# Topological model for q-deformed rational number and categorification

Li Fan  Fl: Department of Mathematical Sciences, Tsinghua University, 100084
Beijing, China<EMAIL_ADDRESS>and  Yu Qiu  Qy: Yau Mathematical
Sciences Center and Department of Mathematical Sciences, Tsinghua University,
100084 Beijing, China. & Beijing Institute of Mathematical Sciences and
Applications, Yanqi Lake, Beijing, China<EMAIL_ADDRESS>
###### Abstract.

Let $\mathbf{D}_{3}$ be a bigraded 3-decorated disk with an arc system
${\mathbf{A}}$. We associate a bigraded simple closed arc
$\widehat{\eta}_{\frac{r}{s}}$ on $\mathbf{D}_{3}$ to any rational number
$\frac{r}{s}\in\overline{\mathbb{Q}}=\mathbb{Q}\cup\\{\infty\\}$. We show that
the right (resp. left) $q$-deformed rational numbers associated to
$\frac{r}{s}$, in the sense of [MGO1] (resp. [BBL]) can be naturally
calculated by the $\mathfrak{q}$-intersection between
$\widehat{\eta}_{\frac{r}{s}}$ and ${\mathbf{A}}$ (resp. dual arc system
${\mathbf{A}}^{*}$). The Jones polynomials of rational knots can be also given
by such intersections. Moreover, the categorification of
$\widehat{\eta}_{\frac{r}{s}}$ is given by the spherical object
$X_{\frac{r}{s}}$ in the Calabi-Yau-$\mathbb{X}$ category of Ginzburg dga of
type $A_{2}$. Reduce to CY-2 case, we recover result of [BBL] with a slight
improvement.

##  1\. Introductions

The notion of (right) $q$-deformed rational numbers $[\frac{r}{s}]^{\sharp}$
was originally introduced by Morier-Genoud and Ovsienko in [MGO1] via
continued fractions. They also developed $q$-deformations to irrational
numbers in [MGO2] by the convergency property. Such $q$-deformations own many
good combinatorial properties and are related to a wide variety of areas, such
as the Farey triangulation, $F$-polynomials of cluster algebras and the Jones
polynomial of rational (two-bridge) knots [MGO1]. Motivated by the study of
compactification of spaces of stability conditions, Bapat, Becker and Licata
[BBL] introduced a twin notion, the left $q$-deformation
$[\frac{r}{s}]^{\flat}$, which also shares all the good properties of
$[\frac{r}{s}]^{\sharp}$. They showed that the two $q$-deformations can be
both described via the action of
$\operatorname{PSL}_{2,\mathfrak{q}}(\mathbb{Z})$ by fractional linear
transformations. Moreover, Farey graph plays an important role in the
definition of $q$-deformations, where the edges are assigned weights according
to some iterative rules [MGO1].

On the other hand, the homotopy classes of simple closed curves on torus with
at most one boundary can be parameterized by
$\overline{\mathbb{Q}}=\mathbb{Q}\cup\\{\infty\\}$. We aim to give a
topological realization of $q$-deformations and their categorification. The
topological model we use is the decorated surface
${\mathbf{S}}_{\bigtriangleup}$ with bigrading introduced by Khovanov and
Seidel in [KS]. The bigrading of arcs provides bi-indices for their
intersections, which we call $\mathfrak{q}$-intersections. We consider the
$A_{2}$ case, where ${\mathbf{S}}_{\bigtriangleup}=\mathbf{D}_{3}$ is a disk
with three decorations and the set of simple closed arcs on $\mathbf{D}_{3}$
can be parameterized by $\overline{\mathbb{Q}}$. We show that the right/left
$q$-deformed rationals can be naturally calculated by the
$\mathfrak{q}$-intersections between corresponding arcs (Theorem 3.18 and
Theorem 3.21). The topological realization directly implies many combinatorial
properties of $q$-deformations, including positivity and specialization
(Corollary 3.23). Surprisingly, the bi-index always collapses into one, which
is not obvious from the construction/definition of
$\mathfrak{q}$-intersection.

For the categorification, we consider the Calabi-Yau-$\mathbb{X}$ category
$\operatorname{\mathcal{D}}_{\mathbb{X}}({\mathbf{S}}_{\bigtriangleup})$
associated to ${\mathbf{S}}_{\bigtriangleup}$ (cf. [KS, IQZ]), which is the
perfect valued derived category of the bigraded Ginzburg algebra constructed
from ${\mathbf{S}}_{\bigtriangleup}$. The $\mathbb{X}$-spherical objects in
$\operatorname{\mathcal{D}}_{\mathbb{X}}({\mathbf{S}}_{\bigtriangleup})$
correspond to the bigraded simple closed arcs in
${\mathbf{S}}_{\bigtriangleup}$, and their $\mathfrak{q}$-dimensions of
$\operatorname{Hom}$-spaces equal to the $\mathfrak{q}$-intersections between
the corresponding arcs [KS, IQZ]. One can specialize $\mathbb{X}=N$, and
$\operatorname{\mathcal{D}}_{\mathbb{X}}({\mathbf{S}}_{\bigtriangleup})$
becomes a Calabi-Yau-$N$ category, for any integer $N\geq 2$. When $N=3$,
$\operatorname{\mathcal{D}}_{3}({\mathbf{S}}_{\bigtriangleup})$ provides an
additive categeorification of cluster algebras of surface type (cf. e.g. [Q]).
When ${\mathbf{S}}_{\bigtriangleup}=\mathbf{D}_{3}$ and $N=2$, we recover
[BBL]’s result (with a slight improvement).

The paper is organized as follows. In section 2, we recall several equivalent
definitions of left and right $\mathfrak{q}$-deformed rationals from [MGO1,
BBL], via continued fractions, braid twist action and $\mathfrak{q}$-weighted
Farey graph respectively. In section 3, we recall the graded decorated surface
in the sense of [KS, IQZ] and prove the main results. In section 4, we give
the categorification and in section 5, we discuss reduction and the relation
with Jones polynomials.

### Acknowledgment

Fl is grateful to Qy for leading her into this research area and providing her
lots of help and supervisions. This work is inspired by the work of Morier-
Genoud, Ovsienko [MGO1] and Bapat, Becker, Licata [BBL]. Qy is supported by
National Key R&D Program of China (No.2020YFA0713000) and National Natural
Science Foundation of China (No.12031007 and No.12271279).

##  2\. $\mathfrak{q}$-deformed rationals and Farey graph

In the paper, we fix the following conventions.

######  Conventions.

  * •

Let $\mathfrak{q}$ be a formal parameter.

  * •

A rational number always belongs to
$\overline{\mathbb{Q}}:=\mathbb{Q}\cup\\{\infty\\}$. We also denote
$\overline{\mathbb{Q}^{\geq 0}}:=\mathbb{Q}^{\geq 0}\cup\\{\infty\\}$. We
usually state the results for $\overline{\mathbb{Q}}$ but prove the non-
negative case since the negative case holds by symmetry.

  * •

We denote a rational number by $\frac{r}{s}$, including the exceptional cases
when $0=\frac{0}{1}$ and $\infty=\frac{1}{0}$. We assume that $\frac{r}{s}$ is
irreducible.

###  2.1. Right and left $\mathfrak{q}$-deformed rationals

We first recall the definitions of right and left $\mathfrak{q}$-deformations
of rational numbers via finite continued fractions and formulate their basic
properties. For a positive rational $\frac{r}{s}$, it can be expressed as an
expansion of continued fraction as

$\frac{r}{s}\quad=\quad
a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{\ddots+\cfrac{1}{a_{2m}}}}\colon=[a_{1},\ldots,a_{2m}],$
(2.1)

for $a_{1}\in\mathbb{N}$ and
$a_{2},\cdots,a_{2m}\in\mathbb{N}\setminus\\{0\\}$, which is known as the
(regular) continued fraction (expression). For the exceptional cases, we
denote $0=[-1,1]$ and $\infty=[]$.

For a non-negative integer $a$, the _right $\mathfrak{q}$-deformation_ is
defined as

$[a]_{\mathfrak{q}}^{\sharp}:=\frac{1-\mathfrak{q}^{a}}{1-\mathfrak{q}}=1+\mathfrak{q}+\mathfrak{q}^{2}+\cdots+\mathfrak{q}^{a-1},$

and the corresponding _left $\mathfrak{q}$-deformation_ is defined as

$[a]_{\mathfrak{q}}^{\flat}:=\frac{1-\mathfrak{q}^{a-1}+\mathfrak{q}^{a}-\mathfrak{q}^{a+1}}{1-\mathfrak{q}}=1+\mathfrak{q}+\cdots+\mathfrak{q}^{a-2}+\mathfrak{q}^{a}.$

######  Definition 2.1 ([MGO1, BBL]).

Let $\frac{r}{s}\in\mathbb{Q}^{+}$ be a rational number with continued
fraction expansion $[a_{1},\ldots,a_{2m}]$.

  1. 1∘.

We define its _right $\mathfrak{q}$-deformation_ by the following formula:

$\Big{[}\frac{r}{s}\Big{]}^{\sharp}_{\mathfrak{q}}:=[a_{1}]^{\sharp}_{\mathfrak{q}}+\cfrac{\mathfrak{q}^{a_{1}}}{[a_{2}]^{\sharp}_{\mathfrak{q}^{-1}}+\cfrac{\mathfrak{q}^{-a_{2}}}{[a_{3}]^{\sharp}_{\mathfrak{q}}+\cfrac{\mathfrak{q}^{a_{3}}}{[a_{4}]^{\sharp}_{\mathfrak{q}^{-1}}+\cfrac{\mathfrak{q}^{-a_{4}}}{\cfrac{\ddots}{[a_{2m-1}]^{\sharp}_{\mathfrak{q}}+\cfrac{\mathfrak{q}^{a_{2m-1}}}{[a_{2m}]^{\sharp}_{\mathfrak{q}^{-1}}}}}}}}\quad.$
(2.2)

  2. 2∘.

We define its _left $\mathfrak{q}$-deformation_ by the following formula:

$\Big{[}\frac{r}{s}\Big{]}_{\mathfrak{q}}^{\flat}:=[a_{1}]^{\sharp}_{\mathfrak{q}}+\cfrac{\mathfrak{q}^{a_{1}}}{[a_{2}]^{\sharp}_{\mathfrak{q}^{-1}}+\cfrac{\mathfrak{q}^{-a_{2}}}{[a_{3}]^{\sharp}_{\mathfrak{q}}+\cfrac{\mathfrak{q}^{a_{3}}}{[a_{4}]^{\sharp}_{\mathfrak{q}^{-1}}+\cfrac{\mathfrak{q}^{-a_{4}}}{\cfrac{\ddots}{[a_{2m-1}]^{\sharp}_{\mathfrak{q}}+\cfrac{\mathfrak{q}^{a_{2m-1}}}{[a_{2m}]^{\flat}_{\mathfrak{q}^{-1}}}}}}}}\quad.$
(2.3)

We normalize them as

$\left[\frac{r}{s}\right]_{\mathfrak{q}}^{\sharp}=\frac{\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}(r/s)}{\textbf{S}_{\mathfrak{q}}^{\sharp}(r/s)},\quad\left[\frac{r}{s}\right]_{\mathfrak{q}}^{\flat}=\frac{\operatorname{\textbf{R}}_{\mathfrak{q}}^{\flat}(r/s)}{\textbf{S}_{\mathfrak{q}}^{\flat}(r/s)},$

so that the denominators are polynomials of $\mathfrak{q}$ with lowest non-
zero constant term. For $0$ and $\infty$, we set

$\displaystyle\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}(0)=0,\quad\textbf{S}_{\mathfrak{q}}^{\sharp}(0)=1,\quad\operatorname{\textbf{R}}_{\mathfrak{q}}^{\flat}(0)=\mathfrak{q}-1,\quad\textbf{S}_{\mathfrak{q}}^{\flat}(0)=\mathfrak{q};$
$\displaystyle\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}(\infty)=1,\quad\textbf{S}_{\mathfrak{q}}^{\sharp}(\infty)=0,\quad\operatorname{\textbf{R}}_{\mathfrak{q}}^{\flat}(\infty)=1,\quad\textbf{S}_{\mathfrak{q}}^{\flat}(\infty)=1-\mathfrak{q}.$

Next, we consider the group

$\operatorname{PSL}_{2}(\mathbb{Z})=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}|a,b,c,d\in\mathbb{Z},ad-bc=1\right\\},$

which is generated by

$t_{1}:=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix},\quad
t_{2}:=\begin{pmatrix}1&0\\\ -1&1\end{pmatrix}.$

It acts on rational numbers $\overline{\mathbb{Q}}$ by linear fractional
transformation as

$\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\cdot\bigg{(}\cfrac{r}{s}\bigg{)}=\frac{ar+bs}{cr+ds},$ (2.4)

where $\frac{r}{s}\in\overline{\mathbb{Q}}$ and $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{PSL}_{2}(\mathbb{Z})$. For a rational number
$\frac{r}{s}\in\overline{\mathbb{Q}^{\geq 0}}$ with continued fraction
expansion as (2.1), it is well-known that (cf. [BBL, Proposition 2.2])

$\cfrac{r}{s}=t_{1}^{a_{1}}t_{2}^{-a_{2}}t_{1}^{a_{3}}t_{2}^{-a_{4}}\cdots
t_{1}^{a_{2m-1}}t_{2}^{-a_{2m}}(\cfrac{1}{0}).$ (2.5)

######  Proposition/Definition 2.2 ([MGO1, BBL]).

Consider the $\mathfrak{q}$-deformation
$\operatorname{PSL}_{2,\mathfrak{q}}(\mathbb{Z})$ of
$\mathrm{PSL}_{2}(\mathbb{Z})$, which is generated by

$\displaystyle t_{1,\mathfrak{q}}=\begin{pmatrix}\mathfrak{q}&1\\\ 0&1\\\
\end{pmatrix},\quad t_{2,\mathfrak{q}}=\begin{pmatrix}1&0\\\
-\mathfrak{q}&\mathfrak{q}\\\ \end{pmatrix}.$

For a rational number $\frac{r}{s}\in\overline{\mathbb{Q}^{\geq 0}}$ with
expression (2.1), we have

$\begin{cases}\left[\displaystyle\frac{r}{s}\right]^{\sharp}_{\mathfrak{q}}&=t_{1,\mathfrak{q}}^{a_{1}}t_{2,\mathfrak{q}}^{-a_{2}}t_{1,\mathfrak{q}}^{a_{3}}t_{2,\mathfrak{q}}^{-a_{4}}\cdots
t_{1,\mathfrak{q}}^{a_{2m-1}}t_{2,\mathfrak{q}}^{-a_{2m}}\left(\displaystyle\frac{1}{0}\right),\\\
\left[\displaystyle\frac{r}{s}\right]_{\mathfrak{q}}^{\flat}&=t_{1,\mathfrak{q}}^{a_{1}}t_{2,\mathfrak{q}}^{-a_{2}}t_{1,\mathfrak{q}}^{a_{3}}t_{2,\mathfrak{q}}^{-a_{4}}\cdots
t_{1,\mathfrak{q}}^{a_{2m-1}}t_{2,\mathfrak{q}}^{-a_{2m}}\left(\displaystyle\frac{1}{1-\mathfrak{q}}\right).\end{cases}$
(2.6)

###  2.2. $\mathfrak{q}$-deformations via Farey graph

The classical Farey graph $\operatorname{FG}$ is an infinite graph with
vertices set

$\operatorname{FG}_{0}=\overline{\mathbb{Q}}.$

There is an edge between $\frac{p}{q}$ and $\frac{u}{v}$ if and only if $pv-
uq=\pm 1$ (see Figure 1). If $\frac{p}{q}$ and $\frac{u}{v}$ are connected by
an edge, we define their _Farey sum_ by

$\frac{p}{q}\oplus\frac{u}{v}:=\frac{p+u}{q+v}.$

$\frac{1}{3}$$0$$\frac{1}{2}$$\frac{2}{3}$$1$$\frac{3}{2}$$-1$$-\frac{1}{2}$$-2$$\infty$$3$$2$
Figure 1. The Farey graph

Moreover, $\operatorname{FG}_{0}$ is parametrized by homotopy classes of
simple closed arcs on torus with at most one boundary and the edges are those
arcs with intersection number one. $\operatorname{PSL}_{2}(\mathbb{Z})$ acts
on $\operatorname{FG}$ by (2.4) taking one edge to another. In particular, if
$T\in\operatorname{PSL}_{2}(\mathbb{Z})$ takes the form

$T_{\frac{r}{s}}=\begin{pmatrix}1+rs&-r^{2}\\\ s^{2}&1-rs\end{pmatrix},$

then it is a rotation which fixes $\frac{r}{s}$.

######  Lemma/Definition 2.3 ([MGO1, Section 2.2]).

Let $\frac{r}{s}\in\mathbb{Q}^{+}$ be any rational number with continued
fraction expansion as (2.1). Then it can be uniquely written as Farey sum of
two rationals $\frac{p}{q},\frac{u}{v}\in\overline{\mathbb{Q}^{\geq 0}}$, i.e.

$\frac{r}{s}=\frac{p}{q}\oplus\frac{u}{v},$

with $uq-pv=1$ and $\frac{p}{q}<\frac{r}{s}<\frac{u}{v}$. In fact,

$\frac{p}{q}=\begin{cases}[a_{1},a_{2},\ldots,a_{2m-2}+1],&~{}\mbox{if}~{}a_{2m-1}=1\mbox{and}~{}m>1;\\\
[a_{1},a_{2},\ldots,a_{2m-1}-1,1],&~{}\mbox{otherwise},\end{cases}$ (2.7)

and

$\frac{u}{v}=\begin{cases}[a_{1},a_{2},\ldots,a_{2m-1},a_{2m}-1],&~{}\mbox{if}~{}a_{2m}\geq
2;\\\ [a_{1},a_{2},\ldots,a_{2m-2}],&~{}\mbox{if}~{}a_{2m}=1.\end{cases}$
(2.8)

Moreover, there is an associated integer defined as

$l=l(\frac{r}{s})=\begin{cases}0,&~{}\mbox{if}~{}a_{2m}\geq 2;\\\
a_{2m-1},&~{}\mbox{if}~{}a_{2m}=1.\end{cases}$ (2.9)

In particular, we have $l(n+1=\frac{n}{1}\oplus\frac{1}{0})=n-1$.

On the other hand, $l$ can also be defined for an edge in $\operatorname{FG}$
connecting $\frac{p}{q}$ and $\frac{u}{v}$, provided
$\frac{p}{q}<\frac{u}{v}$. More precisely,
$l(\frac{p}{q},\frac{u}{v}):=l(\frac{p}{q}\oplus\frac{u}{v})$.

As in [MGO1], we assign a weight to each edge of the Farey graph, that goes
along with the right or left $\mathfrak{q}$-deformations associated to
vertices. Then the right and left $\mathfrak{q}$-deformations can also be
defined via $\mathfrak{q}$-Farey sum.

######  Proposition/Definition 2.4 ([MGO1]).

Let $\frac{r}{s}\in\mathbb{Q}^{+}$ be a rational with the decomposition
$\frac{r}{s}=\frac{p}{q}\oplus\frac{u}{v}$ and $l=l(\frac{p}{q},\frac{u}{v})$
as above. For right $\mathfrak{q}$-deformation, we have __

$\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}(\frac{r}{s}):=\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}(\frac{p}{q})+\mathfrak{q}^{l+1}\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}(\frac{u}{v}),\quad\textbf{S}_{\mathfrak{q}}^{\sharp}(\frac{r}{s}):=\textbf{S}_{\mathfrak{q}}^{\sharp}(\frac{p}{q})+\mathfrak{q}^{l+1}\textbf{S}_{\mathfrak{q}}^{\sharp}(\frac{u}{v}).$
(2.10)

For left $\mathfrak{q}$-deformation, if we set
_$\overline{\textbf{R}}_{\mathfrak{q}}^{\flat}(0)=\textbf{S}_{\mathfrak{q}}^{\flat}(0),\overline{\textbf{R}}_{\mathfrak{q}}^{\flat}(\infty)=\textbf{S}_{\mathfrak{q}}^{\flat}(\infty)$_
and define __

$\overline{\textbf{R}}_{\mathfrak{q}}^{\flat}(\frac{r}{s}):=\overline{\textbf{R}}_{\mathfrak{q}}^{\flat}(\frac{p}{q})+(\mathfrak{q}^{-1})^{l+1}\overline{\textbf{R}}_{\mathfrak{q}}^{\flat}(\frac{u}{v}),\quad\overline{\textbf{S}}_{\mathfrak{q}}^{\flat}(\frac{r}{s}):=\overline{\textbf{S}}_{\mathfrak{q}}^{\flat}(\frac{p}{q})+(\mathfrak{q}^{-1})^{l+1}\overline{\textbf{S}}_{\mathfrak{q}}^{\flat}(\frac{u}{v}),$
(2.11)

then we have __

$\left[\frac{r}{s}\right]_{\mathfrak{q}}^{\flat}=\frac{\overline{\textbf{R}}_{\mathfrak{q}}^{\flat}(r/s)}{\overline{\textbf{S}}_{\mathfrak{q}}^{\flat}(r/s)}.$

We label a weight to each edge and a right or left $\mathfrak{q}$-deformation
to each vertex in Farey graph, which are drawn in Figure 2 and Figure 3
respectively. Here the integer $l=l(\frac{p}{q},\frac{u}{v})$ as above.

$\displaystyle\left[\frac{0}{1}\right]_{\mathfrak{q}}^{\sharp}$$\displaystyle\left[\frac{1}{0}\right]_{\mathfrak{q}}^{\sharp}$$\displaystyle\left[\frac{1}{1}\right]_{\mathfrak{q}}^{\sharp}$$1$$1$$\mathfrak{q}^{-1}$$\displaystyle\frac{\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}(p/q)}{\textbf{S}_{\mathfrak{q}}^{\sharp}(p/q)}$$\displaystyle\frac{\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}(u/v)}{\textbf{S}_{\mathfrak{q}}^{\sharp}(u/v)}$$\displaystyle\frac{\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}(r/s)}{\textbf{S}_{\mathfrak{q}}^{\sharp}(r/s)}$$1$$\mathfrak{q}^{l+1}$$\mathfrak{q}^{l}$
Figure 2. The right $\mathfrak{q}$-deformation via $\mathfrak{q}$-Farey sum
$\displaystyle\left[\frac{0}{1}\right]_{\mathfrak{q}}^{\flat}$$\displaystyle\left[\frac{1}{0}\right]_{\mathfrak{q}}^{\flat}$$\displaystyle\left[\frac{1}{1}\right]_{\mathfrak{q}}^{\flat}$$1$$1$$q$$\displaystyle\frac{\overline{\textbf{R}}_{\mathfrak{q}}^{\flat}(p/q)}{\overline{\textbf{S}}_{\mathfrak{q}}^{\flat}(p/q)}$$\displaystyle\frac{\overline{\textbf{R}}_{\mathfrak{q}}^{\flat}(u/v)}{\overline{\textbf{S}}_{\mathfrak{q}}^{\flat}(u/v)}$$\displaystyle\frac{\overline{\textbf{R}}_{\mathfrak{q}}^{\flat}(r/s)}{\overline{\textbf{S}}_{\mathfrak{q}}^{\flat}(r/s)}$$1$$(\mathfrak{q}^{-1})^{l+1}$$(\mathfrak{q}^{-1})^{l}$
Figure 3. The left $\mathfrak{q}$-deformation via $\mathfrak{q}$-Farey sum

######  Remark 2.5.

For $\frac{r}{s}\in\mathbb{Q}^{+}$ with continued fraction expression
$[a_{1},\ldots,a_{2m}]$, we have

$-\cfrac{r}{s}=[-a_{1},\ldots,-a_{2m}],$ (2.12)

and

$-\cfrac{r}{s}=t_{1}^{-a_{1}}t_{2}^{a_{2}}t_{1}^{-a_{3}}t_{2}^{a_{4}}\cdots
t_{1}^{-a_{2m-1}}t_{2}^{a_{2m}}(\cfrac{1}{0}).$ (2.13)

The right and left $\mathfrak{q}$-deformations for negative rational numbers
are defined as:

$\begin{cases}\left[\displaystyle-\frac{r}{s}\right]^{\sharp}_{\mathfrak{q}}&=t_{1,\mathfrak{q}}^{-a_{1}}t_{2,\mathfrak{q}}^{a_{2}}t_{1,\mathfrak{q}}^{-a_{3}}t_{2,\mathfrak{q}}^{a_{4}}\cdots
t_{1,\mathfrak{q}}^{-a_{2m-1}}t_{2,\mathfrak{q}}^{a_{2m}}\left(\displaystyle\frac{1}{0}\right),\\\
\left[\displaystyle-\frac{r}{s}\right]_{\mathfrak{q}}^{\flat}&=t_{1,\mathfrak{q}}^{-a_{1}}t_{2,\mathfrak{q}}^{a_{2}}t_{1,\mathfrak{q}}^{-a_{3}}t_{2,\mathfrak{q}}^{a_{4}}\cdots
t_{1,\mathfrak{q}}^{-a_{2m-1}}t_{2,\mathfrak{q}}^{a_{2m}}\left(\displaystyle\frac{1}{1-\mathfrak{q}}\right).\end{cases}$
(2.14)

In fact, we can obtain $\mathfrak{q}$-deformations of negative rationals from
positive ones by the following formula:

$\left[-\frac{r}{s}\right]_{\mathfrak{q}}^{*}:=-\mathfrak{q}^{-1}\left[\frac{r}{s}\right]_{\mathfrak{q}^{-1}}^{*},$
(2.15)

where $*\in\\{\sharp,\flat\\}$.

##  3\. The topological model

In this section, we introduce decorated (marked) surfaces as the topological
model which we will use. We first summarize the setting and results in [KS,
IQZ] and then show that the $\mathfrak{q}$-intersections of certain arcs
describe the left/right $\mathfrak{q}$-deformations for rational numbers.

###  3.1. Decorated surfaces

Let $\mathbf{S}$ be an oriented surface with non-empty boundary
$\partial\mathbf{S}$ and we denote its interior by
$\mathbf{S}^{\circ}=\mathbf{S}\setminus(\partial\mathbf{S})$. We decorate
$\mathbf{S}$ with a finite set $\bigtriangleup$ of points (_decorations_) in
$\mathbf{S}^{\circ}$, denoted by ${\mathbf{S}}_{\bigtriangleup}$.

Let
${\mathbf{S}}_{\bigtriangleup}^{\circ}=\mathbf{S}\setminus(\partial\mathbf{S}\cup\bigtriangleup)$.
An _arc_ $c$ in ${\mathbf{S}}_{\bigtriangleup}$ is a curve
$c:[0,1]\rightarrow\mathbf{S}$ such that
$c(t)\in{\mathbf{S}}_{\bigtriangleup}^{\circ}$ for any $t\in(0,1)$. The
_inverse_ $\overline{c}$ of an arc $c$ is defined as $\overline{c}(t)=c(1-t)$
for any $t\in[0,1]$.

######  Definition 3.1.

A _closed arc_ $c$ is an arc whose endpoints $c(0)$ and $c(1)$ are in
$\bigtriangleup$. It is _simple_ if moreover it satisfies $c(0)\neq c(1)$,
without self-intersections in ${\mathbf{S}}_{\bigtriangleup}^{\circ}$. We
denote by $\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})$ the set of simple
closed arcs.

In this paper, we always consider arcs up to taking inverse and homotopy
relative to endpoints and exclude the arcs which are isotopic to a point in
${\mathbf{S}}_{\bigtriangleup}$. Two arcs are in _minimal position_ if their
intersection is minimal in the homotopy class. For three simple closed arcs,
they form a _contractible triangle_ if they bound a disk which is
contractible. For
$\sigma,\tau\in\operatorname{CA}({\mathbf{S}}_{\bigtriangleup}),$ we use the
notations

$\operatorname{Int}_{\Pi}(\sigma,\tau):=\left|\cap(\sigma,\tau)\cap\Pi\right|,$

where $\Pi=\bigtriangleup$ or ${\mathbf{S}}_{\bigtriangleup}^{\circ}$, a
subset of ${\mathbf{S}}_{\bigtriangleup}$, and

$\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\sigma,\tau):=\frac{1}{2}\cdot\operatorname{Int}_{\bigtriangleup}(\sigma,\tau)+\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}^{\circ}}(\sigma,\tau).$
(3.1)

The _mapping class group_ $\operatorname{MCG}({\mathbf{S}}_{\bigtriangleup})$
of a decorated surface ${\mathbf{S}}_{\bigtriangleup}$ consists of the isotopy
classes of the homeomorphisms of $\mathbf{S}$ which fix $\partial\mathbf{S}$
pointwise and fix $\triangle$ setwise. For any
$\alpha\in\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})$, the associated
_braid twist_ $B_{\alpha}\in\operatorname{MCG}({\mathbf{S}}_{\bigtriangleup})$
is defined in Figure 4. We have the formula

$B_{\Psi(\alpha)}=\Psi\circ B_{\alpha}\circ\Psi^{-1}$ (3.2)

for any $\alpha\in\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})$ and
$\Psi\in\operatorname{MCG}({\mathbf{S}}_{\bigtriangleup})$. We define
$\operatorname{BT}({\mathbf{S}}_{\bigtriangleup})$ to be the subgroup of
$\operatorname{MCG}({\mathbf{S}}_{\bigtriangleup})$ generated by $B_{\alpha}$
for $\alpha\in\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})$.

$\bullet$$\circ$$\bullet$$\circ$$B_{\alpha}$

α$\bullet$$\circ$$\bullet$$\circ$

Figure 4. The braid twist

The braid twist can be illustrated by smoothing out.

######  Construction 3.2.

For any $\sigma,\tau\in\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})$ with
$\sigma(0)=\tau(0)=z\in\bigtriangleup$. The extension $\sigma\wedge\tau$ of
$\sigma$ by $\tau$ (with respect to the common starting point) is defined in
Figure 5, which is the operation of smoothing out the intersection moving from
$\sigma$ to $\tau$ clockwisely.

$\tau$$\sigma$$\sigma\wedge\tau$$\bullet$$\circ$$z$ Figure 5. The extension as
smoothing out

Notice that if
$\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\sigma,\tau)=\frac{1}{2}$,
i.e. they only intersect at one decoration, then

$\sigma\wedge\tau=B_{\tau}(\sigma)=B^{-1}_{\sigma}(\tau).$ (3.3)

######  Lemma 3.3.

Assume that $|\bigtriangleup|\geq 3$. For any
$\eta\in\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})$, we have

$\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})=\operatorname{BT}({\mathbf{S}}_{\bigtriangleup})\cdot\\{\eta\\}.$
(3.4)

###### Proof.

Let $\xi$ be a simple closed arc in
$\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})$. We notice that the
intersection
$\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\eta,\xi)\in\cfrac{1}{2}\cdot\mathbb{Z}^{+}$.
When $\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\eta,\xi)>0$, we use
induction on it. For the starting case when
$\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\eta,\xi)=\frac{1}{2}$, we
take
$\alpha=\xi\wedge\eta\in\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})$, and
then $\xi=B_{\alpha}(\eta)$ by (3.3).

Now suppose the assertion holds when
$\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\eta,\xi)<k$ and consider
the case when
$\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\eta,\xi)=k\in\cfrac{1}{2}\cdot\mathbb{Z}^{+}$.
There exists some decoration $z$ which is not the endpoint of $\xi$ (if the
endpoints of $\eta$ and $\xi$ are not coincide, we take $z$ to be an endpoint
of $\eta$) and we connect $z$ to $\xi$ by $l$ such that it intersects $\xi$ at
$p$ (see Figure 6). We cut $\xi$ at $p$ and connect two parts with $l$
respectively. Then we obtain
$\xi_{1},\xi_{2}\in\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})$ such that
$\xi=\xi_{1}\wedge\xi_{2}$ and

$\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\eta,\xi)=\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\eta,\xi_{1})+\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\eta,\xi_{2}).$
(3.5)

By assumption, there exists
$b\in\operatorname{BT}({\mathbf{S}}_{\bigtriangleup})$ such that
$\xi_{1}=b(\eta)$. Thus $\xi=B_{\xi_{2}}(\xi_{1})=(B_{\xi_{2}}\cdot b)(\eta)$.

$\bullet$$p$$l$$\bullet$$\circ$$\bullet$$\circ$$z$$\bullet$$\circ$$\xi_{2}$$\xi_{1}$$\xi$
Figure 6. Decomposing $\xi$

Finally, we consider the case when
$\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\eta,\xi)=0$. We can
choose a simple closed arc $\alpha$ such that
$\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\eta,\alpha)=\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\xi,\alpha)=\frac{1}{2}$.
By the starting case, we have
$\alpha\in\operatorname{BT}({\mathbf{S}}_{\bigtriangleup})\cdot\eta$ and
$\xi\in\operatorname{BT}({\mathbf{S}}_{\bigtriangleup})\cdot\alpha\subset\operatorname{BT}({\mathbf{S}}_{\bigtriangleup})\cdot\eta$.
∎

#### The branched double cover

Let $\Sigma_{\Delta}$ be a branched double cover of
${\mathbf{S}}_{\bigtriangleup}$ branching at decorations. When there exists
extra structure (e.g. a quadratic differential), $\Sigma_{\Delta}$ can be
constructed as the spectral cover (cf. [KS]). We denote the covering map by
$\pi:\Sigma_{\Delta}\rightarrow{\mathbf{S}}_{\bigtriangleup}$.

We consider the special case when
${\mathbf{S}}_{\bigtriangleup}=\mathbf{D}_{3}$ is a disk and
$\bigtriangleup=\\{z_{\infty},z_{*},z_{0}\\}$. We fix two initial simple
closed arcs $\eta_{0}$ and $\eta_{\infty}$ such that
$\eta_{0}\cap\eta_{\infty}=\\{z_{*}\\}$, see Figure 7. Notice that there is an
anti-clockwise angle from $\eta_{\infty}$ to $\eta_{0}$. The branched double
cover $\Sigma_{\Delta}$ is a torus with one boundary $\partial_{\Sigma}$. For
simplicity, we draw $\partial_{\Sigma}$ as a puncture in the figures.

$z_{0}$$z_{*}$$z_{\infty}$$\eta_{0}$$\eta_{\infty}$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$
Figure 7. $\mathbf{D}_{3}$ is a disk with three decorations

We take the $\mathbb{Z}^{2}$-covering $\widetilde{\Sigma_{\Delta}}$ of
$\Sigma_{\Delta}$, where the white area is a fundamental domain (see Figure
8), and we denote the covering map by
$\widetilde{\pi}:\widetilde{\Sigma_{\Delta}}\rightarrow\Sigma_{\Delta}$. When
forgetting the punctures and decorations of $\widetilde{\Sigma_{\Delta}}$, it
is the universal cover of torus. Hence we embed $\widetilde{\Sigma_{\Delta}}$
into $\mathbb{R}^{2}$ such that all decorations and punctures are integer
points, where its fundamental domain is a unit square. For each line (it is
not allowed to pass through the punctures) in $\widetilde{\Sigma_{\Delta}}$
with rational slope $\frac{r}{s}\in\overline{\mathbb{Q}}$, it becomes a simple
closed curve $C_{\frac{r}{s}}$ in $\Sigma_{\Delta}$ under the map
$\widetilde{\pi}$.

$z_{\infty}$$z_{0}$$z_{\infty}$$z_{0}$$z_{*}$$z_{\infty}$$z_{0}$$z_{\infty}$$z_{0}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$
Figure 8. The $\mathbb{Z}^{2}$-covering of $\Sigma_{\Delta}$ and its
fundamental domain

######  Lemma 3.4 ([KQ, §10]).

The set $\operatorname{CA}(\mathbf{D}_{3})$ of simple closed arcs in
$\mathbf{D}_{3}$ can be parameterized by rational numbers, i.e there is a
bijection

$\operatorname{CA}(\mathbf{D}_{3})$$\operatorname{Br}_{3}/\operatorname{Z}(\operatorname{Br}_{3})$$\iota$$\cong$$\overline{\mathbb{Q}}$$\operatorname{PSL}_{2}(\mathbb{Z})$
(3.6)

sending $\eta_{\frac{r}{s}}$ to $\frac{r}{s}$.

###### Proof.

We lift simple closed arcs in
$\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})$ to simple closed curves in
$\Sigma_{\Delta}$, which can be parametrized by rational numbers in
$\overline{\mathbb{Q}}$. That is, for any
$\frac{r}{s}\in\overline{\mathbb{Q}}$, there exists an
$\eta_{\frac{r}{s}}\in\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})$ which
corresponds to a simple closed curve $C_{\frac{r}{s}}$ in $\Sigma_{\Delta}$.
Notice that the homology group
$H_{1}(\Sigma_{\Delta})=\mathbb{Z}[C_{\infty}]\oplus\mathbb{Z}[C_{0}]$. Thus
the homology class $[C_{\frac{r}{s}}]$ corresponds to $(r,s)$ in
$H_{1}(\Sigma_{\Delta})\cong\mathbb{Z}^{2}$. Notice that the braid twist
$\operatorname{BT}(\mathbf{D}_{3})\cong\operatorname{Br}_{3}$ lifts to Dehn
twist
$\operatorname{DT}(\Sigma_{\Delta})\subset\operatorname{MCG}(\Sigma_{\Delta})$,
which is generated by $C_{0}$ and $C_{\infty}$. By the identification
$\operatorname{DT}(\Sigma_{\Delta})/\operatorname{Z}(\operatorname{DT}(\Sigma_{\Delta}))\cong\operatorname{PSL}_{2}(\mathbb{Z})$,
the lemma holds. ∎

######  Remark 3.5.

Here is a consequence of the lemma above. For a rational number
$\frac{r}{s}\in\mathbb{Q}^{+}$ with expression (2.5), the corresponding arc in
$\operatorname{CA}(\mathbf{D}_{3})$ is

$\eta_{\frac{r}{s}}=B_{\eta_{\infty}}^{a_{1}}B_{\eta_{0}}^{-a_{2}}B_{\eta_{\infty}}^{a_{3}}B_{\eta_{0}}^{-a_{4}}\cdots
B_{\eta_{\infty}}^{a_{2m-1}}B_{\eta_{0}}^{-a_{2m}}(\eta_{\infty}).$ (3.7)

Moreover,

$\eta_{-\frac{r}{s}}=B_{\eta_{\infty}}^{-a_{1}}B_{\eta_{0}}^{a_{2}}B_{\eta_{\infty}}^{-a_{3}}B_{\eta_{0}}^{a_{4}}\cdots
B_{\eta_{\infty}}^{-a_{2m-1}}B_{\eta_{0}}^{a_{2m}}(\eta_{\infty}).$ (3.8)

###  3.2. Bigraded arcs and $\mathfrak{q}$-intersections

Let ${\mathbf{S}}_{\bigtriangleup}$ be a decorated surface. In this section,
we define the bigraded arcs and their $\mathfrak{q}$-intersections. Let
$\mathbb{P}T\mathbf{S}_{\bigtriangleup}=\mathbb{P}T(\mathbf{S}\setminus\bigtriangleup)$
be the real projectivization of the tangent bundle of
$\mathbf{S}\setminus\bigtriangleup$. We want to introduce a particular
covering of $\mathbb{P}T\mathbf{S}_{\bigtriangleup}$ with covering group
$\mathbb{Z}\oplus\mathbb{Z}\mathbb{X}\cong\mathbb{Z}^{2}$. A _grading_
$\Lambda:{\mathbf{S}}_{\bigtriangleup}\rightarrow\mathbb{P}T\mathbf{S}_{\bigtriangleup}$
on ${\mathbf{S}}_{\bigtriangleup}$ is determined by a class in
$\operatorname{H}_{1}(\mathbb{P}T\mathbf{S}_{\bigtriangleup},\mathbb{Z}\oplus\mathbb{Z}\mathbb{X})$,
with value $1$ on each anti-clockwise loop
$\\{p\\}\times\mathbb{R}\mathbb{P}^{1}$ on
$\mathbb{P}{T_{p}}{\mathbf{S}}_{\bigtriangleup}$ for $p\notin\bigtriangleup$
and value $-2+\mathbb{X}$ on each anti-clockwise loop $l_{z}\times\\{x\\}$ on
${\mathbf{S}}_{\bigtriangleup}$ around any
$z\in\bigtriangleup,x\in\mathbb{R}\mathbb{P}^{1}$. For any simple loop
$\alpha$ on ${\mathbf{S}}_{\bigtriangleup}$, we denote $\Lambda_{1}(\alpha)$
the $\mathbb{Z}$ part of $\Lambda(\alpha)$ and denote $\Lambda_{2}(\alpha)$
the $\mathbb{Z}\mathbb{X}$ part of $\Lambda(\alpha)$. In fact, the first
grading is a line field $\lambda$ of ${\mathbf{S}}_{\bigtriangleup}$ which is
determined by a class in
$\operatorname{H}_{1}(\mathbb{P}T\mathbf{S}_{\bigtriangleup},\mathbb{Z})$.
Define $\widehat{\mathbb{P}T\mathbf{S}_{\bigtriangleup}}$ to be the
$\mathbb{Z}\oplus\mathbb{Z}\mathbb{X}$ covering of
$\mathbb{P}T\mathbf{S}_{\bigtriangleup}$ classified by the grading $\Lambda$,
and denote the $\mathbb{Z}\oplus\mathbb{Z}\mathbb{X}$ action on
$\widehat{\mathbb{P}T\mathbf{S}_{\bigtriangleup}}$ by $\chi$.

######  Definition 3.6.

A _graded decorated surface_ ${\mathbf{S}}_{\bigtriangleup}^{\Lambda}$
consists of a decorated surface ${\mathbf{S}}_{\bigtriangleup}$ and a grading
$\Lambda$ on ${\mathbf{S}}_{\bigtriangleup}$.

Let ${\mathbf{S}}_{\bigtriangleup}^{\Lambda}$ be a graded decorated surface
and $c:[0,1]\rightarrow\mathbf{S}$ be an arc in $\mathbf{S}$. There is a
canonical section
$s_{c}:c\setminus\bigtriangleup\rightarrow\mathbb{P}T\mathbf{S}_{\bigtriangleup}$
given by $s_{c}(z)=T_{z}c$. A _bigrading_ on $c$ is given by a lift
$\widehat{c}$ of $s_{c}$ to
$\widehat{\mathbb{P}T\mathbf{S}_{\bigtriangleup}}$. The pair $(c,\widehat{c})$
is called a _bigraded arc_ , and we usually denote it by $\widehat{c}$. Note
that there are $\mathbb{Z}^{2}$ lifts of $c$, which are related by the
$\mathbb{Z}^{2}$-action defined by $\chi$. One is the shift grading such that
$\widehat{c}[m](t)=\chi(m,0)\widehat{c}(t)$ and the other is the
$\mathbb{X}$-grading such that $\widehat{c}\\{m\\}(t)=\chi(0,m)\widehat{c}(t)$
for any $m\in\mathbb{Z}$. For any
$\eta\in\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})$, we call any of its
lifts $\widehat{\eta}$ in $\widehat{\mathbb{P}T\mathbf{S}_{\bigtriangleup}}$ a
_bigraded simple closed arc_. Denote by
$\widehat{\operatorname{CA}}({\mathbf{S}}_{\bigtriangleup})$ the set of
bigraded simple closed arcs.

For any bigraded arcs $\widehat{\sigma}$ and $\widehat{\tau}$ which are in
minimal position with respect to each other, let
$p=\sigma(t_{1})=\tau(t_{2})\in\mathbf{S}^{\circ}$ be the point where $\sigma$
and $\tau$ intersect transversally. Fix a small circle
$a\subset\mathbf{S}\setminus\bigtriangleup$ around $p$. Let
$\alpha:[0,1]\rightarrow a$ be an embedded arc which moves anti-clockwise
around $p$, such that $\alpha$ intersects $\sigma$ and $\tau$ at $\alpha(0)$
and $\alpha(1)$, respectively (cf. Figure 9). If $p\in\bigtriangleup$, then
$\alpha$ is unique up to a change of parametrization (see the left one of
Figure 9); otherwise there are two possibilities, which are distinguished by
their endpoints (see the right one of Figure 9). Take a smooth path
$\rho:[0,1]\rightarrow\mathbb{P}T\mathbf{S}_{\bigtriangleup}$ with
$\rho(t)\in\mathbb{P}T_{\alpha(t)}{\mathbf{S}}_{\bigtriangleup}$ for all $t$,
going from $\rho(0)=T_{\alpha(0)}\sigma$ to $\rho(1)=T_{\alpha(1)}\tau$, such
that $\rho(t)\neq T_{\alpha(t)}l$ for all $t$. Lift $\rho$ to a path
$\widehat{\rho}:[0,1]\rightarrow\widehat{\mathbb{P}T\mathbf{S}_{\bigtriangleup}}$
with $\widehat{\rho}(0)=\widehat{\sigma}(\alpha(0))$. Then there exist some
integers $\varrho,\varsigma\in\mathbb{Z}$ such that
$\widehat{\tau}(\alpha(1))=\chi(\varrho+\varsigma\mathbb{X})\widehat{\rho}(1)$.

$\alpha$$\quad\sigma$$\tau\quad$$\bullet$$\bullet$$\circ$$p$$\alpha$$\alpha$$\quad\sigma$$\tau\quad$$\bullet$$p$
Figure 9. Intersection at $p$ when $p$ is a decoration or not

######  Definition 3.7 ([KS]).

For any bigraded arcs $\widehat{\sigma}$ and $\widehat{\tau}$ in
${\mathbf{S}}_{\bigtriangleup}^{\Lambda}$, we call an intersection $p$ of
$\widehat{\sigma}$ and $\widehat{\tau}$ with _bi-index_

$\operatorname{ind}_{p}^{\mathbb{Z}^{2}}(\widehat{\sigma},\widehat{\tau})=\operatorname{ind}_{p}(\widehat{\sigma},\widehat{\tau})+\operatorname{ind}_{p}^{\mathbb{X}}(\widehat{\sigma},\widehat{\tau})\mathbb{X},$
(3.9)

where $\operatorname{ind}_{p}(\widehat{\sigma},\widehat{\tau}):=\varrho$ and
$\operatorname{ind}^{\mathbb{X}}_{p}(\widehat{\sigma},\widehat{\tau}):=\varsigma$
defined as above.

We have the following equations among bi-indices, which will be used later.

######  Lemma 3.8 ([IQZ, Lemma 2.6]).

Let $\widehat{\sigma},\widehat{\tau}$ be bigraded arcs in
${\mathbf{S}}_{\bigtriangleup}^{\Lambda}$ with an intersection
$p\in\mathbf{S}^{\circ}$. If $p\notin\bigtriangleup$, we have

$\operatorname{ind}^{\mathbb{Z}^{2}}_{p}(\widehat{\sigma},\widehat{\tau})+\operatorname{ind}^{\mathbb{Z}^{2}}_{p}(\widehat{\tau},\widehat{\sigma})=1.$
(3.10)

If $p\in\bigtriangleup$, we have

$\operatorname{ind}^{\mathbb{Z}^{2}}_{p}(\widehat{\sigma},\widehat{\tau})+\operatorname{ind}^{\mathbb{Z}^{2}}_{p}(\widehat{\tau},\widehat{\sigma})=\mathbb{X}.$
(3.11)

######  Lemma 3.9 ([IQZ, Lemma 2.7]).

Let $\widehat{\sigma},\widehat{\tau},\widehat{\alpha}$ be bigraded arcs in
${\mathbf{S}}_{\bigtriangleup}^{\Lambda}$. If they are in the case in the left
one of Figure 10, we have

$\operatorname{ind}^{\mathbb{Z}^{2}}_{p}(\widehat{\sigma},\widehat{\alpha})=\operatorname{ind}^{\mathbb{Z}^{2}}_{p}(\widehat{\sigma},\widehat{\gamma})+\operatorname{ind}^{\mathbb{Z}^{2}}_{p}(\widehat{\gamma},\widehat{\alpha}).$
(3.12)

If they are in the case in the left one of Figure 9, we have

$\displaystyle\begin{split}\operatorname{ind}^{\mathbb{Z}^{2}}_{p}(\widehat{\sigma},\widehat{\tau})&=\operatorname{ind}^{\mathbb{Z}^{2}}_{\alpha(0)}(\widehat{\sigma},\widehat{\alpha})-\operatorname{ind}^{\mathbb{Z}^{2}}_{\alpha(1)}(\widehat{\tau},\widehat{\alpha})\\\
&=\operatorname{ind}^{\mathbb{Z}^{2}}_{\alpha(1)}(\widehat{\alpha},\widehat{\tau})-\operatorname{ind}^{\mathbb{Z}^{2}}_{\alpha(0)}(\widehat{\alpha},\widehat{\sigma}).\end{split}$
(3.13)

$\widehat{\tau}$$\bullet$$\widehat{\alpha}$$\widehat{\sigma}$$p$$\bullet$$\widehat{\tau}$$\bullet$$\circ$$\widehat{\alpha}$$\widehat{\sigma}$$z$$\widehat{\theta}$
Figure 10. Bigraded arcs intersect at the same point (or decoration) in anti-
clockwise

######  Lemma 3.10.

Let $\widehat{\sigma},\widehat{\tau},\widehat{\alpha}$ be bigraded arcs on
${\mathbf{S}}_{\bigtriangleup}^{\Lambda}$ which share the same decoration $z$
and sitting in anti-clockwise order in the right one of Figure 10. We have

$\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\sigma},\widehat{\alpha})=\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\sigma},\widehat{\tau})+\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\tau},\widehat{\alpha}).$
(3.14)

###### Proof.

Fix a small circle $a\subset\mathbf{S}\setminus\bigtriangleup$ around $z$. Let
$\widehat{\theta}:[0,1]\rightarrow a$ be an embedded bigraded arc winding
anti-clockwisely at $z$ such that the underlying arc $\theta$ intersect
$\sigma,\tau$ and $\alpha$ at $\theta(0),\theta(\frac{1}{2})$ and $\theta(1)$
respectively (see the right one of Figure 10). The arc $\widehat{\theta}$ is
unique up to a change of parametrization. By Lemma 3.9, we have

$\begin{split}\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\sigma},\widehat{\alpha})&=\operatorname{ind}^{\mathbb{Z}^{2}}_{\theta(0)}(\widehat{\sigma},\widehat{\theta})-\operatorname{ind}^{\mathbb{Z}^{2}}_{\theta(1)}(\widehat{\alpha},\widehat{\theta})\\\
&=[\operatorname{ind}^{\mathbb{Z}^{2}}_{\theta(0)}(\widehat{\sigma},\widehat{\theta})-\operatorname{ind}^{\mathbb{Z}^{2}}_{\theta(\frac{1}{2})}(\widehat{\tau},\widehat{\theta})]+[\operatorname{ind}^{\mathbb{Z}^{2}}_{\theta(\frac{1}{2})}(\widehat{\tau},\widehat{\theta})-\operatorname{ind}^{\mathbb{Z}^{2}}_{\theta(1)}(\widehat{\alpha},\widehat{\theta})]\\\
&=\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\sigma},\widehat{\tau})+\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\tau},\widehat{\alpha}).\end{split}$

∎

######  Definition 3.11.

For
$\widehat{\tau},\widehat{\eta}\in\widehat{\operatorname{CA}}({\mathbf{S}}_{\bigtriangleup})$
satisfying that
$\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\tau,\eta)=\frac{1}{2}$
and $\operatorname{ind}^{\mathbb{X}}_{z}(\widehat{\tau},\widehat{\eta})=a$,
let $z\in\bigtriangleup$ be their common endpoint. Denote by
$\widehat{\tau}\wedge\widehat{\eta}$ to be the bigraded arc in
$\widehat{\operatorname{CA}}({\mathbf{S}}_{\bigtriangleup})$ whose underlying
arc is obtained by the smoothing out
$\widehat{\tau}\cup\widehat{\eta}[(a-1)\mathbb{X}]$ at $z$ and whose grading
inherits from $\widehat{\tau}$. That is, we have
$\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\tau},\widehat{\tau}\wedge\widehat{\eta})=0$,
cf. Figure 11.

######  Proposition 3.12.

For $\widehat{\tau},\widehat{\eta}$ and $\widehat{\tau}\wedge\widehat{\eta}$
in $\widehat{\operatorname{CA}}({\mathbf{S}}_{\bigtriangleup})$ as in
Definition 3.11, we have

$\operatorname{ind}^{\mathbb{Z}^{2}}_{v_{2}}(\widehat{\tau},\widehat{\tau}\wedge\widehat{\eta})+\operatorname{ind}^{\mathbb{Z}^{2}}_{v_{1}}(\widehat{\tau}\wedge\widehat{\eta},\widehat{\eta})+\operatorname{ind}^{\mathbb{Z}^{2}}_{v_{3}}(\widehat{\eta},\widehat{\tau})=1.$

###### Proof.

$\bullet$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$v_{2}$$v_{1}$$v_{3}$$\widehat{\tau}\wedge\widehat{\eta}$$\widehat{\tau}$$\widehat{\eta}$
Figure 11. The sum of bi-indices of intersections between bigraded arcs via
smoothing out

We calculate the two gradings separately. For the first grading, it is a line
field which is determined by
$\operatorname{H}_{1}(\mathbb{P}T\mathbf{S}_{\bigtriangleup},\mathbb{Z})$. We
can identify all the projectization of the tangent space of any point in the
contractible triangle formed by the three arcs (except the decorations)
simultaneously. Hence, the sum of the first grading is 1 (rotating anti-
clockwisely). For the second grading, we use the log surface in the sense of
[IQZ, §2.4]. By Definition 3.11, the segments of
$\widehat{\tau}\wedge\widehat{\eta}$ and $\widehat{\eta}[(a-1)\mathbb{X}]$
near $v_{1}$ are in the same sheet of $\log({\mathbf{S}}_{\bigtriangleup})$
and the anti-clockwise angle does not cross the cut (cf. [IQZ, Figure 5]).
Thus we have

$\begin{split}\operatorname{ind}^{\mathbb{X}}_{v_{2}}(\widehat{\tau},\widehat{\tau}\wedge\widehat{\eta})+\operatorname{ind}^{\mathbb{X}}_{v_{1}}(\widehat{\tau}\wedge\widehat{\eta},\widehat{\eta})+\operatorname{ind}^{\mathbb{X}}_{v_{3}}(\widehat{\eta},\widehat{\tau})&=0+(a-1)\mathbb{X}+(1-\operatorname{ind}^{\mathbb{X}}_{v_{3}}(\widehat{\tau},\widehat{\eta}))\\\
&=(a-1)+(1-a)=0\end{split}$

as required, where $\operatorname{ind}_{?}^{\mathbb{X}}$ denote the second
grading. ∎

######  Notations 3.13.

For $(\varrho,\varsigma)\in\mathbb{Z}^{2}$, we denote
$\cap^{\varrho+\varsigma\mathbb{X}}(\widehat{\sigma},\widehat{\tau})$ the set
of intersections between $\widehat{\sigma}$ and $\widehat{\tau}$ with bi-index
$\varrho+\varsigma\mathbb{X}$. We will use the notations

$\displaystyle\operatorname{Int}^{\varrho+\varsigma\mathbb{X}}_{\Pi}(\widehat{\sigma},\widehat{\tau})\colon=\left|\cap^{\varrho+\varsigma\mathbb{X}}(\widehat{\sigma},\widehat{\tau})\cap\Pi\right|,$
$\displaystyle\operatorname{Int}^{\varrho+\varsigma\mathbb{X}}_{{\mathbf{S}}_{\bigtriangleup}}(\widehat{\sigma},\widehat{\tau})\colon=\frac{1}{2}\cdot\operatorname{Int}^{\varrho+\varsigma\mathbb{X}}_{\bigtriangleup}(\widehat{\sigma},\widehat{\tau})+\operatorname{Int}^{\varrho+\varsigma\mathbb{X}}_{{\mathbf{S}}_{\bigtriangleup}^{\circ}}(\widehat{\sigma},\widehat{\tau}),$

for the bi-index $(\varrho+\varsigma\mathbb{X})$ intersection numbers at any
proper subset $\Pi$ of ${\mathbf{S}}_{\bigtriangleup}$ and at all of
$\mathbf{S}^{\circ}$ respectively. The _total intersection_

$\operatorname{Int}_{?}(\widehat{\sigma},\widehat{\tau})=\sum_{\varrho,\varsigma\in\mathbb{Z}}\operatorname{Int}^{\varrho+\varsigma\mathbb{X}}_{?}(\widehat{\sigma},\widehat{\tau})$

is the sum over all bi-indices, where $?=\bigtriangleup$ or
${\mathbf{S}}_{\bigtriangleup}^{\circ}$.

######  Definition 3.14 ([KS, IQZ]).

Let $q_{1}$ and $q_{2}$ be two formal parameters. The _$\mathbb{Z}^{2}$
-graded $\mathfrak{q}$-intersection_ of
$\widehat{\sigma},\widehat{\tau}\in\widehat{\operatorname{CA}}({\mathbf{S}}_{\bigtriangleup})$
is defined to be

$\operatorname{Int}^{\mathfrak{q}}(\widehat{\sigma},\widehat{\tau})=\displaystyle\sum_{\varrho,\varsigma\in\mathbb{Z}}q_{1}^{\varrho}q_{2}^{\varsigma}\cdot\operatorname{Int}_{\bigtriangleup}^{\varrho+\varsigma\mathbb{X}}(\widehat{\sigma},\widehat{\tau})+(1+q_{1}^{-1}q_{2})\displaystyle\sum_{\varrho,\varsigma\in\mathbb{Z}}q_{1}^{\varrho}q_{2}^{\varsigma}\cdot\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}^{\circ}}^{\varrho+\varsigma\mathbb{X}}(\widehat{\sigma},\widehat{\tau}).$
(3.15)

Note that we have
$\operatorname{Int}^{\mathfrak{q}}(-,-)\mid_{q_{1}=q_{2}=1}=2\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(-,-)=\operatorname{Int}_{\Sigma_{\Delta}}(-,-)$,
where $\Sigma_{\Delta}$ is the branched double cover of
${\mathbf{S}}_{\bigtriangleup}$.

###  3.3. Left $\mathfrak{q}$-deformations as $\mathfrak{q}$-intersections

Recall that $\mathbf{D}_{3}$ is a disk and
$\bigtriangleup=\\{z_{\infty},z_{*},z_{0}\\}$. By Lemma 3.3, we can label
simple closed arcs by $\overline{\mathbb{Q}}$ as follows. We fix two initial
bigraded simple closed arcs and denote them $\widehat{\eta}_{0}$ and
$\widehat{\eta}_{\infty}$ such that
$\operatorname{ind}_{z_{*}}(\widehat{\eta}_{\infty},\widehat{\eta}_{0})=1$
(see Figure 13).

######  Construction 3.15.

We define a map

$\begin{split}\widehat{\eta}:\overline{\mathbb{Q}}&\rightarrow\widehat{\operatorname{CA}}(\mathbf{D}_{3})\\\
\frac{r}{s}&\mapsto\widehat{\eta}_{\frac{r}{s}},\end{split}$ (3.16)

as follows. For any rational $\frac{r}{s}\in\mathbb{Q}^{+}$, by 2.3 we know
that it can be uniquely written as

$\frac{r}{s}=\frac{p}{q}\oplus\frac{u}{v}.$

We iteratively define that

$\widehat{\eta}_{\frac{r}{s}}:=B_{\eta_{\frac{u}{v}}}(\widehat{\eta}_{\frac{p}{q}})=\widehat{\eta}_{\frac{p}{q}}\wedge\widehat{\eta}_{\frac{u}{v}},$
(3.17)

noticing that
$\operatorname{Int}_{\mathbf{D}_{3}}(\eta_{\frac{p}{q}},\eta_{\frac{u}{v}})=\frac{1}{2}$.
For negative case, we set $\widehat{\eta}_{-\infty}:=\widehat{\eta}_{\infty}$
and define

$\widehat{\eta}_{-\frac{r}{s}}:=B_{\eta_{-\frac{p}{q}}}(\widehat{\eta}_{-\frac{u}{v}}).$
(3.18)

Thus, we get some bigraded closed arcs in
$\widehat{\operatorname{CA}}(\mathbf{D}_{3})$ which are
$\widehat{\eta}_{\frac{r}{s}}[\varrho+\varsigma\mathbb{X}]$, where
$\varrho,\varsigma\in\mathbb{Z}$ and $\frac{r}{s}\in\overline{\mathbb{Q}}$.

Let $\frac{r}{s}\in\mathbb{Q}^{+}$ with
$\frac{r}{s}=\frac{p}{q}\oplus\frac{u}{v}$. By Definition 3.11, the grading of
the new arc $\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}}$ inherits the
grading of $\widehat{\eta}_{\frac{p}{q}}$. That is, for any bigraded arc
$\widehat{\sigma}$ intersecting $\widehat{\eta}_{\frac{r}{s}}$ and
$\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}}$ at
$p_{1},p_{2}\in\mathbf{S}^{\circ}$ respectively (see Figure 12), we have

$\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}})=0,\quad\operatorname{ind}^{\mathbb{Z}^{2}}_{p_{1}}(\widehat{\sigma},\widehat{\eta}_{\frac{p}{q}})=\operatorname{ind}^{\mathbb{Z}^{2}}_{p_{2}}(\widehat{\sigma},\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}}),$
(3.19)

where $z\in\bigtriangleup$ is the common endpoint of
$\widehat{\eta}_{\frac{p}{q}}$ and
$\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}}$. The second equation in (3.19)
follows from the fact that
$\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}}$
and $\widehat{\eta}_{\frac{u}{v}}$ form a contractible triangle.

$\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}}$$\widehat{\eta}_{\frac{r}{s}}$$\widehat{\sigma}$$p_{1}$$p_{2}$
Figure 12. The bigrading inherited from braid twist

From Proposition 3.12, we know that these three simple closed arcs
$\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}}$
and $\widehat{\eta}_{\frac{u}{v}}$ (take
$\widehat{\tau}=\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}=\widehat{\eta}_{\frac{u}{v}}$
in Figure 11) satisfy

$\operatorname{ind}^{\mathbb{Z}^{2}}_{v_{2}}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}})+\operatorname{ind}^{\mathbb{Z}^{2}}_{v_{1}}(\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}},\widehat{\eta}_{\frac{u}{v}})+\operatorname{ind}^{\mathbb{Z}^{2}}_{v_{3}}(\widehat{\eta}_{\frac{u}{v}},\widehat{\eta}_{\frac{p}{q}})=1,$
(3.20)

where $v_{1},v_{2},v_{3}$ are the corresponding intersecting decorations.

We fix the following setting.

######  Setting 3.16.

Recall that in 2.3, any fraction $\frac{r}{s}\in\mathbb{Q}^{+}$ can be
uniquely written as

$\frac{r}{s}=\frac{p}{q}\oplus\frac{u}{v},$

with $\frac{p}{q},\frac{u}{v}\in\overline{\mathbb{Q}^{\geq 0}},uq-pv=1$ and an
associated integer $l(\frac{p}{q},\frac{u}{v})$. The corresponding arcs in
$\widehat{\operatorname{CA}}(\mathbf{D}_{3})$ of these fractions are
$\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}},\widehat{\eta}_{\frac{u}{v}}$,
where $\widehat{\eta}_{\frac{p}{q}}$ and $\widehat{\eta}_{\frac{u}{v}}$
intersect at only one decoration $z$ in $\bigtriangleup$. We do not
distinguish $\frac{r}{s}$ and $\frac{p}{q}\oplus\frac{u}{v}$ in the following.

######  Lemma 3.17.

For any two arcs $\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{u}{v}}$
in $\widehat{\operatorname{CA}}(\mathbf{D}_{3})$ in 3.16, the bi-index is of
the form $l(1-\mathbb{X})$ where $l\in\mathbb{N}$, i.e.

$\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{u}{v}})=l(1-\mathbb{X}),$
(3.21)

except the special case

$\operatorname{ind}^{\mathbb{Z}^{2}}_{z_{*}}(\widehat{\eta}_{0},\widehat{\eta}_{\infty})=\mathbb{X}-1.$
(3.22)

Here $l=l(\frac{p}{q},\frac{u}{v})$ in 2.3.

###### Proof.

For the special case, we know that

$\operatorname{ind}^{\mathbb{Z}^{2}}_{z_{*}}(\widehat{\eta}_{0},\widehat{\eta}_{\infty})=\mathbb{X}-\operatorname{ind}^{\mathbb{Z}^{2}}_{z_{*}}(\widehat{\eta}_{\infty},\widehat{\eta}_{0})=\mathbb{X}-1,$

from (3.11). In general, we prove the lemma by induction on $l$. For initial
bigraded simple closed arcs $\widehat{\eta}_{0},\widehat{\eta}_{\frac{1}{1}}$
and $\widehat{\eta}_{\infty}$ in
$\widehat{\operatorname{CA}}(\mathbf{D}_{3})$, we have
$\operatorname{ind}_{z_{0}}(\widehat{\eta}_{0},\widehat{\eta}_{\frac{1}{1}})=\operatorname{ind}_{z_{\infty}}(\widehat{\eta}_{\frac{1}{1}},\widehat{\eta}_{\infty})=0$
and thus the lemma holds obviously. We assume that (3.21) holds for $l$ and
consider the $l+1$ case. For any
$\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{u}{v}}$ in
$\widehat{\operatorname{CA}}(\mathbf{D}_{3})$ in 3.16, we assume that they
intersect at only one decoration $v_{3}$ with

$\operatorname{ind}^{\mathbb{Z}^{2}}_{v_{3}}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{u}{v}})=l(1-\mathbb{X}),$
(3.23)

where $l\in\mathbb{N}$. By (3.11), we have

$\operatorname{ind}^{\mathbb{Z}^{2}}_{v_{3}}(\widehat{\eta}_{\frac{u}{v}},\widehat{\eta}_{\frac{p}{q}})=(l+1)\mathbb{X}-l.$

Since the grading of $\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}}$ inherits
the grading of $\widehat{\eta}_{\frac{p}{q}}$, we have

$\operatorname{ind}^{\mathbb{Z}^{2}}_{v_{1}}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}})=0.$
(3.24)

By (3.20), we deduce that

$\begin{split}\operatorname{ind}^{\mathbb{Z}^{2}}_{v_{2}}(\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}},\widehat{\eta}_{\frac{u}{v}})&=1-0-[(l+1)\mathbb{X}-l]\\\
&=(l+1)-(l+1)\mathbb{X}.\end{split}$ (3.25)

Finally, combining (3.24) and (3.25), the lemma is true. ∎

######  Theorem 3.18.

For any rational number $\frac{r}{s}\in\overline{\mathbb{Q}}$, we have

$\Big{[}\frac{r}{s}\Big{]}^{\flat}_{\mathfrak{q}}=\frac{\varepsilon\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{r}{s}},\widehat{\eta}_{0})}{\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{r}{s}},\widehat{\eta}_{\infty})}\bigg{|}_{\mathfrak{q}=q_{1}^{-1}q_{2}},$
(3.26)

corresponding to the left $\mathfrak{q}$-deformation of $\frac{r}{s}$, where

$\varepsilon=\begin{cases}q_{1}^{-1},&\frac{r}{s}\geq 0,\\\
-1,&\frac{r}{s}<0.\end{cases}$

In particular, for $\frac{r}{s}\in\overline{\mathbb{Q}^{\geq 0}}$, we have __

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}\overline{\textbf{R}}_{\mathfrak{q}}^{\flat}(\frac{r}{s})&=q_{1}^{-1}\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{r}{s}},\widehat{\eta}_{0})\big{|}_{\mathfrak{q}=q_{1}^{-1}q_{2}},\\\
\overline{\textbf{S}}_{\mathfrak{q}}^{\flat}(\frac{r}{s})&=\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{r}{s}},\widehat{\eta}_{\infty})\big{|}_{\mathfrak{q}=q_{1}^{-1}q_{2}}.\end{array}\right.\end{split}$
(3.27)

$z_{\infty}$$z_{*}$$z_{0}$$\widehat{\eta}_{\frac{3}{2}}$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\widehat{\gamma}_{\infty}$$\widehat{\gamma}_{0}$$z_{\infty}$$z_{*}$$z_{0}$$\widehat{\eta}_{-\frac{2}{1}}$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\widehat{\gamma}_{\infty}$$\widehat{\gamma}_{0}$$z_{\infty}$$z_{*}$$z_{0}$$\widehat{\eta}_{\infty}$$\widehat{\eta}_{0}$$\widehat{\eta}_{1}$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\bullet$$\bullet$$\widehat{\gamma}_{\infty}$$\widehat{\gamma}_{0}$

Figure 13. Red arcs: examples of bigraded closed arcs in
$\widehat{\operatorname{CA}}(\mathbf{D}_{3})$

###### Proof.

For $0$ and $\infty$, we have

$\operatorname{Int}^{\mathfrak{q}}_{z_{*}}(\widehat{\eta}_{0},\widehat{\eta}_{\infty})=q_{1}^{-1}q_{2}~{}{\rm
and}~{}\operatorname{Int}^{\mathfrak{q}}_{z_{*}}(\widehat{\eta}_{\infty},\widehat{\eta}_{0})=q_{1}$

by definition. We set

$\overline{\textbf{R}}_{\mathfrak{q}}^{\flat}(0)=q_{2}-q_{1}~{}{\rm
and}~{}\overline{\textbf{S}}_{\mathfrak{q}}^{\flat}(\infty)=1-q_{1}^{-1}q_{2}.$

We prove the non-negative case by induction using 3.16. At starting step, the
fractions in the theorem are $\frac{q_{1}^{-1}q_{2}-1}{q_{1}^{-1}q_{2}}$ and
$\frac{1}{1-q_{1}^{-1}q_{2}}$, which coincide the left
$\mathfrak{q}$-deformation of $0$ and $\infty$ respectively when
$\mathfrak{q}=q_{1}^{-1}q_{2}$ We assume that the formula (3.26) holds for
$\frac{p}{q},\frac{u}{v}\in\overline{\mathbb{Q}^{\geq 0}}$ in 3.16 which
satisfies

$\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{u}{v}})=l(1-\mathbb{X}),$

where $l=l(\frac{p}{q}\oplus\frac{u}{v})\in\mathbb{N}$ and $z$ is the common
endpoint of $\widehat{\eta}_{\frac{p}{q}}$ and $\widehat{\eta}_{\frac{u}{v}}$.
We use the $\mathbb{Z}^{2}$-covering $\widetilde{\Sigma_{\Delta}}$ to compute
the $\mathfrak{q}$-intersection between
$\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}}$ and $\widehat{\eta}_{0}$ (and
$\widehat{\eta}_{\infty}$ is similar). We lift the arcs in
$\widehat{\operatorname{CA}}(\mathbf{D}_{3})$ to lines in
$\widetilde{\Sigma_{\Delta}}$. Then $\widehat{\eta}_{0}$ becomes a series of
horizontal lines which pass through $z_{*}$ and $z_{0}$. The topological
triangle $T$ in $\mathbf{D}_{3}$ bounded by
$\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{u}{v}}$ and
$\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}}$ becomes a triangle
$\widetilde{T}$ in $\mathbb{R}^{2}$ up to translation (or reflection). We may
assume that the vertices of $\widetilde{T}$ are
$\widetilde{z},\widetilde{z}+(p,q),\widetilde{z}+(r,s)$, which says that
$z^{\prime},z,z^{\prime\prime}$ in $T$. The area of $\widetilde{T}$ equals to
$\frac{1}{2}$ by the relation $uq-pv=1$. By Pick’s theorem, there are no
decorations or punctures in $\widetilde{T}$. The intersections between
$\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}}$ and $\widehat{\eta}_{0}$
consist of two parts (see Figure 14):

  * •

intersections below and on $a$, which inherit from
$\widehat{\eta}_{\frac{p}{q}}$, and

  * •

intersections above $a$, which is induced from $\widehat{\eta}_{\frac{u}{v}}$.

$\bullet$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\widehat{\eta}_{\frac{p}{q}}$$\widehat{\eta}_{\frac{u}{v}}$$\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}}$$\widehat{\eta}_{0}$$z^{\prime}$$z$$z^{\prime\prime}$$a$$p$
Figure 14. Intersections whose bi-indices inherit (resp. are induced from) the
one between $\widehat{\eta}_{\frac{p}{q}}$ (resp.
$\widehat{\eta}_{\frac{u}{v}}$) and $\widehat{\eta}_{0}$

For the first case, if the two intersections are both either in the interior
or at decorations, the bi-indexes are same. For the second case, the two bi-
indexes differ from
$\operatorname{ind}^{\mathbb{Z}^{2}}_{z^{\prime\prime}}(\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}},\widehat{\eta}_{\frac{u}{v}})=(l+1)\cdot(1-\mathbb{X})$.
Thus we have

$\operatorname{Int}^{\mathfrak{q}}_{\mathbf{D}_{3}\backslash\\{p\\}}(\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}},\widehat{\eta}_{0})=\operatorname{Int}^{\mathfrak{q}}_{\mathbf{D}_{3}\backslash\\{z\\}}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{0})+q_{1}^{l+1}q_{2}^{-l-1}\cdot\operatorname{Int}^{\mathfrak{q}}_{\mathbf{D}_{3}\backslash\\{z\\}}(\widehat{\eta}_{\frac{u}{v}},\widehat{\eta}_{0}).$
(3.28)

For the intersection on $a$, by inheritance, we have bi-index given directly
by

$\operatorname{ind}^{\mathbb{Z}^{2}}_{p}(\widehat{\eta}_{0},\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}})=\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{0},\widehat{\eta}_{\frac{p}{q}}).$
(3.29)

Hence we have

$\begin{split}\operatorname{ind}^{\mathbb{Z}^{2}}_{p}(\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}},\widehat{\eta}_{0})&=1-\operatorname{ind}^{\mathbb{Z}^{2}}_{p}(\widehat{\eta}_{0},\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}})=1-\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{0},\widehat{\eta}_{\frac{p}{q}})\\\
&=1-\mathbb{X}+\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{0}).\end{split}$

Thus, the bi-index in $\mathbf{D}_{3}^{\circ}$ contributes
$(1-\mathbb{X})+\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{0})$
and
$\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{0})$
in $\mathfrak{q}$-intersection from (3.15).

On the other hand, we have

$\begin{split}\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{\frac{u}{v}},\widehat{\eta}_{0})&=\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{0})-\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{u}{v}})\\\
&=[\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{0})+(1-\mathbb{X})]-[\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{u}{v}})+(1-\mathbb{X})]\\\
&=[\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{0})+(1-\mathbb{X})]-(l+1)\cdot(1-\mathbb{X}).\end{split}$

Thus, we deduce that

$\operatorname{Int}^{\mathfrak{q}}_{z}(\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}},\widehat{\eta}_{0})=\operatorname{Int}^{\mathfrak{q}}_{z}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{0})+q_{1}^{l+1}q_{2}^{-l-1}\cdot\operatorname{Int}^{\mathfrak{q}}_{z}(\widehat{\eta}_{\frac{u}{v}},\widehat{\eta}_{0}).$
(3.30)

Therefore, we have

$\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}},\widehat{\eta}_{0})=\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{0})+q_{1}^{l+1}q_{2}^{-l-1}\cdot\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{u}{v}},\widehat{\eta}_{0}),$
(3.31)

which coincides with
$\overline{\textbf{R}}_{\mathfrak{q}}^{\flat}(\frac{r}{s})$ with a
multiplication of $q_{1}^{-1}$ if we take $\mathfrak{q}=q_{1}^{-1}q_{2}$.
Similarly, we deduce that

$\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}},\widehat{\eta}_{\infty})=\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\infty})+q_{1}^{l+1}q_{2}^{-l-1}\cdot\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{u}{v}},\widehat{\eta}_{\infty}).$
(3.32)

If we take $q_{1}^{-1}q_{2}=\mathfrak{q}$, the fraction we get in the theorem
coincides the left $\mathfrak{q}$-deformation of
$\frac{r}{s}=\frac{p}{q}\oplus\frac{u}{v}$, which completes the proof. ∎

######  Example 3.19.

We give examples of $\frac{3}{2}$ and $-2$ (see Figure 13) for the left
$\mathfrak{q}$-deformations. We compute that

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{3}{2}},\widehat{\eta}_{0})&=q_{1}+q_{2}+q_{1}^{3}q_{2}^{-2},\\\
\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{3}{2}},\widehat{\eta}_{\infty})&=1+q_{1}^{2}q_{2}^{-2};\end{array}\right.\end{split}$

and

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{-\frac{2}{1}},\widehat{\eta}_{0})&=q_{1}^{3}q_{2}^{-2}+q_{1},\\\
\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{-\frac{2}{1}},\widehat{\eta}_{\infty})&=q_{2}.\end{array}\right.\end{split}$

By Theorem 3.18, we have

$\Big{[}\frac{3}{2}\Big{]}^{\flat}_{\mathfrak{q}}=\frac{q_{1}^{-1}\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{3}{2}},\widehat{\eta}_{0})}{\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{3}{2}},\widehat{\eta}_{\infty})}\bigg{|}_{\mathfrak{q}=q_{1}^{-1}q_{2}}=\frac{\mathfrak{q}^{3}+\mathfrak{q}^{2}+1}{\mathfrak{q}^{2}+1},$

and

$\Big{[}-\frac{2}{1}\Big{]}^{\flat}_{\mathfrak{q}}=-\frac{\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{-\frac{2}{1}},\widehat{\eta}_{0})}{\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{-\frac{2}{1}},\widehat{\eta}_{\infty})}\bigg{|}_{\mathfrak{q}=q_{1}^{-1}q_{2}}=-\frac{\mathfrak{q}^{2}+1}{\mathfrak{q}^{3}}.$

###  3.4. Right $\mathfrak{q}$-deformations as $\mathfrak{q}$-intersections

In this subsection, we add a finite set $\mathbf{M}$ of _(open) marked points_
to $\partial\mathbf{S}$ satisfying $|\mathbf{M}|=|\bigtriangleup|$ and get a
decorated marked surface (or _DMS_ for short). We still denote the DMS by
${\mathbf{S}}_{\bigtriangleup}$. An arc $c$ is called _open_ if $c(0)$ and
$c(1)$ are in $\mathbf{M}$, without self-intersections in
${\mathbf{S}}_{\bigtriangleup}^{\circ}$. We call two open arcs do not cross
each other if they do not have intersections in
${\mathbf{S}}_{\bigtriangleup}^{\circ}$.

We also have bigraded open arcs as before. For an open arc $\gamma$, we define
the _$\mathbb{Z}^{2}$ -graded $\mathfrak{q}$-intersection_ between a lift
$\widehat{\gamma}$ of $\gamma$ and
$\widehat{\tau}\in\widehat{\operatorname{CA}}({\mathbf{S}}_{\bigtriangleup})$
to be

$\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma},\widehat{\tau})=\displaystyle\sum_{\varrho,\varsigma\in\mathbb{Z}}q_{1}^{\varrho}q_{2}^{\varsigma}\cdot\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}^{\circ}}^{\varrho+\varsigma\mathbb{X}}(\widehat{\gamma},\widehat{\tau}).$
(3.33)

Note that we have
$\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma},\widehat{\tau})\mid_{q_{1}=q_{2}=1}=\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}}(\gamma,\tau)$.
We define a special class of open (bigraded) arcs.

######  Definition 3.20.

An _open full formal arc system_
${\mathbf{A}}=\\{\gamma_{1},\cdots,\gamma_{n}\\}$ of a DMS
${\mathbf{S}}_{\bigtriangleup}$ is a collection of pairwise non-crossing open
arcs, such that it divides the surface ${\mathbf{S}}_{\bigtriangleup}$ into
polygons, called _${\mathbf{A}}$ -polygons_, satisfying that each
${\mathbf{A}}$-polygon contains exactly one decoration. We call
$\sigma\in\operatorname{CA}({\mathbf{S}}_{\bigtriangleup})$ the _dual_ to
$\gamma_{i}$, if $\gamma_{i}$ intersects it once and $\gamma_{j}$ does not
intersect it for any $j\neq i$. Denote $s_{i}$ the dual to $\gamma_{i}$ and
${\mathbf{A}}^{*}=\\{s_{1},\cdots,s_{n}\\}$.

Let
$\widehat{\gamma}_{1},\cdots,\widehat{\gamma}_{n},\widehat{s}_{1},\cdots,\widehat{s}_{n}$
be their bigraded lifts with

$\operatorname{ind}^{\mathbb{Z}^{2}}(\widehat{\gamma}_{i},\widehat{s}_{j})=\delta_{ij}.$
(3.34)

We add three open marked points to the boundary of $\mathbf{D}_{3}$ in Section
3.3. Let $\gamma_{0}$, $\gamma_{\infty}$ be two open arcs which form an open
arc system in $\mathbf{D}_{3}$ and intersect with $\eta_{0}$ and
$\eta_{\infty}$ transitively only once respectively. Let
$\widehat{\gamma}_{0},\widehat{\gamma}_{\infty}$ be their bigraded lifts
respectively which satisfy

$\operatorname{ind}^{\mathbb{Z}^{2}}(\widehat{\gamma}_{\infty},\widehat{\eta}_{\infty})=\operatorname{ind}^{\mathbb{Z}^{2}}(\widehat{\gamma}_{0},\widehat{\eta}_{0})=0.$
(3.35)

We draw them as blue arcs in Figure 13.

######  Theorem 3.21.

For any rational number $\frac{r}{s}\in\overline{\mathbb{Q}}$, we have

$\Big{[}\frac{r}{s}\Big{]}^{\sharp}_{\mathfrak{q}}=\frac{\varepsilon\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{\infty},\widehat{\eta}_{\frac{r}{s}})}{\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{0},\widehat{\eta}_{\frac{r}{s}})}\bigg{|}_{\mathfrak{q}=q_{1}^{-1}q_{2}},$
(3.36)

corresponding to the right $\mathfrak{q}$-deformation of $\frac{r}{s}$, where

$\varepsilon=\begin{cases}1,&\frac{r}{s}\geq 0,\\\
-q_{1}^{-1},&\frac{r}{s}<0,\end{cases}$

and the polynomials in the numerator and denominator are polynomials in
$\mathbb{Z}[q_{1}^{-1}q_{2}]$. In particular, for
$\frac{r}{s}\in\overline{\mathbb{Q}^{\geq 0}}$, we have __

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}(\frac{r}{s})&=\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{\infty},\widehat{\eta}_{\frac{r}{s}})\big{|}_{\mathfrak{q}=q_{1}^{-1}q_{2}},\\\
\textbf{S}_{\mathfrak{q}}^{\sharp}(\frac{r}{s})&=\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{0},\widehat{\eta}_{\frac{r}{s}})\big{|}_{\mathfrak{q}=q_{1}^{-1}q_{2}}.\end{array}\right.\end{split}$
(3.37)

###### Proof.

The theorem follows the same way of the left version and we prove the non-
negative case by induction using 3.16. For the starting case, we have

$\begin{split}\operatorname{Int}(\widehat{\gamma}_{\infty},\widehat{\eta}_{0})&=\operatorname{Int}(\widehat{\gamma}_{0},\widehat{\eta}_{\infty})=0,\\\
\operatorname{Int}(\widehat{\gamma}_{\infty},\widehat{\eta}_{\infty})&=\operatorname{Int}(\widehat{\gamma}_{0},\widehat{\eta}_{0})=q_{1}^{0}q_{2}^{0}=1.\end{split}$

Thus, they coincide with the right $\mathfrak{q}$-deformation of 0 and
$\infty$ respectively. We assume that the formula (3.36) holds for
$\frac{p}{q},\frac{u}{v}\in\overline{\mathbb{Q}^{\geq 0}}$ in 3.16 which
satisfy

$\operatorname{ind}^{\mathbb{Z}^{2}}_{z}(\widehat{\eta}_{\frac{p}{q}},\widehat{\eta}_{\frac{u}{v}})=l(1-\mathbb{X}),$

where $l=l(\frac{p}{q}\oplus\frac{u}{v})\in\mathbb{N}$. Similar as the left
version, the intersections between $\widehat{\gamma}_{0}$ and
$\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}}$ consist of two parts. One
inherits from $\widehat{\eta}_{\frac{p}{q}}$, whose bi-indices are same; and
the other one is induced from $\widehat{\eta}_{\frac{u}{v}}$, whose bi-indices
differ from
$-\operatorname{ind}^{\mathbb{Z}^{2}}_{z^{\prime\prime}}(\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}},\widehat{\eta}_{\frac{u}{v}})=(l+1)\cdot(\mathbb{X}-1)$.
Notice that the intersections are all in the interior, which simplifies things
a lot. Thus, we have

$\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{0},\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}})=\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{0},\widehat{\eta}_{\frac{p}{q}})+q_{1}^{-l-1}q_{2}^{l+1}\cdot\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{0},\widehat{\eta}_{\frac{u}{v}}),$
(3.38)

which coincides denominator of the right $\mathfrak{q}$-deformation of
$\frac{p}{q}\oplus\frac{u}{v}$ if we take $q_{1}^{-1}q_{2}=\mathfrak{q}$.
Similarly, we deduce that

$\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{\infty},\widehat{\eta}_{\frac{p}{q}\oplus\frac{u}{v}})=\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{\infty},\widehat{\eta}_{\frac{p}{q}})+q_{1}^{-l-1}q_{2}^{l+1}\cdot\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{\infty},\widehat{\eta}_{\frac{u}{v}}).$
(3.39)

Thus, we finish the proof. ∎

######  Example 3.22.

We continue the examples of $\frac{3}{2}$ and $-2$ (see Figure 13) for the
right $\mathfrak{q}$-deformations. We compute that

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{\infty},\widehat{\eta}_{\frac{3}{2}})&=1+q_{1}^{-1}q_{2}+q_{1}^{-2}q_{2}^{2},\\\
\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{0},\widehat{\eta}_{\frac{3}{2}})&=1+q_{1}^{-1}q_{2};\end{array}\right.\end{split}$

and

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{\infty},\widehat{\eta}_{-\frac{2}{1}})&=1+q_{1}^{-1}q_{2},\\\
\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{0},\widehat{\eta}_{-\frac{2}{1}})&=q_{1}^{-3}q_{2}^{2}.\end{array}\right.\end{split}$

By Theorem 3.21, we have

$\Big{[}\frac{3}{2}\Big{]}^{\sharp}_{\mathfrak{q}}=\frac{\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{\infty},\widehat{\eta}_{\frac{3}{2}})}{\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{0},\widehat{\eta}_{\frac{3}{2}})}\bigg{|}_{\mathfrak{q}=q_{1}^{-1}q_{2}}=\frac{\mathfrak{q}^{2}+\mathfrak{q}+1}{\mathfrak{q}+1},$

and

$\Big{[}-\frac{2}{1}\Big{]}^{\sharp}_{\mathfrak{q}}=-\frac{q_{1}^{-1}\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{\infty},\widehat{\eta}_{-\frac{2}{1}})}{\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{0},\widehat{\eta}_{-\frac{2}{1}})}\bigg{|}_{\mathfrak{q}=q_{1}^{-1}q_{2}}=-\frac{\mathfrak{q}+1}{\mathfrak{q}^{2}}.$

###  3.5. Combinatorial properties via $\mathfrak{q}$-intersections

Next, we give some topological explanation of some properties in [MGO1] via
$\mathfrak{q}$-intersections. We draw $\gamma_{0}$ and $\gamma_{\infty}$ as
foliations in the branched double cover $\Sigma_{\Delta}$, which intersect
$C_{0}$ and $C_{\infty}$ only once respectively (see Figure 15). We have the
following corollary.

$z_{\infty}$$z_{0}$$z_{\infty}$$z_{0}$$z_{*}$$\gamma_{\infty}$$\gamma_{0}$$\gamma_{\infty}$$\gamma_{0}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$
Figure 15. The branched double cover $\Sigma_{\Delta}$ of $\mathbf{D}_{3}$ and
its foliations

######  Corollary 3.23 ([MGO1]).

For any rational number $\frac{r}{s}\in\overline{\mathbb{Q}^{\geq 0}}$, its
right and left $\mathfrak{q}$-deformations satisfy the following properties.

Positivity:

The polynomials
_$\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}(r/s),\textbf{S}_{\mathfrak{q}}^{\sharp}(r/s),\operatorname{\textbf{R}}_{\mathfrak{q}}^{\flat}(r/s)$_
and _$\textbf{S}_{\mathfrak{q}}^{\flat}(r/s)$_ have positive integer
coefficients.

Specialization:

If we take $\mathfrak{q}=1$, we have __

$\displaystyle\begin{split}\left\\{\begin{array}[]{ll}\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}(r/s)|_{\mathfrak{q}=1}&=\operatorname{\textbf{R}}_{\mathfrak{q}}^{\flat}(r/s)|_{\mathfrak{q}=1}=r,\\\
\textbf{S}_{\mathfrak{q}}^{\sharp}(r/s)|_{\mathfrak{q}=1}&=\textbf{S}_{\mathfrak{q}}^{\flat}(r/s)|_{\mathfrak{q}=1}=s.\end{array}\right.\end{split}$
(3.40)

###### Proof.

The positivity follows from Theorem 3.21, Theorem 3.18 and that intersection
numbers are all positive.

For specialization, we consider the branched double covering $\Sigma_{\Delta}$
of $\mathbf{D}_{3}$. Then the closed arc $\eta_{\frac{r}{s}}$ becomes the
simple closed curve $C_{\frac{r}{s}}$, whose preimage under $\widetilde{\pi}$
is a line with slope $\frac{r}{s}$, on $\Sigma_{\Delta}$ through these
corresponding decorations which are endpoints of $\eta_{\frac{r}{s}}$. When we
take $q_{1}=q_{2}=1,$ the $\mathfrak{q}$-intersection degenerates to usual
intersection. By construction above, we have

$\displaystyle\operatorname{Int}_{\mathbf{D}_{3}}(\eta_{\frac{r}{s}},\eta_{0})=\frac{1}{2}\cdot\operatorname{Int}_{\Sigma_{\Delta}}(C_{\frac{r}{s}},C_{0})=r,$
$\displaystyle\quad\operatorname{Int}_{\mathbf{D}_{3}}(\eta_{\frac{r}{s}},\eta_{\infty})=\frac{1}{2}\cdot\operatorname{Int}_{\Sigma_{\Delta}}(C_{\frac{r}{s}},C_{\infty})=s;$
$\displaystyle\operatorname{Int}_{\mathbf{D}_{3}}(\gamma_{\infty},\eta_{\frac{r}{s}})=\frac{1}{2}\cdot\operatorname{Int}_{\Sigma_{\Delta}}(\gamma_{\infty},C_{\frac{r}{s}})=r,$
$\displaystyle\quad\operatorname{Int}_{\mathbf{D}_{3}}(\gamma_{0},\eta_{\frac{r}{s}})=\frac{1}{2}\cdot\operatorname{Int}_{\Sigma_{\Delta}}(\gamma_{0},C_{\frac{r}{s}})=s.$

Therefore, the results follows from Theorem 3.18 and Theorem 3.21. ∎

######  Example 3.24.

We notice that in the example of $\frac{3}{s}$, $\eta_{\frac{3}{2}}$ hits
$\gamma_{\infty}$ three times and hits $\gamma_{0}$ twice in
${\mathbf{S}}_{\bigtriangleup}$, which implies that
$\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}(3/2)|_{\mathfrak{q}=1}=3$
and $\textbf{S}_{\mathfrak{q}}^{\sharp}(3/2)|_{\mathfrak{q}=1}=2$.

##  4\. Categorification

###  4.1. Ginzburg algebra and derived categories

######  Definition 4.1 ([Ke, IQ]).

Let $Q=(Q_{0},Q_{1})$ be a finite quiver with vertices set
$Q_{0}=\\{1,2,\ldots,n\\}$ and arrows set $Q_{1}$. The _Ginzburg Calabi-Yau-
$\mathbb{X}$ ddg algebra_ $\Gamma_{\mathbb{X}}Q:=(\mathbf{k}\overline{Q},d)$
is defined as follows. We define a
$\mathbb{Z}\oplus\mathbb{Z}\mathbb{X}$-graded quiver $\overline{Q}$ with the
same vertices set as $Q_{0}$ and the following arrows:

  * •

original arrows $a:i\rightarrow j\in Q_{1}$ with degree 0;

  * •

opposite arrows $a^{*}:j\rightarrow i\in Q_{1}$ associated to $a\in Q_{1}$
with degree $2-\mathbb{X}$;

  * •

a loop $e_{i}^{*}$ for each $i\in Q_{0}$ with degree $1-\mathbb{X}$, where
$e_{i}$ is the idempotent at $i$.

Let $\mathbf{k}\overline{Q}$ be a
$\mathbb{Z}\oplus\mathbb{Z}\mathbb{X}$-graded path algebra of $\overline{Q}$,
and define a differential
$d:\mathbf{k}\overline{Q}\rightarrow\mathbf{k}\overline{Q}$ of degree 1 by

  * •

$da=da^{*}=0$ for $a\in Q_{1}$;

  * •

$de^{*}_{i}=e_{i}\big{(}\sum_{a\in Q_{1}}(aa^{*}-a^{*}a)\big{)}e_{i}$.

We denote by
$\operatorname{\mathcal{D}}_{\mathbb{X}}(Q):=\operatorname{pvd}\Gamma_{\mathbb{X}}Q$
the perfect value derived category of $\Gamma_{\mathbb{X}}Q$, which is the
same as the finite-dimensional derived category of $\Gamma_{\mathbb{X}}Q$. We
consider the $A_{2}$ case, where $A_{2}$ is a quiver with vertices set
$\\{1,2\\}$ and an arrow $1\rightarrow 2$, and the corresponding category is
denoted by $\operatorname{\mathcal{D}}_{\mathbb{X}}(A_{2})$.

###  4.2. Rational case via spherical objects

In this subsection, we aim to find spherical objects in some certain category
which correspond to rational numbers and represent their hom space via right
and left $\mathfrak{q}$-deformations. We particularly consider the case of the
Calabi-Yau-$\mathbb{X}$ category of the $A_{2}$ quiver. Recall that a
triangulated category $\operatorname{\mathcal{D}}$ is called _Calabi-Yau-
$\mathbb{X}$_ if for any objects $L,M$ in $\operatorname{\mathcal{D}}$, we
have a natural isomorphism

$\operatorname{Hom}_{\operatorname{\mathcal{D}}}(L,M)\cong
D\operatorname{Hom}_{\operatorname{\mathcal{D}}}(M,L[\mathbb{X}]),$

where $D=\operatorname{Hom}(-,\mathbf{k})$ is the dual functor and
$\mathbf{k}$ is an algebraically closed field. In particular,
$\operatorname{\mathcal{D}}_{\mathbb{X}}(A_{2})$ is a Calabi-Yau-$\mathbb{X}$
category with a distinguish auto-equivalence

$\mathbb{X}:\operatorname{\mathcal{D}}_{\mathbb{X}}(A_{2})\rightarrow\operatorname{\mathcal{D}}_{\mathbb{X}}(A_{2}).$

######  Definition 4.2 ([IQZ]).

For any $M,N\in\operatorname{\mathcal{D}}$, we define the _bigraded Hom_ as

$\operatorname{Hom}^{\mathbb{Z}^{2}}(M,N):=\bigoplus_{\varrho,\varsigma\in\mathbb{Z}}\operatorname{Hom}_{\operatorname{\mathcal{D}}}(M,N[\varrho+\varsigma\mathbb{X}]),$

and its $\mathfrak{q}$-dimension as

$\displaystyle\operatorname{dim}^{\mathfrak{q}}\operatorname{Hom}^{\mathbb{Z}^{2}}(M,N)\colon=\sum_{\varrho,\varsigma\in\mathbb{Z}}q_{1}^{\varrho}q_{2}^{\varsigma}\cdot\operatorname{dim}\operatorname{Hom}_{\operatorname{\mathcal{D}}}(M,N[\varrho+\varsigma\mathbb{X}]).$
(4.1)

When $M=N$, $\operatorname{Hom}^{\mathbb{Z}^{2}}(M,M)$ becomes a
$\mathbb{Z}^{2}$-graded algebra, called the Ext-algebra of $M$ and denoted by
$\operatorname{Ext}^{\mathbb{Z}^{2}}(M,M)$.

By definition, we directly have

$\operatorname{dim}\operatorname{Hom}^{\mathbb{Z}^{2}}(M,N)=\operatorname{dim}^{\mathfrak{q}}\operatorname{Hom}^{\mathbb{Z}^{2}}(M,N)\mid_{q_{1}=q_{2}=1}.$
(4.2)

######  Definition 4.3 ([IQZ]).

An object $S$ is called _$\mathbb{X}$ -spherical_ if
$\operatorname{Hom}^{\bullet}(S,S)=\mathbf{k}\oplus\mathbf{k}[-\mathbb{X}]$.

For any spherical object $S$ in an Calabi-Yau-$\mathbb{X}$ category
$\operatorname{\mathcal{D}}$, there is an associated auto-equivalence, namely
the _twist functor_
$\phi_{S}:\operatorname{\mathcal{D}}\rightarrow\operatorname{\mathcal{D}}$,
defined by

$\phi_{S}(X)=\operatorname{Cone}(S\otimes\operatorname{Hom}^{\bullet}(S,X)\rightarrow
X)$

with inverse

$\phi_{S}^{-1}(X)=\operatorname{Cone}(X\rightarrow
S\otimes\operatorname{Hom}^{-\bullet}(X,S))[-1].$

By [ST, Lemma 2.11], we have the formula

$\phi_{\psi(M)}=\psi\circ\phi_{M}\circ\psi^{-1}$ (4.3)

for any spherical object $M$ and any automorphism $\psi$ in
$\operatorname{Aut}\operatorname{\mathcal{D}}$. We define
$\widehat{\operatorname{Sph}}(\Gamma_{\mathbb{X}}A_{2})$ to be the set of all
spherical objects in $\operatorname{\mathcal{D}}_{\mathbb{X}}(A_{2})$ which
are simples in some hearts (cf. [KQ, Section 10]). Let

$\operatorname{Sph}(\Gamma_{\mathbb{X}}A_{2}):=\widehat{\operatorname{Sph}}(\Gamma_{\mathbb{X}}A_{2})/\langle[1],[\mathbb{X}]\rangle.$

We define $\operatorname{ST}(A_{2})$ to be the subgroup of
$\operatorname{Aut}\operatorname{\mathcal{D}}_{\mathbb{X}}(A_{2})$ generated
by $\phi_{S}$ for any
$S\in\widehat{\operatorname{Sph}}(\Gamma_{\mathbb{X}}A_{2})$.

Let $\mathbf{D}_{3}$ be the three decorated disk as before. There are
reachable spherical objects up to shifts $[1]$ and $\mathbb{X}$ in
$\operatorname{\mathcal{D}}_{\mathbb{X}}(A_{2})$ corresponding to rational
numbers. We have a categorification of Lemma 3.4 as follows.

######  Proposition 4.4 ([IQZ, §4]).

There is a bijection $X$ and an isomorphism $\iota$ which fits into the
following:

$\widehat{\operatorname{CA}}(\mathbf{D}_{3})$$\operatorname{BT}(\mathbf{D}_{3})$$\iota$$X$$\widehat{\operatorname{Sph}}(\Gamma_{\mathbb{X}}A_{2})$$\operatorname{ST}(A_{2})$
(4.4)

sending $\widehat{\eta}_{\pm\frac{r}{s}}$ to $X_{\pm\frac{r}{s}}$ and
$B_{\eta_{\pm\frac{r}{s}}}$ to $\phi_{X_{\pm\frac{r}{s}}}$, satisfying

$X_{B_{\eta_{\frac{u}{v}}}(\widehat{\eta}_{\frac{p}{q}})}=\phi_{X_{\frac{u}{v}}}(X_{\frac{p}{q}}),$
(4.5)

and

$X_{B_{\eta_{-\frac{p}{q}}}(\widehat{\eta}_{-\frac{u}{v}})}=\phi_{X_{-\frac{p}{q}}}(X_{-\frac{u}{v}})$
(4.6)

for $\frac{r}{s}\in\overline{\mathbb{Q}^{\geq 0}}$. Hence, (3.21) translates
to the triangle

${X_{\frac{p}{q}}}$${X_{\frac{p}{q}\oplus\frac{u}{v}}}$${X_{\frac{u}{v}}[(l+1)(1-\mathbb{X})]}$${X_{\frac{p}{q}}[1],}$
(4.7)

where $l=l(\frac{p}{q}\oplus\frac{u}{v})$ is the integer in 3.16.

In fact, when we draw
$\\{X_{\frac{r}{s}}\\}_{\frac{r}{s}\in\overline{\mathbb{Q}^{\geq 0}}}$ on the
weighted Farey graph in the right one in Figure 16, $\mathfrak{q}^{l},l\geq 0$
represents that there is a morphism of degree $l(1-\mathbb{X})$ between the
connected spherical objects.

$\displaystyle X_{0}$$\displaystyle X_{\infty}$$\displaystyle
X_{\frac{1}{1}}$$1$$1$

$\displaystyle X_{\frac{p}{q}}$$\displaystyle X_{\frac{u}{v}}$$\displaystyle
X_{\frac{p}{q}\oplus\frac{u}{v}}$$1$$\mathfrak{q}^{l+1}$$\mathfrak{q}^{l}$

Figure 16. The categorification

Here is the iterative construction of Proposition 4.4. Let $X_{0},X_{\infty}$
be two spherical objects which are simples in some canonical heart with
$\operatorname{Ext}^{1}(X_{\infty},X_{0})\neq 0$. Let
$X_{\frac{1}{1}}=\phi_{X_{\infty}}(X_{0})$, and we deduce that
$X_{\frac{1}{1}}$ is also a spherical object. We have a triangle

${X_{0}}$${X_{\frac{1}{1}}}$${X_{\infty}}$${X_{0}[1]}$

by construction. We assume that the triangle in (4.7) holds for $l$ and
consider the $l+1$ case. By Calabi-Yau-$\mathbb{X}$ duality, we have non-zero
morphisms $X_{\frac{p}{q}\oplus\frac{u}{v}}\rightarrow
X_{\frac{p}{q}}[\mathbb{X}]$ and $X_{\frac{u}{v}}\rightarrow
X_{\frac{p}{q}\oplus\frac{u}{v}}[(l+1)\mathbb{X}-l]$. Hence we can extend them
to triangles:

${X_{\frac{p}{q}}}$${Y}$${X_{\frac{p}{q}\oplus\frac{u}{v}}[1-\mathbb{X}]}$${X_{\frac{p}{q}}[1]}$

and

${X_{\frac{p}{q}\oplus\frac{u}{v}}}$${Y^{\prime}}$${X_{\frac{u}{v}}[(l+1)(1-\mathbb{X})]}$${X_{\frac{p}{q}\oplus\frac{u}{v}}[1],}$

where these new spherical objects
$Y=\phi_{X_{\frac{p}{q}\oplus\frac{u}{v}}[1-\mathbb{X}]}(X_{\frac{p}{q}})=\phi_{X_{\frac{p}{q}\oplus\frac{u}{v}}}(X_{\frac{p}{q}})$
and
$Y^{\prime}=\phi_{X_{\frac{u}{v}}[(l+1)(1-\mathbb{X})]}(X_{\frac{p}{q}\oplus\frac{u}{v}})=\phi_{X_{\frac{u}{v}}}(X_{\frac{p}{q}\oplus\frac{u}{v}})$.
Thus we construct the spherical objects associated to non negative rational
numbers and the negative case is similar.

######  Theorem 4.5 ([IQZ, Lemma 3.4, Proposition 4.6]).

For any
$\widehat{\eta},\widehat{\eta}^{\prime}\in\widehat{\operatorname{CA}}(\mathbf{D}_{3})$
satisfying
$\operatorname{Int}_{\mathbf{D}_{3}^{\circ}}(\widehat{\eta},\widehat{\eta}^{\prime})=0$,
and $\widehat{\gamma}_{i}\in{\mathbf{A}}$, we have

$\displaystyle\operatorname{dim}^{\mathfrak{q}}\operatorname{Hom}^{\mathbb{Z}^{2}}(P_{i},X_{\widehat{\eta}})=\operatorname{Int}^{\mathfrak{q}}(\widehat{\gamma}_{i},\widehat{\eta}),$
(4.8)

and

$\displaystyle\operatorname{dim}^{\mathfrak{q}}\operatorname{Hom}^{\mathbb{Z}^{2}}(X_{\widehat{\eta}},X_{\widehat{\eta}^{\prime}})=\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta},\widehat{\eta}^{\prime}),$
(4.9)

where $P_{i}$ is the indecomposable projective module corresponding to
$\widehat{\gamma}_{i}$.

By Theorem 3.18, Theorem 3.21 and Theorem 4.5, we have the following
corollaries directly.

######  Corollary 4.6.

For any rational number $\frac{r}{s}\in\mathbb{Q}\setminus\\{0\\}$, the
fraction

$\frac{\varepsilon\operatorname{dim}^{\mathfrak{q}}\operatorname{Hom}^{\mathbb{Z}^{2}}(X_{\frac{r}{s}},X_{0})}{\operatorname{dim}^{\mathfrak{q}}\operatorname{Hom}^{\mathbb{Z}^{2}}(X_{\frac{r}{s}},X_{\infty})}\bigg{|}_{\mathfrak{q}=q_{1}^{-1}q_{2}}$
(4.10)

corresponds to the left $\mathfrak{q}$-deformation of $\frac{r}{s}$, where

$\varepsilon=\begin{cases}q_{1}^{-1},&\frac{r}{s}\geq 0,\\\
-1,&\frac{r}{s}<0.\end{cases}$

######  Corollary 4.7.

For any rational number $\frac{r}{s}\in\overline{\mathbb{Q}}$, the fraction

$\frac{\varepsilon\operatorname{dim}^{\mathfrak{q}}\operatorname{Hom}^{\mathbb{Z}^{2}}(P_{\infty},X_{\frac{r}{s}})}{\operatorname{dim}^{\mathfrak{q}}\operatorname{Hom}^{\mathbb{Z}^{2}}(P_{0},X_{\frac{r}{s}})}\bigg{|}_{\mathfrak{q}=q_{1}^{-1}q_{2}}$
(4.11)

corresponds to the right $\mathfrak{q}$-deformation of $\frac{r}{s}$, where
$P_{0}$ and $P_{\infty}$ are the corresponding indecomposable projectives
satisfying that
$\operatorname{dim}^{\mathfrak{q}}\operatorname{Hom}^{\mathbb{Z}^{2}}(P_{i},X_{j})=\delta_{i,j},i,j\in\\{0,\infty\\}$.
Here

$\varepsilon=\begin{cases}1,&\frac{r}{s}\geq 0,\\\
-q_{1}^{-1},&\frac{r}{s}<0.\end{cases}$

##  5\. Applications

###  5.1. Reduction to single grading as foliations

Let $N\geq 2$ be an integer. We collapse the double grading $\Lambda$ on
${\mathbf{S}}_{\bigtriangleup}$ to single grading $\lambda$, which is a line
field (or foliation) in $\mathbb{P}T\mathbf{S}_{\bigtriangleup}$ by setting
$\mathbb{X}=N$. More precisely, a double grading
$(a,b)\in\mathbb{Z}\oplus\mathbb{Z}\mathbb{X}$ collapses into
$a+bN\in\mathbb{Z}$. The foliations in such cases are given by quadratic
differentials

$(z^{3}+az+b)^{N-2}\mathrm{d}z^{\otimes 2}$

on $\mathbb{C}\mathbb{P}^{1}$ with real blow-up at $\infty$, cf. Figure 17 for
$N=2,3,4$ and ${\mathbf{S}}_{\bigtriangleup}=\mathbf{D}_{3}$. Notice that the
foliations come from quadratic differential (cf. [I, BQS]). Then
$q_{2}=q_{1}^{N}$ and the $\mathfrak{q}$-intersection formula (3.15) reduces
to

$\operatorname{Int}^{\mathfrak{q}}(\widehat{\sigma},\widehat{\tau})=\displaystyle\sum_{k\in\mathbb{Z}}q_{1}^{k}\cdot\operatorname{Int}_{\bigtriangleup}^{k}(\widehat{\sigma},\widehat{\tau})+(1+q_{1}^{N-1})\displaystyle\sum_{k\in\mathbb{Z}}q_{1}^{k}\cdot\operatorname{Int}_{{\mathbf{S}}_{\bigtriangleup}^{\circ}}^{k}(\widehat{\sigma},\widehat{\tau}).$
(5.1)

for
$\widehat{\sigma},\widehat{\tau}\in\widehat{\operatorname{CA}}({\mathbf{S}}_{\bigtriangleup})$.
Moreover, $\mathfrak{q}=q_{1}^{N-1}$ in Theorem 3.18 and Theorem 3.21. When
$N=2$, $\mathfrak{q}=q_{1}$ and no specialization is required.

$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$
$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$
$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$

Figure 17. The foliations of the CY-2,3,4 case

###  5.2. Relation with BBL’s results

In Definition 4.1, if we replace $\mathbb{X}$ by an integer $N\geq 2$, we
obtain $N$-Ginzburg dg algebra $\Gamma_{N}Q$ and the corresponding Calabi-
Yau-$N$ category $\operatorname{\mathcal{D}}_{N}(Q)$. That is, there is a
projection

$\pi_{N}:\Gamma_{\mathbb{X}}Q\rightarrow\Gamma_{N}Q$ (5.2)

collapsing the double grading $\mathbb{Z}\oplus\mathbb{Z}\mathbb{X}$ into
$\mathbb{Z}$ by setting $\mathbb{X}=N$ similar as above. It induces a functor
$\pi_{N}:\operatorname{\mathcal{D}}_{\mathbb{X}}(Q)\rightarrow\operatorname{\mathcal{D}}_{N}(Q).$
We consider the case when $Q=A_{2}$. For
$\frac{r}{s}\in\overline{\mathbb{Q}^{\geq 0}}$, we claim that

$\operatorname{dim}^{\mathfrak{q}}\operatorname{Hom}^{\mathbb{Z}^{2}}(P_{?},X_{\frac{r}{s}})=\sum_{k\in\mathbb{Z}}m(\frac{r}{s},k)q_{1}^{-k}q_{2}^{k},$

where $m(\frac{r}{s},k)$ is the occurrence times of $X_{?}[k-k\mathbb{X}]$ in
the HN-filtration of $X_{\frac{r}{s}}$, $?\in\\{0,\infty\\}$. It holds
naturally from the fact that

$\operatorname{dim}^{\mathfrak{q}}\operatorname{Hom}^{\mathbb{Z}^{2}}(P_{i},X_{j}[\varrho+\varsigma\mathbb{X}])=\delta_{i,j}\delta_{\varrho,0}\delta_{\varsigma,0}$
(5.3)

where $i,j\in\\{0,\infty\\}$ and induction on $l(\frac{r}{s})$.

Consider the case when $N=2$ again with $\mathfrak{q}=q_{1}$ and we write
$q_{1}=q$. In [BBL], they define two functionals

$\operatorname{occ}_{q},\overline{\operatorname{hom}}_{q}:\widehat{\operatorname{Sph}}^{\mathbb{Z}}(\Gamma_{2}A_{2})\times\widehat{\operatorname{Sph}}^{\mathbb{Z}}(\Gamma_{2}A_{2})\rightarrow\mathbb{Z}[q^{-1},q],$

where $\widehat{\operatorname{Sph}}^{\mathbb{Z}}(\Gamma_{2}A_{2})$ is the set
of spherical objects in $\operatorname{\mathcal{D}}_{2}(A_{2})$. The former
one $\operatorname{occ}_{q}(X_{?},X_{\frac{r}{s}})$ counts the occurrences of
$X_{?}$ in the HN-filtration of $X_{\frac{r}{s}}$ for $?\in\\{0,\infty\\}$. By
(5.3), we deduce that

$\operatorname{occ}_{q}(X_{?},X)=\operatorname{dim}^{\mathfrak{q}}\operatorname{Hom}^{\mathbb{Z}^{2}}(P_{?},X)\mid_{q_{1}^{-1}q_{2}=q^{-1}}.$

The latter one is

$\overline{\operatorname{hom}}_{q}(L,M):=\begin{cases}q^{k}(q^{-2}-q^{-1}),&M\cong
L[k],\\\
\sum_{k\in\mathbb{Z}}\operatorname{dim}\operatorname{Hom}(L,M[k])q^{-k},&\mbox{otherwise}.\end{cases}$

Recall that in Remark 2.5, the $q$-deformations for negative rational numbers
are defined as:

$\left[-\frac{r}{s}\right]_{q}^{*}:=-q^{-1}\left[\frac{r}{s}\right]_{q^{-1}}^{*},$

where $\frac{r}{s}\in\mathbb{Q}^{+}\cup\\{\infty\\}$ and
$*\in\\{\sharp,\flat\\}$.

######  Corollary 5.1.

When specializing $\mathbb{X}=2$, the formulae (4.11) and (4.10) in Corollary
4.7 and Corollary 4.6 become the formulae in [BBL, Theorem 3.7, Theorem 3.8]
respectively. Notice that the condition $X\geq 0$ corresponds to our
$X_{-\frac{r}{s}}$ with $\frac{r}{s}\geq 0$.

###  5.3. Grothendieck group interpretation

Recall that $\operatorname{\mathcal{D}}_{\mathbb{X}}(A_{2})$ is a Calabi-
Yau-$\mathbb{X}$ category. The Grothendieck group
$\operatorname{K}(\operatorname{\mathcal{D}}_{\mathbb{X}}(A_{2}))$ admits a
basis $\\{[X_{0}],[X_{\infty}]\\}$ and is a $\mathbb{Z}[\mathfrak{q}^{\pm
1}]$-module defined by the action

$q_{1}^{l}q_{2}^{k}\cdot[E]:=[E[-l-k\mathbb{X}]].$ (5.4)

We have the following result.

######  Proposition 5.2.

For any $\frac{r}{s}\in\overline{\mathbb{Q}}$, we have

$[X_{\frac{r}{s}}]=\textbf{R}_{q_{1}^{-1}q_{2}}^{\sharp}(\frac{r}{s})[X_{0}]+\textbf{S}_{q_{1}^{-1}q_{2}}^{\sharp}(\frac{r}{s})[X_{\infty}],$
(5.5)

where $\textbf{R}_{q_{1}^{-1}q_{2}}^{\sharp}$ $($resp.
$\textbf{S}_{q_{1}^{-1}q_{2}}^{\sharp})$ is a polynomial of $q_{1}^{-1}q_{1}$
if we take $\mathfrak{q}=q_{1}^{-1}q_{1}$ in
$\operatorname{\textbf{R}}_{\mathfrak{q}}^{\sharp}$ $($resp.
$\textbf{S}_{\mathfrak{q}}^{\sharp})$.

###### Proof.

We only prove the non-negative case by induction on $l(\frac{r}{s})$. For the
initial case, it holds for $X_{0}$ and $X_{\infty}$ obviously. We assume that
it holds for $X_{\frac{p}{q}}$ and $X_{\frac{u}{v}}$ where
$\frac{p}{q},\frac{u}{v}$ are in 3.16. For $X_{\frac{p}{q}\oplus\frac{u}{v}}$,
we have

$\begin{split}[X_{\frac{p}{q}\oplus\frac{u}{v}}]&=[X_{\frac{p}{q}}]+[X_{\frac{u}{v}}[(l+1)(1-\mathbb{X})]]\\\
&=[X_{\frac{p}{q}}]+q_{1}^{-l-1}q_{2}^{l+1}[X_{\frac{u}{v}}]\\\
&=\textbf{R}_{q_{1}^{-1}q_{2}}^{\sharp}(\frac{p}{q})[X_{0}]+\textbf{S}_{q_{1}^{-1}q_{2}}^{\sharp}(\frac{p}{q})[X_{\infty}]+q_{1}^{-l-1}q_{2}^{l+1}\big{(}\textbf{R}_{q_{1}^{-1}q_{2}}^{\sharp}(\frac{u}{v})[X_{0}]+\textbf{S}_{q_{1}^{-1}q_{2}}^{\sharp}(\frac{u}{v})[X_{\infty}]\big{)}\\\
&=\textbf{R}_{q_{1}^{-1}q_{2}}^{\sharp}(\frac{p}{q}\oplus\frac{u}{v})[X_{0}]+\textbf{S}_{q_{1}^{-1}q_{2}}^{\sharp}(\frac{p}{q}\oplus\frac{u}{v})[X_{\infty}],\end{split}$
(5.6)

which implies the result. ∎

###  5.4. Relation to Jones polynomials for rational case

For every rational number $\frac{r}{s}>1$, there is an associated rational
(two-bridge) knot $C(\frac{r}{s})$ in the sense of [LS]. The _Jones
polynomial_ of $C(\frac{r}{s})$ is defined via the skein relation in [LS],
which is denoted by

$V_{\frac{r}{s}}(t)\in
t^{\frac{1}{2}}\mathbb{Z}[t,t^{-1}]\cup\mathbb{Z}[t,t^{-1}].$

When we take $\mathfrak{q}=-t^{-1}$ and multiply the leading term, we get a
polynomial $J_{\frac{r}{s}}(\mathfrak{q})\in\mathbb{Z}[\mathfrak{q}]$, which
is called the _normalized Jones polynomial_. There is a corollary directly
from [BBL, Theorem A.3] and Theorem 3.18.

######  Corollary 5.3.

For every rational $\frac{r}{s}>1$, we have
$J_{\frac{r}{s}}(\mathfrak{q})=|\operatorname{Int}^{\mathfrak{q}}(\widehat{\eta}_{\frac{r}{s}},\widehat{\eta}_{0})\mid_{\mathfrak{q}=q_{1}^{-1}q_{2}}|$,
where $|\cdot|$ is the normalized one.

## References

  * [BBL] A. Bapat, L. Becker A. M. Licata, $q$-deformed rational numbers and the $2$-Calabi–Yau category of type $A_{2}$, arXiv:2202.07613.
  * [BQS] T. Bridgeland, Y. Qiu T. Sutherland, Stability conditions and A2 quivers, Adv. Math. 365 (2020) 107049. (arXiv:1406.2566)
  * [I] A. Ikeda, Stability conditions on CY-N categories associated to $A_{n}$-quivers and period maps, Math. Ann. 367 (2017) 1–49. (arXiv:1405.5492)
  * [IQ] A. Ikeda Y. Qiu, $q$-Stability conditions on Calabi-Yau-$\mathbb{X}$ categories, arXiv:1807.00469.
  * [IQZ] A. Ikeda, Y. Qiu Y. Zhou, Graded decorated marked surfaces: Calabi-Yau-X categories of gentle algebras, arXiv:2006.00009.
  * [Ke] B. Keller with an appendix by M.Van den Bergh, Deformed Calabi-Yau completions, J. reine angew. Math. 654 (2011) 125–180. (arXiv:0908.3499)
  * [KQ] A. King Y. Qiu, Exchange graphs and Ext quivers, Adv. Math. 285 (2015) 1106–1154. (arXiv:1109.2924)
  * [KS] M. Khovanov P. Seidel, Quivers, floer cohomology and braid group actions, J. Amer. Math. Soc. 15 (2002) 203–271. (arXiv:0006056)
  * [LS] K. Lee R. Schiffler, Cluster algebras and Jones polynomials, Selecta Math. (N.S.) 25 (2019). (arXiv:1710.08063)
  * [MGO1] S. Morier-Genoud V. Ovsienko, $q$-deformed rationals and $q$-continued fractions, Forum of Mathematics, Sigma 8 (2020). (arXiv:1812.00170)
  * [MGO2] S. Morier-Genoud V. Ovsienko, On $q$-deformed real numbers, Experimental Mathematics (2019) 1–9. (arXiv:1908.04365)
  * [Q] Y. Qiu, Decorated marked surfaces: Spherical twists versus braid twists, Math. Ann. 365 (2016) 595–633. (arXiv:1407.0806)
  * [QZ] Y. Qiu Y. Zhou, Decorated marked surfaces II: Intersection numbers and dimensions of Homs, Trans. Amer. Math. Soc. 372 (2019) 635–660. (arXiv:1411.4003)
  * [ST] P. Seidel R. Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001) 37–108. (arXiv:0001043)