}$ is the radius to some puncture $p$, then

$y^{T^{\prime
o}}_{\alpha_{1}}=y^{T^{\prime}}_{\alpha_{1}}=y^{T}_{\alpha_{1}}(y^{T}_{\alpha_{2}^{(p)}})^{b^{T}_{\alpha_{2}^{(p)}\alpha_{1}}}(1\oplus
y^{T}_{\alpha_{2}^{(p)}})^{-b^{T}_{\alpha_{2}^{(p)}\alpha_{1}}}=y^{T^{o}}_{\alpha_{1}}(1\oplus(y^{T^{o}}_{\alpha})^{-1})^{-b^{T^{o}}_{\alpha\alpha_{1}}}=y^{T^{o}}_{\alpha_{1}}(1\oplus
y^{T^{\prime o}}_{\alpha^{\prime}})^{-b^{T^{o}}_{\alpha\alpha_{1}}}.$

If $\alpha_{1}$ is a side of a self-folded triangle in $T^{o}$, but $\alpha$
is not a side of a self-folded triangle in $T^{o}$, then $\alpha_{1}$ is a
loop of a self-folded triangle with radius $r$ to a puncture $p$. Thus we have

$y^{T^{\prime
o}}_{\alpha_{1}}=y^{T^{\prime}}_{r^{(p)}}=y^{T}_{r^{(p)}}(y^{T}_{\alpha})^{b^{T}_{\alpha
r^{(p)}}}(1\oplus y^{T}_{\alpha})^{-b^{T}_{\alpha
r^{(p)}}}=y^{T^{o}}_{\alpha_{1}}(1\oplus(y^{T^{o}}_{\alpha})^{-1})^{-b^{T^{o}}_{\alpha\alpha_{1}}}=y^{T^{o}}_{\alpha_{1}}(1\oplus
y^{T^{\prime o}}_{\alpha^{\prime}})^{-b^{T^{o}}_{\alpha\alpha_{1}}}.$

If $\alpha_{1}$ and $\alpha$ are sides of some self-folded triangles in
$T^{o}$, then $\alpha_{1},\alpha$ are loops of some self-folded triangles.
Assume that $\alpha_{2}=\alpha_{3}$ is the radius to some puncture $p$ and
$\alpha_{1}$ is the loop of a self-folded triangle with radius $r$ to a
puncture $q$. Thus we have

$y^{T^{\prime
o}}_{\alpha_{1}}=y^{T^{\prime}}_{r^{(q)}}=y^{T}_{r^{(q)}}(y^{T}_{\alpha_{2}^{(p)}})^{b^{T}_{\alpha_{2}^{(p)}r^{(p)}}}(1\oplus
y^{T}_{\alpha_{2}^{(p)}})^{-b^{T}_{\alpha_{2}^{(p)}r^{(p)}}}=y^{T^{o}}_{\alpha_{1}}(1\oplus(y^{T^{o}}_{\alpha})^{-1})^{-b^{T^{o}}_{\alpha\alpha_{1}}}=y^{T^{o}}_{\alpha_{1}}(1\oplus
y^{T^{\prime o}}_{\alpha^{\prime}})^{-b^{T^{o}}_{\alpha\alpha_{1}}}.$

The result follows.

The statement (b) follows immediately by
$y_{\alpha_{1}}^{T^{\prime}}=y_{\alpha_{1}}^{T}$.

(4) As
$x_{\alpha^{\prime}}=\frac{y^{T^{o}}_{\alpha}x_{\alpha_{2}}x_{\alpha_{4}}+x_{\alpha_{1}}x_{\alpha_{3}}}{x_{\alpha}\cdot(1\oplus
y^{T^{o}}_{\alpha})}$, we have

$x_{\alpha}=\frac{y^{T^{o}}_{\alpha}x_{\alpha_{2}}x_{\alpha_{4}}+x_{\alpha_{1}}x_{\alpha_{3}}}{x_{\alpha^{\prime}}\cdot(1\oplus
y^{T^{o}}_{\alpha})}=\frac{x_{\alpha_{2}}x_{\alpha_{4}}+(y^{T^{o}}_{\alpha})^{-1}x_{\alpha_{1}}x_{\alpha_{3}}}{x_{\alpha^{\prime}}\cdot((y^{T^{o}}_{\alpha})^{-1}\oplus
1)}=\frac{x_{\alpha_{2}}x_{\alpha_{4}}+x_{\alpha_{1}}x_{\alpha_{3}}y^{T^{\prime
o}}_{\alpha^{\prime}}}{x_{\alpha^{\prime}}\cdot(1\oplus y^{T^{\prime
o}}_{\alpha^{\prime}})}.$

$\square$

The following lemma follows by Definition 6.1 (2) (3).

######  Lemma 6.4.

Let $P,Q$ be two perfect matchings. If $P<Q$ are related by a twist on a tile
with diagonal labeled $\tau_{i}$ then
$\frac{y^{T}(Q)}{y^{T}(P)}=y^{T^{o}}_{\tau_{i}}.$

The following theorem is the first main result of this paper.

######  Theorem 6.5.

Let $\Sigma$ be an orbifold without orbifold points of weight $2$. Let $T^{o}$
be an ideal triangulation of $\Sigma$ and $\beta$ be an arc. Then in the
quantum cluster algebra $\mathcal{A}_{v}(\Sigma)$ we have

$X_{\beta}=\sum_{P\in\mathcal{L}(T^{o},\widetilde{\beta})}v^{w(P)}X^{T}(P),$

where $w:\mathcal{L}(T^{o},\widetilde{\beta})\to\mathbb{Z}$ is given by
Proposition 5.3.

###  6.2. Expansion formula for one end tagged notched arcs.

Suppose that $q$ is a puncture and $q\neq p$. Thus $\widetilde{\beta}=\beta$.
Assume $T$ contains no arc tagged notched at $q$. In this section, we give a
quantum expansion formula for $\beta^{(q)}=\widetilde{\beta}^{(q)}$.

For any triangle $\Delta_{j}(q)$ incident to $q$, recall that the three sides
are $\tau_{j-1}(q),\tau_{j}(q)$ and $\tau_{[j]}(q)$.

######  Definition 6.6.

With the notation in Section 4.2. For any
$(P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})$ with
$P\in\mathcal{P}(G_{T^{o},\widetilde{\beta}})$ and $j\in\\{1,\cdots,t\\}$,

  1. $(1)$

The _weight_ $x^{T}(\Delta_{j}(q))$ of $\Delta_{j}(q)$ is defined to be

$x^{T}(\Delta_{j}(q)):=x^{-1}_{\tau_{j-1}(q)}x_{\tau_{[j]}(q)}x^{-1}_{\tau_{j}(q)}.$

  2. $(2)$

The _weight_ $x^{T}(P,\Delta_{j}(q))$ of $(P,\Delta_{j}(q))$ is defined to be

(15)
$x^{T}(P,\Delta_{j}(q)):=x^{T}(P)x^{T}(\Delta_{j}(q))=x^{T}(P)x^{-1}_{\tau_{j-1}(q)}x_{\tau_{[j]}(q)}x^{-1}_{\tau_{j}(q)}.$

  3. $(3)$

The _height monomial_ $h^{T}(P,\Delta_{j}(q))$ of $(P,\Delta_{j}(q))$ is
defined as follows:

    1. (a)

if $\widetilde{\beta}(=\beta)\notin T^{o}$ then

(16) $h^{T}(P,\Delta_{j}(q)):=\begin{cases}h^{T}(P),&\mbox{if $E_{1}(q)\in P$
and $j=1$},\\\ h^{T}(P)h^{T}_{\tau_{1}(q)}h^{T}_{\tau_{2}(q)}\cdots
h^{T}_{\tau_{t-1}(q)}h^{T}_{\tau_{t}(q)},&\mbox{if $E_{2}(q)\in P$ and
$j=1$},\\\ h^{T}(P)h^{T}_{\tau_{1}(q)}h^{T}_{\tau_{2}(q)}\cdots
h^{T}_{\tau_{j-1}(q)},&\mbox{otherwise},\end{cases}$

where $E_{1}(q)$ and $E_{2}(q)$ are the edges of $G_{c}$ determined by (6) in
Section 4.

    2. (b)

if $\widetilde{\beta}(=\beta)\in T^{o}$ then

$h^{T}(P,\Delta_{j}(q)):=h^{T}_{\tau_{1}(q)}h^{T}_{\tau_{2}(q)}\cdots
h^{T}_{\tau_{j-1}(q)}.$

  4. $(4)$

The _specialized height monomial_ $y^{T}(P,\Delta_{j}(q))$ of
$(P,\Delta_{j}(q))$ is defined to be

(17) $y^{T}(P,\Delta_{j}(q)):=\Phi(h^{T}(P,\Delta_{j}(q))),$

where $\Phi$ is given by (14) in Definition 6.1.

  5. $(5)$

The _quantum weight_ $X^{T}(P,\Delta_{j}(q))$ of $(P,\Delta_{j}(q))$ is
defined to be the element $X^{a(P,\Delta_{j}(q))}$ such that

$X^{a(P,\Delta_{j}(q))}|_{v=1}=\frac{x^{T}(P,\Delta_{j}(q))y^{T}(P,\Delta_{j}(q))}{\bigoplus_{(R,\Delta)\in\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})}y^{T}(R,\Delta)}.$

The following lemmas follow by Definition 6.6.

######  Lemma 6.7.

Assume $P>Q$ are related by a twist on a tile $G$ with diagonal labeled $\tau$
and $(P,\Delta_{j}(q))$ covers $(Q,\Delta_{j}(q))$. Then
$\frac{y^{T}(P,\Delta_{j}(q))}{y^{T}(Q,\Delta_{j}(q))}=y^{T^{o}}_{\tau}$

######  Lemma 6.8.

If $(P,\Delta_{j+1}(q))$ covers $(P,\Delta_{j}(q))$ then
$\frac{y^{T}(P,\Delta_{j+1}(q))}{y^{T}(P,\Delta_{j}(q))}=y^{T^{o}}_{\tau_{j}(q)}.$

The following theorem is the second main result of this paper.

######  Theorem 6.9.

Let $\Sigma$ be an orbifold without orbifold points of weight $2$. Let $T^{o}$
be an ideal triangulation of $\Sigma$ and $T$ be the corresponding tagged
triangulation. For any $\beta$, let $\widetilde{\beta}$ be the curve
associated with $\beta$ by (4). Assume that $\widetilde{\beta}$ connects $p$
and a puncture $q$ with $q\neq p$. If $T$ contains no arc tagged notched at
$q$, then in the quantum cluster algebra $\mathcal{A}_{v}(\Sigma)$ we have

$X_{\beta^{(q)}}=\sum_{(P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})}v^{w(P,\Delta_{j}(q))}X^{T}(P,\Delta_{j}(q)),$

where $w:\mathcal{L}(T^{o},\widetilde{\beta}^{(q)})\to\mathbb{Z}$ is given by
Proposition 5.8.

###  6.3. Expansion formula for two ends tagged notched arcs.

Suppose that $p$ and $q$ are punctures. Assume that $T$ contains no arc tagged
notched at $p$ or $q$. If $\beta$ is a pending arc incident to puncture $p$,
let $X_{\widetilde{\beta}^{(p,q)}}=X_{\beta^{(p)}}$. Herein, we give an
expansion formula for $\widetilde{\beta}^{(p,q)}$.

For any triangle $\Delta_{i}(p)$ incident to $p$, recall that the three sides
are $\tau_{i-1}(p),\tau_{i}(p)$ and $\tau_{[i]}(p)$.

######  Definition 6.10.

With the notation in Section 4.3. Assume that $p$ and $q$ are punctures. For
any
$(\Delta_{i}(p),P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})$
with $P\in\mathcal{P}(G_{T^{o},\widetilde{\beta}})$ and
$i\in\\{1,\cdots,s\\},j\in\\{1,\cdots,t\\}$,

  1. $(1)$

The _weight_ $x^{T}(\Delta_{i}(p))$ of $\Delta_{i}(p)$ is defined to be

$x^{T}(\Delta_{i}(p)):=x^{-1}_{\tau_{i-1}(p)}x_{\tau_{[i]}(p)}x^{-1}_{\tau_{i}(p)}.$

  2. $(2)$

The _weight_ $x^{T}(\Delta_{j}(q))$ of $\Delta_{j}(q)$ is defined to be

$x^{T}(\Delta_{j}(q)):=x^{-1}_{\tau_{j-1}(q)}x_{\tau_{[j]}(q)}x^{-1}_{\tau_{j}(q)}.$

  3. $(3)$

The _weight_ $x^{T}\left(\Delta_{i}(p),P,\Delta_{j}(q)\right)$ of
$(\Delta_{i}(p),P,\Delta_{j}(q))$ is defined to be

(18)
$x^{T}\left(\Delta_{i}(p),P,\Delta_{j}(q)\right):=x^{-1}_{\tau_{i-1}(p)}x_{\tau_{[i]}(p)}x^{-1}_{\tau_{i}(p)}x^{T}(P)x^{-1}_{\tau_{j-1}(q)}x_{\tau_{[j]}(q)}x^{-1}_{\tau_{j}(q)}.$

  4. $(4)$

The _height monomial_ $h^{T}(\Delta_{i}(p),P,\Delta_{j}(q))$ of
$(\Delta_{i}(p),P,\Delta_{j}(q))$ is defined as follows:

    1. (a)

if $\widetilde{\beta}\notin T^{o}$ then

$h^{T}(\Delta_{i}(p),P,\Delta_{j}(q)):=\begin{cases}h^{T}(P,\Delta_{j}(q)),&\mbox{if
$E_{1}(p)\in P$ and $i=1$},\\\
h^{T}(P,\Delta_{j}(q))h^{T}_{\tau_{1}(p)}h^{T}_{\tau_{2}(p)}\cdots
h^{T}_{\tau_{s-1}(p)}h^{T}_{\tau_{s}(p)},&\mbox{if $E_{2}(p)\in P$ and
$i=1$},\\\ h^{T}(P,\Delta_{j}(q))h^{T}_{\tau_{1}(p)}h^{T}_{\tau_{2}(p)}\cdots
h^{T}_{\tau_{i-1}(p)},&\mbox{otherwise},\end{cases}$

where $E_{1}(p)$ is the edge of $G_{1}$ determined by (8) in Section 4.

    2. (b)

if $\widetilde{\beta}\in T^{o}$ and $s,t\geq 2$ then

(19) $h^{T}(\Delta_{i}(p),P_{\beta},\Delta_{j}(q)):=\begin{cases}1,&\mbox{if
$i=1$ and $j=t$},\\\ h^{T}_{\tau_{1}(p)}h^{T}_{\tau_{2}(p)}\cdots
h^{T}_{\tau_{s-1}(p)}h^{T}_{\tau_{1}(q)}\cdots
h^{T}_{\tau_{t-1}(q)}(h^{T}_{\beta})^{2},&\mbox{if $i=s$ and $j=1$},\\\
h^{T}_{\tau_{1}(p)}h^{T}_{\tau_{2}(p)}\cdots
h^{T}_{\tau_{i-1}(p)}h^{T}_{\tau_{1}(q)}\cdots
h^{T}_{\tau_{j-1}(q)}h^{T}_{\beta},&\mbox{otherwise}.\end{cases}$

  5. $(5)$

The _specialized height monomial_ $y^{T}(\Delta_{i}(p),P,\Delta_{j}(q))$ of
$(\Delta_{i}(p),P,\Delta_{j}(q))$ is defined to be

$y^{T}(\Delta_{i}(p),P,\Delta_{j}(q)):=\Phi(h^{T}(\Delta_{i}(p),P,\Delta_{j}(q))),$

where $\Phi$ is given by (14) in Definition 6.1.

  6. $(6)$

The _quantum weight_ $X^{T}(\Delta_{i}(p),P,\Delta_{j}(q))$ of
$(\Delta_{i}(p),P,\Delta_{j}(q))$ is defined to be the element
$X^{a(\Delta_{i}(p),P,\Delta_{j}(q))}$ such that

$X^{a(\Delta_{i}(p),P,\Delta_{j}(q))}|_{v=1}=\frac{x^{T}(\Delta_{i}(p),P,\Delta_{j}(q))y^{T}(\Delta_{i}(p),P,\Delta_{j}(q))}{{\bigoplus_{(\Delta,R,\Delta^{\prime})\in\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})}y^{T}(\Delta,R,\Delta^{\prime})}}.$

From the definition of weight
$x^{T}\left(\Delta_{i}(p),P,\Delta_{j}(q)\right)$ and
$x^{T}\left(P,\Delta_{j}(q)\right)$, we have

$x^{T}\left(\Delta_{i}(p),P,\Delta_{j}(q)\right)=x^{T}(\Delta_{i}(p))x^{T}\left(P,\Delta_{j}(q)\right).$

The following lemmas follow by Definition 6.10 (4) (5).

######  Lemma 6.11.

Let $P,Q$ be two perfect matchings such that $P>Q$ are related by a twist on a
tile $G$ with diagonal labeled $\tau$. If $(\Delta_{i}(p),P,\Delta_{j}(q))$
covers $(\Delta_{i}(p),Q,\Delta_{j}(q))$ then

$\frac{y^{T}(\Delta_{i}(p),P,\Delta_{j}(q))}{y^{T}(\Delta_{i}(p),Q,\Delta_{j}(q))}=y^{T^{o}}_{\tau}.$

######  Lemma 6.12.

If $(\Delta_{i+1}(p),P,\Delta_{j}(q))$ covers
$(\Delta_{i}(p),P,\Delta_{j}(q))$ then

$\frac{y^{T}(\Delta_{i+1}(p),P,\Delta_{j}(q))}{y^{T}(\Delta_{i}(p),P,\Delta_{j}(q))}=y^{T^{o}}_{\tau_{i}(p)}.$

######  Lemma 6.13.

If $(\Delta_{i}(p),P,\Delta_{j+1}(q))$ covers
$(\Delta_{i}(p),P,\Delta_{j}(q))$ then

$\frac{y^{T}(\Delta_{i}(p),P,\Delta_{j+1}(q))}{y^{T}(\Delta_{i}(p),P,\Delta_{j}(q))}=y^{T^{o}}_{\tau_{j}(q)}.$

The following theorem is the third main result of this paper.

######  Theorem 6.14.

Let $\Sigma$ be an orbifold without orbifold points of weight $2$. Let $T^{o}$
be an ideal triangulation of $\Sigma$ and $\beta$ be an arc. Let $T$ be the
corresponding tagged triangulation of $T^{o}$. Let $\widetilde{\beta}$ be the
curve associated with $\beta$ given by (4). Assume that $\widetilde{\beta}$
connects two punctures $p$ and $q$. If $T$ contains no arc tagged notched at
$p$ or $q$, then in the quantum cluster algebra $\mathcal{A}_{v}(\Sigma)$ we
have

$X_{\widetilde{\beta}^{(p,q)}}=\sum_{(\Delta_{i}(p),P,\Delta_{j}(q))\in\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})}v^{w(\Delta_{i}(p),P,\Delta_{j}(q))}X^{T}(\Delta_{i}(p),P,\Delta_{j}(q)),$

where $w:\mathcal{L}(T^{o},\widetilde{\beta}^{(p,q)})\to\mathbb{Z}$ is given
by Proposition 5.15.

As an immediate corollary of Theorem 6.5, Theorem 6.9, and Theorem 6.14, we
obtain the positivity for quantum cluster algebras from orbifolds.

######  Theorem 6.15.

Positivity conjecture holds for quantum cluster algebras from orbifolds
without orbifold points of weight $2$.

######  Remark 6.16.

In case $\Sigma$ be an orbifold with or without orbifold points of weight $2$,
our method can be also applied to its associated orbifold $\hat{\Sigma}$,
which is obtained from $\Sigma$ by making all orbifold points of weight $2$ to
punctures.

##  7\. Partition bijection between perfect matchings

Let $T^{o}$ be an ideal triangulation and $\gamma$ be a curve connecting two
marked points $p$ and $q$. Choose an orientation of $\gamma$, assume that
$\gamma$ crosses $T^{o}$ at $p_{1},\cdots,p_{c}$ sequentially. Denote
$p_{0}=p$ and $p_{c+1}=q$. Suppose that $p_{j}\in\tau_{i_{j}}$ for
$j=1,\cdots,c$. Given a non-self-folded arc $\alpha\in T^{o}$, denote