#  $q$-deformed rational numbers and the 2-Calabi–Yau category of type $A_{2}$

Asilata Bapat  Mathematical Sciences Institute, Australian National University
<EMAIL_ADDRESS>,  Louis Becker  Australian National University
<EMAIL_ADDRESS>and  Anthony M. Licata  Mathematical Sciences
Institute, Australian National University<EMAIL_ADDRESS>
We describe a family of compactifications of the space of Bridgeland stability
conditions of any triangulated category following earlier work by Bapat,
Deopurkar, and Licata. We particularly consider the case of the 2-Calabi–Yau
category of the $A_{2}$ quiver. The compactification is the closure of an
embedding (depending on $q$) of the stability space into an infinite-
dimensional projective space.

In the $A_{2}$ case, the three-strand braid group $B_{3}$ acts on this
closure. We describe two distinguished braid group orbits in the boundary,
points of which can be identified with certain rational functions in $q$.
Points in one of the orbits are exactly the $q$-deformed rational numbers
recently introduced by Morier-Genoud and Ovsienko, while the other orbit gives
a new $q$-deformation of the rational numbers. Specialising $q$ to a positive
real number, we obtain a complete description of the boundary of the
compactification.

##  1\. Introduction

###  1.1. Deformed rational numbers

The theory of $q$-deformations of rational numbers has an extremely recent
history. Deformed rational numbers were originally introduced in [MGO20] and
developed in [MGO19]. The definition considered in these papers is via
deformations of continued fraction expansions of rational numbers. The
resulting $q$-rationals enjoy several pleasant properties, and the authors
discuss connections to a wide variety of classical topics including the Farey
triangulation, cluster algebras, and the Jones polynomial.

In this paper we present an enhancement of the notion of $q$-deformed
rationals, from which the $q$-rationals of [MGO20] can be easily read off. For
any fixed positive $q$ and a rational number $r/s$, we describe a pair of real
numbers, denoted $[r/s]^{\flat}$ and $[r/s]^{\sharp}$. The number
$[r/s]^{\sharp}$ is precisely the $q$-deformation of $r/s$ considered in
[MGO20]. We argue that $[r/s]^{\flat}$ can also reasonably be thought of as a
$q$-deformation of $r/s$.

For example, the number $[1/0]^{\sharp}$, which is the $q$-deformation of
$\infty$ in [MGO20], is just $\infty$ itself. On the other hand,
$[1/0]^{\flat}=1/(1-q)$. For other rationals $r/s$, the two $q$-deformations
can both be described via the action of a group
$\operatorname{PSL}_{2,q}(\mathbb{Z})\subset\operatorname{PSL}_{2}(\mathbb{R})$,
which acts on $\mathbb{R}\cup\\{\infty\\}$ by fractional linear
transformations. We define the groups $\operatorname{PSL}_{2,q}(\mathbb{Z})$
and give the construction of the two $q$-deformations in parallel in Section
2.

###  1.2. A compactification of the space of stability conditions

The motivation for our two constructions of $q$-deformed rational numbers
comes from homological algebra. To explain this, in Section 4, we associate to
each positive real $q$ a compactification of the space of Bridgeland stability
conditions on $\mathcal{C}_{2}$, the 2-Calabi–Yau (2-CY) triangulated category
for the $A_{2}$ quiver. This compactification is a generalisation of a
construction of [BDL20], which corresponds to the case $q=1$, and proceeds by
embedding the space of stability conditions into an infinite dimensional
projective space, and taking the closure there. We set $M_{q}$ to be the image
of the space of stability conditions under this embedding, $\overline{M_{q}}$
to be its closure, and $\partial\overline{M_{q}}$ to be
$\overline{M_{q}}\backslash M_{q}$. In the construction, the ambient
projective space is independent of $q$, but the embedding itself depends on
$q$.

When $q=1$, the spaces involved can be identified with basic objects of
hyperbolic geometry: the space of stability conditions
$\operatorname{Stab}(\mathcal{C}_{2})/\mathbb{C}$ can be identified with the
upper half plane $\mathbb{H}^{2}$ (homeomorphic to the open disk), and the
compactification is then the closed disk
$\mathbb{H}^{2}\cup\mathbb{R}\cup\\{\infty\\}$. The points of the boundary
don’t correspond to stability conditions, but they can be regarded as
translation-invariant metrics on the category $\mathcal{C}_{2}$. Moreover,
there is a canonical bijection between the spherical objects of
$\mathcal{C}_{2}$ up to shift and the rationals
$\mathbb{Q}\cup\\{\infty\\}\subset\mathbb{R}\cup\\{\infty\\}$. It is a theorem
of Rouquier–Zimmermann [RZ03] that this bijection intertwines the braid group
action on the spherical objects with the $\operatorname{PSL}_{2}(\mathbb{Z})$
action on $\mathbb{Q}\cup\\{\infty\\}$. Details about this construction may be
found in [BDL20] and references therein.

One of the main goals of this paper is to establish analogues of this result
for the case $q\neq 1$. When we vary $q$, the space $M_{q}$ remains an open
disk, but something interesting happens at the boundary: each spherical object
in the category corresponds not to a rational point, but rather to an entire
interval in $\partial\overline{M}_{q}$. We elaborate on this statement in
Section 3.4, by associating two projectivised linear functionals to each
spherical object $X$. These functionals, called
$\overline{\hom}_{q}(X,\cdotp)$ and $\operatorname{occ}_{q}(X,\cdotp)$, each
map the set of spherical objects to $\mathbb{R}$. The former sends $Y$ to the
Laurent polynomial whose $q^{k}$-coefficient is the number of degree-$k$
morphisms from $X$ to $Y$. The definition of
$\operatorname{occ}_{q}(X,\cdotp)$ is a little more involved, but is also
intrinsically well-motivated. At $q=1$, these two functionals agree and lie on
the boundary, coinciding with the rational point in
$\mathbb{R}\cup\\{\infty\\}$ corresponding to the spherical object $X$. When
$q\neq 1$, they are distinct functionals, and the entire closed interval of
convex combinations of the two functionals lies on the boundary. We call this
interval $I_{q,X}$. We summarise these statements in the following theorem.

######  Theorem 1.1.

  1. (1)

For all $q\in(0,\infty)$, the space of stability conditions maps
homeomorphically onto ${M}_{q}$, which is in turn homeomorphic to an open
disk.

  2. (2)

The boundary of $\overline{M_{q}}$ is $\partial\overline{M}_{q}$, and is
homeomorphic to $\mathbb{R}\cup\\{\infty\\}$.

  3. (3)

The union of the intervals $I_{q,X}$, as $X$ ranges over the spherical objects
of $\mathcal{C}_{2}$, is a dense subset of the boundary
$\partial\overline{M}_{q}$.

Further, we conjecture that $\overline{M}_{q}$ is homeomorphic to a closed
disk. We prove this theorem in Section 4. The main tool used in the proof is a
finite state automaton known as a Harder–Narasimhan automaton or HN automaton,
introduced in [BDL20]. This automaton controls how the HN filtration of an
object in the $A_{2}$ category transforms when acted on by a braid. (This
automaton is also of use when $q=1$, where is allows for a simpler proof of
Rouquier–Zimmermann’s original result.)

We expect the role of HN automata in the study of triangulated autoequivalence
groups will be somewhat analogous to the role of Thurston’s train track
automata in the study of mapping class groups. For 2-CY catgories, such
automata have been constructed in type $A_{2}$ and type $\widehat{A}_{1}$ in
[BDL20], and in the PhD thesis of Edmund Heng in all other rank 2 Coxeter
types. Constructing HN automata in higher rank examples and exploiting them to
study stability conditions and autoequivalence groups is an important open
problem. We give the definition of HN automata in Appendix B.

###  1.3. Relationship to the $q$-rational numbers

Recall that at $q=1$, the correspondence of spherical objects to rational
numbers intertwines the braid group action on the spherical objects with the
$\operatorname{PSL}_{2}(\mathbb{Z})$ action on $\mathbb{R}\cup\\{\infty\\}$.
The bulk of Section 3 is devoted to the $q$-analogue of this correspondence,
where the group $\operatorname{PSL}_{2}(\mathbb{Z})$ is replaced by the group
$\operatorname{PSL}_{2,q}(\mathbb{Z})$. We prove the following result (see
Theorems 3.7 and 3.8 for the precise statements).

######  Theorem 1.2.

Consider assignments

$X\mapsto(-q)^{-\epsilon}\frac{\operatorname{occ}_{q}(P_{2},X)}{\operatorname{occ}_{q}(P_{1},X)}\quad\text{and}\quad
X\mapsto
q^{-1}(-q)^{\epsilon}\frac{\overline{\hom}_{q}(X,P_{2})}{\overline{\hom}_{q}(X,P_{1})},$

where $\epsilon$ is either $0$ or $1$ depending on $X$. Each of these
assignments intertwines the $B_{3}$ action on the spherical objects of
$\mathcal{C}_{2}$ with the $\operatorname{PSL}_{2,q}(\mathbb{Z})$ action on
$\mathbb{R}\cup\\{\infty\\}$.

In fact, the first of the two assignments above recovers the $q$-rational
numbers of [MGO20, MGO19].

######  Theorem 1.3.

Let $X$ be a spherical object of $\mathcal{C}_{2}$ corresponding to the
rational number $r/s$. The ratio

$(-q)^{-\epsilon}\frac{\operatorname{occ}_{q}(P_{2},X)}{\operatorname{occ}_{q}(P_{1},X)}$

is the $q$-deformation $[r/s]_{q}^{\sharp}$ in the sense of Morier-
Genoud–Ovsienko [MGO20, MGO19].

This result suggests that the corresponding ratio of $\overline{\hom}_{q}$
functionals

$q^{-1}(-q)^{\epsilon}\frac{\overline{\hom}_{q}(X,P_{2})}{\overline{\hom}_{q}(X,P_{1})}$

should be thought of as another $q$-deformation of the corresponding rational
number; it is in fact the alternative $q$-deformation $[r/s]_{q}^{\flat}$
mentioned above. In Section 2 we give an elementary parallel account of both
of these $q$-deformations, and Theorems 3.7 and 3.8 give precise relationships
between the $q$-deformations and homological algebra.

###  1.4. The Jones polynomial of rational knots

In [MGO20, Appendix A], Morier-Genoud and Ovsienko relate the $q$-deformed
rational numbers to the Jones polynomial of rational (two-bridge) knots. In
particular, they prove that the normalised Jones polynomial is equal to
$q\mathcal{R}(q)+(1-q)\mathcal{S}(q)$, where
$[r/s]^{\sharp}_{q}=\mathcal{R}(q)/\mathcal{S}(q)$. In our Appendix A, we
prove an analogous result involving our new $q$-deformed rational number
$[r/s]^{\flat}_{q}$: the numerator of $[r/s]^{\flat}_{q}$ is equal to the
Jones polynomial of the corresponding rational knot (up to the signs of the
coefficients).

##  2\. $q$-deformed rational numbers

In this section we present two different $q$-deformations of the set
$\mathbb{Q}\cup\\{\infty\\}$, one of which is exactly the $q$-deformed
rational numbers of [MGO20]. We begin by defining the two $q$-deformations via
continued fraction expansions. Although the definitions may appear unmotivated
at first, the remainder of the section provides several viewpoints towards the
two deformations that also serve as motivation. Throughout this paper, a
rational number is considered to be any element of
$\mathbb{Q}\cup\\{\infty\\}$. Also throughout this paper, we treat the case
$0<q\leq 1$. The case $1\leq q<\infty$ is symmetric, and, up to sign, all of
our results are preserved under this symmetry.

###  2.1. Left and right $q$-deformed rational numbers

Recall that every non-zero rational number $r/s$ has a unique even continued
fraction expression $[a_{1},...,a_{2n}]$, where

$\frac{r}{s}=a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}+\cfrac{1}{\cfrac{\ddots}{a_{2n-1}+\cfrac{1}{a_{2n}}}}}},$

and either $a_{1}\in\mathbb{N}$ and
$a_{2},..,a_{2n}\in\mathbb{N}\setminus\\{0\\}$, or $-a_{1}\in\mathbb{N}$ and
$-a_{2},...,-a_{2n}\in\mathbb{N}\setminus\\{0\\}$. The continued fraction
expansion of 0 is $[-1,1]$. The continued fraction expansion of $\infty$ is
the empty expansion $[\,]$.

######  Remark 2.1.

The convention for continued fractions that we have chosen here is slightly
nonstandard. However, we have the following pleasing dichotomy under this
convention:

  1. (1)

strictly positive rational numbers only contain non-negative numbers in their
continued fraction expansions;

  2. (2)

strictly negative rational numbers only contain non-positive numbers in their
continued fraction expansions.

The points 0 and $\infty$ are both negative and positive, and thus
exceptional. The choice of which corresponds to the empty expansion is
equivalent to the choice between odd and even continued fraction expansions.
The odd continued fraction expansion of 0 is $[\,]$, and that of $\infty$ is
$[0,1,-1]$. For even continued fractions, $\infty$ is the ‘basepoint’ in
Proposition 2.2; for odd continued fractions, it is 0.

The continued fraction expansion is related to the action on
$\mathbb{Q}\cup\\{\infty\\}$ by the modular group
$\operatorname{PSL}_{2}(\mathbb{Z})$. Recall that
$\operatorname{PSL}_{2}(\mathbb{R})$ is the quotient of the group of $2\times
2$ real invertible matrices by scalar matrices. The group
$\operatorname{PSL}_{2}(\mathbb{Z})$ is the subgroup of
$\operatorname{PSL}_{2}(\mathbb{R})$ generated by

$\displaystyle\sigma_{1}:=\begin{bmatrix}1&-1\\\
0&1\end{bmatrix},\quad\sigma_{2}:=\begin{bmatrix}1&0\\\ 1&1\end{bmatrix}.$

The group $\operatorname{PSL}_{2}(\mathbb{R})$ acts on each of the sets

$\mathbb{R}\cup\\{\infty\\}\subset\mathbb{C}\cup\\{\infty\\}$

by fractional linear transformations:

(1) $\begin{bmatrix}a&b\\\ c&d\end{bmatrix}\cdot z=\frac{az+b}{cz+d}.$

The modular group $\operatorname{PSL}_{2}(\mathbb{Z})$ also preserves, and is
transitive on, $\mathbb{Q}\cup\\{\infty\\}$. The following is easy to check.

######  Proposition 2.2.

Let $r/s$ be a rational number. Suppose that $r/s$ has an even continued
fraction expansion $[a_{1},\ldots,a_{2n}]$. Then

$\frac{r}{s}=\sigma_{1}^{-a_{1}}\sigma_{2}^{a_{2}}\sigma_{1}^{-a_{3}}\cdots\sigma_{1}^{-a_{2n-1}}\sigma_{2}^{a_{2n}}(\infty),$

under the usual action of $\operatorname{PSL}_{2}(\mathbb{Z})$ on
$\mathbb{Q}\cup\\{\infty\\}$.

In order to obtain $q$-deformations of the rationals, we first define a family
of discrete subgroups of $\operatorname{PSL}_{2}(\mathbb{R})$ indexed by $q$,
which deform the modular group $\operatorname{PSL}_{2}(\mathbb{Z})$.

######  Definition 2.3.

Set $\operatorname{PSL}_{2,q}(\mathbb{Z})$ to be the subgroup of
$\operatorname{PSL}_{2}(\mathbb{R})$ generated by

$\sigma_{1,q}:=\begin{bmatrix}q^{-1}&-q^{-1}\\\
0&1\end{bmatrix},\quad\sigma_{2,q}:=\begin{bmatrix}1&0\\\
1&q^{-1}\end{bmatrix}.$

At $q=1$, the group $\operatorname{PSL}_{2,q}(\mathbb{Z})$ is simply
$\operatorname{PSL}_{2}(\mathbb{Z})$. Recall the three-strand Artin braid
group:

$B_{3}:=\langle\sigma_{1},\sigma_{2}\mid\sigma_{1}\sigma_{2}\sigma_{1}=\sigma_{2}\sigma_{1}\sigma_{2}\rangle.$

The following is easy to check.

######  Proposition 2.4.

The assignment

$\sigma_{1}\mapsto\sigma_{1,q},\quad\sigma_{2}\mapsto\sigma_{2,q}$

gives a homomorphism from $B_{3}$ to $\operatorname{PSL}_{2,q}(\mathbb{Z})$.

Our definitions of the $q$-deformed rational numbers are motivated by
Proposition 2.2, and use the homomorphism from
$B_{3}\to\operatorname{PSL}_{2,q}(\mathbb{Z})$ described above. We first set
up the following notation.

######  Definition 2.5.

Given an even continued fraction expansion

$\mathbf{a}=[a_{1},\ldots,a_{2n}]$

as above, define the braid corresponding to $\mathbf{a}$ as the following
element of $B_{3}$:

$\beta_{\mathbf{a}}=\sigma_{1}^{-a_{1}}\sigma_{2}^{a_{2}}\sigma_{1}^{-a_{3}}\cdots\sigma_{1}^{-a_{2n-1}}\sigma_{2}^{a_{2n}}.$

For any $q\in\mathbb{R}$, denote by $\beta_{\mathbf{a},q}$ the image of
$\beta_{\mathbf{a}}$ in $\operatorname{PSL}_{2,q}(\mathbb{Z})$.

Now we are in a position to define the $q$-deformed rational numbers.

######  Definition 2.6 (Left and right $q$-deformed rationals).

Let $r/s\in\mathbb{Q}\cup\\{\infty\\}$ with continued fraction expansion
$\mathbf{a}$.

  1. (1)

The _left_ $q$-deformed rational number corresponding to $r/s$ is

$\left[\frac{r}{s}\right]^{\flat}_{q}=\beta_{\mathbf{a},q}\left(\frac{1}{1-q}\right).$

  2. (2)

The _right_ $q$-deformed rational number corresponding to $r/s$ is

$\left[\frac{r}{s}\right]^{\sharp}_{q}=\beta_{\mathbf{a},q}\left(\infty\right).$

Note that at $q=1$, the left and right deformed rationals coincide, and by
Proposition 2.2, are both equal to $r/s$. By the results of [MGO20, §2.6], it
is clear that our right $q$-deformed rational numbers are the same as the
$q$-rational numbers considered in [MGO20]. In the next section we motivate
our definition of left $q$-deformed rational numbers.

###  2.2. Motivation via the $q$-Farey tessellation

In this section we recall a $q$-deformation of the classical Farey
tessellation of the hyperbolic plane, originally considered by Morier-
Genoud–Ovsienko in [MGO20]. We first recall the classical Farey tessellation.
The hyperbolic plane has an upper half-plane model:

$\mathbb{H}:=\\{z\in\mathbb{C}\mid\Im(z)>0\\},$

which embeds naturally into $\mathbb{C}\cup\\{\infty\\}$. The group
$\operatorname{PSL}_{2}(\mathbb{R})$, which acts on
$\mathbb{C}\cup\\{\infty\\}$ by fractional linear transformations, preserves
the hyperbolic plane $\mathbb{H}$, and acts on it by hyperbolic isometries.

The hyperbolic plane admits a tessellation by ideal triangles according to the
following iterative construction. We begin with an ‘initial geodesic’ $E$,
namely the straight vertical line from $0$ to $\infty$. Starting with $E$, we
complete every geodesic in the tessellation to a triangle according to the
following rule, called the _Farey rule_. By induction, every geodesic in the
tessellation has rational vertices $r/s$ and $r^{\prime}/s^{\prime}$ (the
integers $r,s,r^{\prime},s^{\prime}$ are uniquely determined by construction).
We add a geodesic from $r/s$ to $(r+r^{\prime})/(s+s^{\prime})$ and a geodesic
from $(r+r^{\prime})/(s+s^{\prime})$ to $r^{\prime}/s^{\prime}$ (see Fig. 1).

$\displaystyle\frac{r}{s}$$\displaystyle\frac{r+r^{\prime}}{s+s^{\prime}}$$\displaystyle\frac{r^{\prime}}{s^{\prime}}$
Figure 1. A standard Farey triangle.

The ‘positive half’ of the hyperbolic plane, those $z\in\mathbb{H}$ such that
$0\leq\Re(z)<\infty$, is covered by applying the rule to the initial geodesic
$E$ with its vertices viewed as the fractions $0/1$ and $1/0$. The ‘negative
half’ is obtained by applying the rule to $E$ with its vertices viewed as the
fractions $0/1$ and $-1/0$. We call the resulting tessellation the Farey
tessellation of the hyperbolic plane. The set of vertices of the Farey
tessellation is exactly $\mathbb{Q}\cup\\{\infty\\}$. Moreover, the Farey
tessellation is preserved by the $\operatorname{PSL}_{2}(\mathbb{Z})$-action
described above (that is, the $\operatorname{PSL}_{2}(\mathbb{Z})$-action
sends triangles to triangles), and the
$\operatorname{PSL}_{2}(\mathbb{Z})$-orbit of any Farey triangle is the entire
hyperbolic plane.

Morier-Genoud and Ovsienko introduced a _$q$ -deformed Farey rule_, which
generates a tessellation of a subset of $\mathbb{H}$. As with the regular
Farey rule, we begin with a geodesic $E$ between $0$ and $\infty$, but it is
now assigned a label $q^{-1}$. We then extend every geodesic in the
tessellation, with vertices $\mathcal{R}(q)/\mathcal{S}(q)$ and
$\mathcal{R}^{\prime}(q)/\mathcal{S}^{\prime}(q)$ (the vertices are now
rational functions evaluated at $q$), to a triangle according to Fig. 2 (see
[MGO20, Definition 2.9]).

$\displaystyle\frac{\mathcal{R}(q)}{\mathcal{S}(q)}$$\displaystyle\frac{\mathcal{R}(q)+q^{k}\mathcal{R}^{\prime}(q)}{\mathcal{S}(q)+q^{k}\mathcal{S}^{\prime}(q)}$$\displaystyle
q^{k-1}$$\displaystyle 1$$\displaystyle
q^{k}$$\displaystyle\frac{\mathcal{R}^{\prime}(q)}{\mathcal{S}^{\prime}(q)}$
Figure 2. A $q$-deformed Farey triangle.

For the positive half, view $E$ as being the top edge in Fig. 2 connecting the
fractions $0/1$ and $1/0$, and for the negative half, as connecting the
fractions $0/q$ and $-1/0$. The bottom left edge of any triangle is always
labelled $1$, and we thus iteratively deduce labels for all other edges. The
set of vertices of the $q$-deformed Farey tessellations is exactly the set of
_right_ $q$-deformed rational numbers $[r/s]^{\sharp}_{q}$. Fig. 3 depicts
what the $q$-deformed Farey tessellations look like at two different values of
$q$.

(a) $q=0.3$

(b) $q=0.7$

Figure 3. The deformed Farey tessellations at two values of $q$.

The action of $\operatorname{PSL}_{2,q}(\mathbb{Z})$ on $\mathbb{H}$ preserves
$q$-deformed Farey triangles (since $\operatorname{PSL}_{2,q}(\mathbb{Z})$ is
a subgroup of $\operatorname{PSL}_{2}(\mathbb{R})$, which preserves
geodesics). Moreover, $\operatorname{PSL}_{2,q}(\mathbb{Z})$ acts on the
$q$-deformed Farey tessellation exactly as
$\operatorname{PSL}_{2}(\mathbb{Z})$ acts on the classical Farey tessellation.
More formally, let $\mathbb{T}$ denote the set of triangles in the Farey
tessellation, and let $\mathbb{T}_{q}$ denote the set of $q$-deformed
triangles. Since the $\operatorname{PSL}_{2}(\mathbb{Z})$-action preserves
Farey triangles, there is an induced
$\operatorname{PSL}_{2}(\mathbb{Z})$-action on $\mathbb{T}$, and similarly
there is an induced $\operatorname{PSL}_{2,q}(\mathbb{Z})$-action on
$\mathbb{T}_{q}$. Let
$\psi\colon\operatorname{PSL}_{2,q}(\mathbb{Z})\to\operatorname{PSL}_{2}(\mathbb{Z})$
be the group homomorphism given by setting $q=1$, and let
$\phi\colon\mathbb{T}_{q}\to\mathbb{T}$ be induced from the map
$\mathcal{R}(q)/\mathcal{S}(q)\mapsto\mathcal{R}(1)/\mathcal{S}(1)$ on the
vertices. Then for any $\beta\in\operatorname{PSL}_{2,q}(\mathbb{Z})$, we have
the following commutative square, as can be proven by comparing the actions on
the vertices of the respective tessellations.

${\mathbb{T}_{q}}$${\mathbb{T}}$${\mathbb{T}_{q}}$${\mathbb{T}}$$\scriptstyle{\beta}$$\scriptstyle{\phi}$$\scriptstyle{\psi(\beta)}$$\scriptstyle{\phi}$

We can now describe how left $q$-rationals arise in the $q$-deformed Farey
tessellation. Consider any rational $r/s$. In the standard Farey tessellation,
$r/s$ is the middle vertex of a unique triangle, and the left (resp. right)
vertex of an infinite sequence of triangles. In the notation of Fig. 4, $r/s$
is the right vertex of $T_{0},T_{1},...$, and the left vertex of
$T_{0}^{{}^{\prime}},T_{1}^{{}^{\prime}},...$ .

$T_{1}$$T_{1}^{\prime}$$T_{2}$$T_{2}^{\prime}$$T_{3}$$T_{3}^{\prime}$$\displaystyle\frac{r}{s}$
Figure 4. A fraction in the standard Farey tessellation.

Assume for convenience that $0<r/s<\infty$, and let $T_{A}$ denote the
triangle with vertices $0,1,\infty$. If the continued fraction expansion of
$r/s$ is $\mathbf{a}$, then

$T_{0}=\beta_{\mathbf{a}}T_{A},\quad
T_{n}=(\beta_{\mathbf{a}}\sigma_{1}^{-1}\beta_{\mathbf{a}}^{-1})^{n}T_{0}.$

Similarly, letting
$\beta_{\mathbf{a}}^{{}^{\prime}}=\beta_{\mathbf{a}}\sigma_{2}^{-1}\sigma_{1}^{-1}$,
we have

$T_{0}^{{}^{\prime}}=\beta_{\mathbf{a}}^{{}^{\prime}}T_{A},\quad
T_{n}^{{}^{\prime}}=(\beta_{\mathbf{a}}^{{}^{\prime}}\sigma_{2}(\beta_{\mathbf{a}}^{{}^{\prime}})^{-1})^{n}T_{0}.$

As suggested by Fig. 4, for any point $x\in T_{0}$,

$\lim_{n\to\infty}(\beta_{\mathbf{a}}\sigma_{1}^{-1}\beta_{\mathbf{a}}^{-1})^{n}x=\frac{r}{s},$

and for any point $x^{\prime}\in T_{0}^{{}^{\prime}}$,

$\lim_{n\to\infty}(\beta_{\mathbf{a}}^{{}^{\prime}}\sigma_{2}(\beta_{\mathbf{a}}^{{}^{\prime}})^{-1})^{n}x^{\prime}=\frac{r}{s}.$

So we can view the rational point $r/s$ in the boundary as the limit of the
sequences of Farey triangles
$(T_{n})_{n\in\mathbb{N}},(T_{n}^{{}^{\prime}})_{n\in\mathbb{N}}$.

$T_{1}$$T_{2}$$T_{3}$$T_{1}^{\prime}$$T_{2}^{\prime}$$T_{3}^{\prime}$$\frac{r}{s}$$T_{1}$$T_{2}$$T_{3}$$T_{1}^{\prime}$$T_{2}^{\prime}$$T_{3}^{\prime}$$\frac{r}{s}$
Figure 5. The sequence of triangles $T_{n}$ converging to the $q$-deformed
rational curve of $r/s$, shown as a dotted line.

When we $q$-deform the Farey tessellation, we obtain corresponding sequences
of triangles
$(T_{q,n})_{n\in\mathbb{N}},(T_{q,n}^{{}^{\prime}})_{n\in\mathbb{N}}$. As we
will partially prove in Section 2.4, based on arguments from [MGO19], the
triangles $T_{q,n}^{{}^{\prime}}$ once again limit to a point, but the
triangles $T_{q,n}$ limit to a geodesic, with endpoints on the boundary (see
Fig. 5). We call this geodesic the _$q$ -deformed rational curve_
corresponding to $r/s$. The left $q$-rational corresponding to $r/s$ is the
left endpoint of a $q$-deformed rational curve, and the right $q$-rational
corresponding to $r/s$ is the right endpoint of the same curve. We denote by
$\mathbb{Q}_{q}$ the union of all of the $q$-deformed rational curves,
together with their endpoints. This motivates our definition of left and right
$q$-rationals. Namely, if $T_{n}\xrightarrow{n\to\infty}r/s$, then the
corresponding $q$-deformed sequence of triangles $T_{n,q}$ limits to the
geodesic with endpoints $[r/s]_{q}^{\flat}$ and $[r/s]_{q}^{\sharp}$.

###  2.3. Alternate descriptions of the deformed rational numbers

The left and right $q$-deformed rational numbers admit several elegant
intrinsic definitions. Recall the classical $q$-deformation of the integers,
which we call the _right_ $q$-deformation:

$[n]^{\sharp}_{q}:=\frac{1-q^{n}}{1-q}=1+q+\cdots+q^{n-1}.$

We introduce a corresponding _left_ $q$-deformation of $n$:

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

The following proposition gives equivalent definitions of the two
$q$-deformations of the rational numbers, and is easy to check. In fact, this
is how the (right) $q$-deformed rational numbers were originally defined (see
[MGO20, Definition 1.1]).

######  Proposition 2.7.

Let $r/s$ be a finite rational number with even continued fraction expansion
$[a_{1},...,a_{2n}]$.

  1. (1)

The corresponding right $q$-deformed rational has the following formula.

$\Big{[}\cfrac{r}{s}\Big{]}^{\sharp}_{q}=[a_{1}]^{\sharp}_{q}+\cfrac{q^{a_{1}}}{[a_{2}]^{\sharp}_{q^{-1}}+\cfrac{q^{-a_{2}}}{[a_{3}]^{\sharp}_{q}+\cfrac{q^{a_{3}}}{[a_{4}]^{\sharp}_{q^{-1}}+\cfrac{q^{-a_{4}}}{\cfrac{\ddots}{[a_{2n-1}]^{\sharp}_{q}+\cfrac{q^{a_{2n-1}}}{[a_{2n}]^{\sharp}_{q^{-1}}}}}}}}$

  2. (2)

The corresponding left $q$-deformed rational has the following formula.

$\Big{[}\cfrac{r}{s}\Big{]}_{q}^{\flat}=[a_{1}]^{\sharp}_{q}+\cfrac{q^{a_{1}}}{[a_{2}]^{\sharp}_{q^{-1}}+\cfrac{q^{-a_{2}}}{\cfrac{\ddots}{[a_{2n-1}]^{\sharp}_{q}+\cfrac{q^{a_{2n-1}}}{[a_{2n}]_{q^{-1}}^{\flat}}}}}$

######  Definition 2.8.

Following [MGO20], we fix the following notation for the rational functions
associated to the left and right $q$-deformed rational numbers:

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

where the polynomials in the numerator and denominator are normalised so that
if $r/s>0$ (resp. $r/s<0$), the denominator has lowest (resp. highest) degree
term equal to 1. If $r/s=0$, we set

$\mathcal{R}^{\flat}(q)=1-q^{-1},\quad\mathcal{S}^{\flat}(q)=1,\quad\mathcal{R}^{\sharp}(q)=0,\quad\mathcal{S}^{\sharp}(q)=1.$

If $r/s=\infty$, we set

$\mathcal{R}^{\flat}(q)=1,\quad\mathcal{S}^{\flat}(q)=1-q,\quad\mathcal{R}^{\sharp}(q)=1,\quad\mathcal{S}^{\sharp}(q)=0.$

######  Remark 2.9.

Note that $\mathcal{R}^{\flat}(q)$, $\mathcal{S}^{\flat}(q)$,
$\mathcal{R}^{\sharp}(q)$, and $\mathcal{S}^{\sharp}(q)$ depend on _both_ $r$
and $s$.

The continued fraction expansion of every $q$-deformed rational number induces
an element of $\operatorname{PSL}_{2,q}(\mathbb{Z})$ analogous to that in
Proposition 2.2. Right-multiplying this element by $\infty$ (for right
$q$-rationals) or $1/(1-q)$ (for left $q$-rationals) returns the original
$q$-rational. The following proposition expresses this element of
$\operatorname{PSL}_{2,q}(\mathbb{Z})$ in terms of the numerators and
denominators of the corresponding $q$-rationals. The second part of this
proposition appears as [MGO20, Proposition 4.3].

######  Proposition 2.10 (Matrix formula for $q$-rationals).

Consider any rational number $r/s$ with continued fraction expansion
$\mathbf{a}=[a_{1},...,a_{2n}]$. Let $r^{\prime}/s^{\prime}$ be the rational
number with continued fraction expansion $[a_{1},...,a_{2n-1}]$.

  1. (1)

We have the following for left $q$-rationals.

$q^{a_{2}+a_{4}+...+a_{2n-2}}\beta_{\mathbf{a},q}\begin{bmatrix}1&1-q^{-1}\\\
1-q&1\end{bmatrix}=\begin{cases}\begin{bmatrix}q^{-a_{2n}}\mathcal{R}^{\flat}(q)&\mathcal{R}^{\flat\prime}(q)\\\
q^{-a_{2n}}\mathcal{S}^{\flat}(q)&\mathcal{S}^{\flat\prime}(q)\end{bmatrix},&\displaystyle
0\leq\frac{r}{s}<\infty,\\\\[12.91663pt]
\begin{bmatrix}q^{-a_{2n}+1}\mathcal{R}^{\flat}(q)&\mathcal{R}^{\flat\prime}(q)\\\
q^{-a_{2n}+1}\mathcal{S}^{\flat}(q)&\mathcal{S}^{\flat\prime}(q)\end{bmatrix},&\displaystyle\frac{r}{s}<0\text{
or }\frac{r}{s}=\infty.\end{cases}$

  2. (2)

We have the following for right $q$-rationals.

$q^{a_{2}+a_{4}+...+a_{2n}}\beta_{\mathbf{a},q}=\begin{cases}\begin{bmatrix}q\mathcal{R}^{\sharp}(q)&\mathcal{R}^{\sharp\prime}(q)\\\
q\mathcal{S}^{\sharp}(q)&\mathcal{S}^{\sharp\prime}(q)\end{bmatrix},&\displaystyle
0\leq\frac{r}{s}<\infty,\\\\[12.91663pt]
\begin{bmatrix}\mathcal{R}^{\sharp}(q)&\mathcal{R}^{\sharp\prime}(q)\\\
\mathcal{S}^{\sharp}(q)&\mathcal{S}^{\sharp\prime}(q)\end{bmatrix},&\displaystyle\frac{r}{s}<0\text{
or }\frac{r}{s}=\infty.\end{cases}$

###  2.4. Limits of $q$-deformed rational numbers

The following theorem partially formalises the description of $q$-deformed
rational numbers depicted in Fig. 4 and Fig. 5. The vertices of the $q$-Farey
tessellation are the right $q$-deformed rationals, while the left $q$-deformed
rational numbers arise as limits of sequences of right $q$-deformed rationals.
This theorem is worth comparing to Morier-Genoud and Ovsienko’s study of
limiting behaviour of $q$-deformed rational numbers, in terms of Taylor
series, in [MGO19], especially Theorem 1 and Remark 3.2. The proof owes its
technique to the proofs in that paper.

######  Theorem 2.11 (Limits of $q$-rationals).

Suppose $q<1$. Consider any convergent sequence of rational numbers
$(r_{m}/s_{m})_{m\in\mathbb{N}}\in\mathbb{Q}^{\mathbb{N}}$.

  1. (1)

If $(r_{m}/s_{m})_{m\in\mathbb{N}}$ converges to an irrational number
$t\in\mathbb{R}\setminus\mathbb{Q}$, then the sequence of right $q$-deformed
rational numbers
$([r_{m}/s_{m}]_{q}^{\sharp})_{m\in\mathbb{N}}\in\mathbb{R}^{\mathbb{N}}$
converges to a limit, uniquely determined by $t$.

  2. (2)

If $(r_{m}/s_{m})_{m\in\mathbb{N}}$ converges to a rational number $r/s$ from
the right, then the sequence of right $q$-deformed rational numbers
$([r_{m}/s_{m}]_{q}^{\sharp})_{m\in\mathbb{N}}\in\mathbb{R}^{\mathbb{N}}$
converges to $[r/s]_{q}^{\sharp}$.

  3. (3)

If $(r_{m}/s_{m})_{m\in\mathbb{N}}$ converges to a rational number $r/s$ from
the left, then the sequence of right $q$-deformed rational numbers
$([r_{m}/s_{m}]_{q}^{\sharp})_{m\in\mathbb{N}}\in\mathbb{R}^{\mathbb{N}}$
converges to $[r/s]_{q}^{\flat}$.

By Theorem 2.11 Part (1), for every irrational number $t$, there is a unique
$q$-deformation $[t]_{q}$, which is the limit of the $q$-deformation of any
rational sequence limiting to $t$. We call the set

$\mathbb{I}_{q}:=\\{[t]_{q}\mid t\in\mathbb{R}\setminus\mathbb{Q}\\}$

the _$q$ -deformed irrational numbers_.

###### Proof.

First, we prove (1).

As noted in [LMGOV21, Section 2], the negative $q$-rationals are the image of
the positive $q$-rationals under the continuous bijection

$x\mapsto-\frac{q}{x},$

with $\infty$ sent to $0$ and vice versa. Thus it is sufficient to give a
proof for positive $t$.

Let $(r_{n}/s_{n})_{n\in\mathbb{N}}$ converge to an irrational number
$t\in\mathbb{R}\setminus\mathbb{Q}$. It is well-known that every irrational
number has a unique infinite continued fraction expansion $[a_{1},a_{2},...]$.
We can approximate the sequence $(r_{n}/s_{n})_{n\in\mathbb{N}}$ by the
‘sequence of convergents’ $([a_{1},a_{2},...,a_{n}])_{n\in\mathbb{N}}$. Since
$[r/s]_{q}^{\sharp}>[r^{\prime}/s^{\prime}]_{q}^{\sharp}$ if and only if
$r/s>r^{\prime}/s^{\prime}$, the right $q$-deformed sequence
$([r_{n}/s_{n}]_{q}^{\sharp})_{n\in\mathbb{N}}$ is also approximated by
$([a_{1},a_{2},...,a_{n}]_{q}^{\sharp})_{n\in\mathbb{N}}$. Thus, we can assume
without loss of generality that $r_{n}/s_{n}=[a_{1},a_{2},...,a_{n}]$ for all
$n\in\mathbb{N}$.

We split $(r_{n}/s_{n})_{n\in\mathbb{N}}$ into two disjoint subsequences.
Define the index set $I\subset\mathbb{N}$ by

$I:=\Big{\\{}n\in\mathbb{N}\mid\frac{r_{n}}{s_{n}}<t\Big{\\}}.$

Both $\lim_{n\in I}r_{n}/s_{n}$ and $\lim_{n\notin I}r_{n}/s_{n}$ exist, as
each sequence is monotone and bounded. Denote the former by $[t]_{q}^{-}$ and
the latter by $[t]_{q}^{+}$.

We would like to show $[t]_{q}^{-}=[t]_{q}^{+}$. It is sufficient to show that

$\lim_{n\to\infty}\Big{|}\Big{[}\frac{r_{n}}{s_{n}}\Big{]}_{q}^{\sharp}-\Big{[}\frac{r_{n-1}}{s_{n-1}}\Big{]}_{q}^{\sharp}\Big{|}=0.$

Let
$[r_{n}/s_{n}]_{q}^{\sharp}=\mathcal{R}^{\sharp}_{n}(q)/\mathcal{S}^{\sharp}_{n}(q)$,
and let
$[r_{n-1}/s_{n-1}]_{q}^{\sharp}=\mathcal{R}^{\sharp}_{n-1}(q)/\mathcal{S}^{\sharp}_{n-1}(q)$,
using the notation of 2.8. By the matrix formula for right $q$-deformed
rationals (Proposition 2.10), we have

$\Big{|}\frac{\mathcal{R}^{\sharp}_{n}(q)}{\mathcal{S}^{\sharp}_{n}(q)}-\frac{\mathcal{R}^{\sharp}_{n-1}(q)}{\mathcal{S}^{\sharp}_{n-1}(q)}\Big{|}=q^{a_{2}+a_{4}+...+a_{n}-1}\frac{|\det\beta_{\mathbf{a},q}|}{\mathcal{S}^{\sharp}_{n}(q)\mathcal{S}^{\sharp}_{n-1}(q)}.$

By construction, $\mathcal{S}^{\sharp}_{n}(q)$ and
$\mathcal{S}^{\sharp}_{n-1}(q)$ both have constant term $1$ and positive
coefficients, so
$\mathcal{S}^{\sharp}_{n}(q),\mathcal{S}^{\sharp}_{n-1}(q)\geq 1$. And one can
compute that

$\det\beta_{\mathbf{a},q}=q^{a_{1}-a_{2}+...+a_{n-1}-a_{n}},$

therefore,

$\Big{|}\frac{\mathcal{R}^{\sharp}_{n}(q)}{\mathcal{S}^{\sharp}_{n}(q)}-\frac{\mathcal{R}^{\sharp}_{n-1}(q)}{\mathcal{S}^{\sharp}_{n-1}(q)}\Big{|}\leq
q^{a_{1}+a_{3}+...+a_{n-1}-1}.$

(Technically this assumes $n$ is even, however an all but identical argument
works for $n$ odd.) Given the assumption that $t$ (and therefore also
$\frac{r_{n}}{s_{n}}$ for sufficiently large $n$) are positive, we have
$a_{1},...,a_{n-1}\geq 0$, so since $q<1$ we deduce that

$\lim_{n\to\infty}\Big{|}\Big{[}\frac{r_{n}}{s_{n}}\Big{]}_{q}^{\sharp}-\Big{[}\frac{r_{n-1}}{s_{n-1}}\Big{]}_{q}^{\sharp}\Big{|}=0.$

To prove that the limit is uniquely determined by $t$, we note again that
$[r/s]_{q}^{\sharp}>[r^{\prime}/s^{\prime}]_{q}^{\sharp}$ if and only if
$r/s>r^{\prime}/s^{\prime}$. Thus, between any two ‘$q$-irrationals’
$[t]_{q},[t^{\prime}]_{q}$ for $t\neq t^{\prime}$, there is a right
$q$-rational $[r/s]_{q}^{\sharp}$, so we cannot have
$[t]_{q}=[t^{\prime}]_{q}$.

Now we prove (2). As in the irrational case, we can assume that $r/s$ is
positive. Let $[a_{1},...,a_{2n}]$ be the continued fraction expansion of
$r/s$. Any right limiting sequence approaching $r/s$ can be approximated by
the sequence $(r_{m}/s_{m})_{m\in\mathbb{N}}$, where $r_{m}/s_{m}$ has
continued fraction expansion $\mathbf{a}_{m}=[a_{1},...,a_{2n}-1,1,m]$ (unless
$a_{2n}=1$, in which case $r_{m}/s_{m}$ has expansion
$\mathbf{a}_{m}=[a_{1},...,a_{2n-1}+1,m]$). Since
$[r/s]_{q}^{\sharp}>[r^{\prime}/s^{\prime}]_{q}^{\sharp}$ if and only if
$r/s>r^{\prime}/s^{\prime}$, the corresponding right $q$-deformed sequence is
also approximated by $([r_{m}/s_{m}]_{q}^{\sharp})_{m\in\mathbb{N}}$.

Let $[{r}/{s}]_{q}^{\sharp}=\mathcal{R}^{\sharp}(q)/\mathcal{S}^{\sharp}(q)$,
and
$[r_{m}/s_{m}]_{q}^{\sharp}=\mathcal{R}_{m}^{\sharp}(q)/\mathcal{S}_{m}^{\sharp}(q)$,
as defined in 2.8 We view the right $q$-deformed rational numbers as vectors
in projective space. Applying the matrix formula for right $q$-rationals
(Proposition 2.10), we have

$\displaystyle\lim_{m\to\infty}\begin{bmatrix}\frac{q^{1-m}}{[m]_{q^{-1}}}\mathcal{R}_{m}^{\sharp}(q)\\\
\frac{q^{1-m}}{[m]_{q^{-1}}}\mathcal{S}_{m}^{\sharp}(q)\end{bmatrix}$
$\displaystyle=\lim_{m\to\infty}\frac{1}{[m]_{q^{-1}}}q^{a_{2}+a_{4}+...+a_{2n}}\beta_{\mathbf{a}_{m},q}\begin{bmatrix}1\\\
0\end{bmatrix}$
$\displaystyle=q^{a_{2}+a_{4}+...+a_{2n}}\sigma_{1,q}^{-a_{1}}\sigma_{2,q}^{a_{2}}...\sigma_{1,q}^{-a_{2n-1}}\sigma_{2,q}^{a_{2n}-1}\sigma_{1,q}^{-1}\begin{bmatrix}0&0\\\
1&q^{-1}-1\end{bmatrix}\begin{bmatrix}1\\\ 0\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\mathcal{R}^{\sharp}(q)\\\
\mathcal{S}^{\sharp}(q)\end{bmatrix}.$

The final claim (3) can be proved in essentially the same way as (2),
replacing the sequence $[a_{1},...,a_{2n}-1,1,m]$ with the sequence
$[a_{1},...,a_{2n},m]$. ∎

######  Remark 2.12.

In the preceding proof, we found that right $q$-deformed rationals correspond
to limits of sequences of the form
$([a_{1},...,a_{2n+1},m]_{q}^{\sharp})_{m\in\mathbb{Z}}$, and left
$q$-deformed rationals correspond to limits of sequences of the form
$([a_{1},...,a_{2n},m]_{q}^{\sharp})_{m\in\mathbb{Z}}$. Note that, for
ordinary continued fractions,

$\displaystyle\lim_{m\to\infty}[a_{1},...,a_{2n+1},m]=[a_{1},...,a_{2n+1}],$
$\displaystyle\lim_{m\to\infty}[a_{1},...,a_{2n},m]=[a_{1},...,a_{2n}].$

Thus, in a sense, right $q$-rationals correspond to odd continued fraction
expansions, and left $q$-rationals to even continued fraction expansions.

###  2.5. Topology of the $q$-Farey tessellation

We briefly summarise here the most important topological feature of the
$q$-deformed Farey tessellation: it is an open disk, and the boundary of its
closure is a circle, consisting of exactly the $q$-deformed rational curves,
and the $q$-deformed irrational numbers. Recall that $\mathbb{Q}_{q}$ denotes
the union of the $q$-deformed rational curves (including their endpoints, the
left and right $q$-deformed rational numbers), and that $\mathbb{I}_{q}$
denotes the $q$-deformed irrational numbers.

######  Proposition 2.13.

Let $L_{q}$ denote the proper subset of $\mathbb{H}$ covered by the
$q$-deformed Farey tessellation.

  1. (1)

The set $L_{q}$ is homeomorphic to an open disk.

  2. (2)

The boundary $\partial\overline{L}_{q}$ is equal to
$\mathbb{Q}_{q}\sqcup\mathbb{I}_{q}$.

  3. (3)

There is a homeomorphism
$\mathbb{Q}_{q}\sqcup\mathbb{I}_{q}\to\mathbb{R}\cup\\{\infty\\}$, which
restricts to the identity on the left and right $q$-deformed rationals, and
the $q$-deformed irrationals. So, by Part (2),
$\partial\overline{L}_{q}\cong\mathbb{R}\cup\\{\infty\\}$.

  4. (4)

The set $\mathbb{Q}_{q}$ is dense in $\partial\overline{L}_{q}$.

We expect the following conjecture to hold as well.

######  Conjecture 2.14.

The closure of the $q$-deformed Farey tessellation $\overline{L}_{q}$ is
homeomorphic to a closed disk.

Because the $q$-deformed rational curves are dense in the boundary of
$\overline{L}_{q}$, these curves are arguably the true $q$-deformed analogues
of the rational numbers, hence the notation $\mathbb{Q}_{q}$ for their union.

###### Proof of Proposition 2.13.

Part (1) of Proposition 2.13 is true by construction; the $q$-deformed Farey
tessellation is homeomorphic to the classical Farey tessellation, which is
homeomorphic to an open disk.

The homological counterpart of Part (2) is Theorem 4.13 below, and the proof
of Part (2) is virtually the same as the proof of Theorem 4.13.

We will prove Part (3). Part (4) follows immediately from Part (3) and Theorem
2.11.

It is not hard to check that for any two real numbers $t<t^{\prime}$ (which
may be rational or irrational), we have the following inequalities:

(2)
$[t]_{q}^{\flat}\leq[t]_{q}^{\sharp}<[t^{\prime}]_{q}^{\flat}\leq[t^{\prime}]_{q}^{\sharp}.$

The left (resp. right) weak inequality is strict if and only if $t$ (resp.
$t^{\prime}$) is rational. Define an embedding
$\iota:\mathbb{Q}_{q}\sqcup\mathbb{I}_{q}\to\mathbb{R}\cup\\{\infty\\}$ as
follows. For $t\in\mathbb{R}\cup\\{\infty\\}$,

$\iota([t]_{q}^{\flat})=[t]_{q}^{\flat},\quad\iota([t]_{q}^{\sharp})=[t]_{q}^{\sharp}.$

Extend $\iota$ to $\mathbb{Q}_{q}$ by continuously mapping the hyperbolic
geodesic between $[r/s]_{q}^{\flat}$ and $[r/s]_{q}^{\sharp}$ onto the real
interval between $[r/s]_{q}^{\flat}$ and $[r/s]_{q}^{\sharp}$. By Eq. 2,
$\iota$ is indeed an embedding.

To complete the proof, we just need to show that $\iota$ is surjective.
Consider any $p\in\mathbb{R}\cup\\{\infty\\}$. For convenience, assume that
$0<p<\infty$. Suppose that $p\notin\iota(\mathbb{Q}_{q})$; we will show that
$p\in\mathbb{I}_{q}$. By assumption, $p$ does not lie in the interval in
$\mathbb{R}\cup\\{\infty\\}$ between $[\infty]_{q}^{\flat}$ and
$[\infty]_{q}^{\sharp}$. By the proof of Theorem 2.11, the sequence
$[n]^{\sharp}_{q}$ for $n\geq 0$ is a monotone increasing sequence converging
to $[\infty]_{q}^{\flat}$. So, there exists $a_{1}\geq 0$ such that

$[a_{1}]_{q}^{\sharp}<p<[a_{1}+1]_{q}^{\sharp}=[a_{1},1]^{\sharp}.$

The sequence $[a_{1},n]_{q}^{\sharp}$ for $n\geq 1$ is a monotone decreasing
sequence converging to $[a_{1}]_{q}^{\sharp}$, by the proof of Theorem 2.11.
Since $p>[a_{1}]_{q}^{\sharp}$, there exists $a_{2}\geq 1$ such that

$[a_{1},a_{2}+1]_{q}^{\sharp}<p<[a_{1},a_{2}]_{q}^{\sharp}.$

Continuing in this manner, we obtain an infinite $q$-deformed continued
fraction expansion $[a_{1},a_{2},...]_{q}^{\sharp}$ converging to $p$. By the
first part of Theorem 2.11, the number $p$ is the $q$-deformation of the
irrational number with continued fraction expansion $[a_{1},a_{2},...]$, which
proves that $p\in\mathbb{I}_{q}$ as desired. This completes the proof. ∎

##  3\. A homological interpretation of $q$-deformed rational numbers

The aim of this section is to give a concrete homological interpretation of
$q$-deformed rational numbers. Our main tool will be an abstract gadget called
a Harder–Narasimhan automaton, whose construction depends on the choice of a
Bridgeland stability condition. We write down a particular Harder–Narasimhan
automaton for the 2-Calabi–Yau category associated to the $A_{2}$ quiver,
which yields the desired application. We recall details about Bridgeland
stability conditions and Harder–Narasimhan automata in Appendix B.

###  3.1. The 2-Calabi–Yau category associated to the $A_{2}$ quiver

We follow the construction of the 2-Calabi–Yau category associated to the
$A_{2}$ quiver given in [ST01, §4a] and [BDL20, §2.1]. Let $P(A_{2})$ denote
the path algebra of the double of the $A_{2}$ quiver. This quiver has two
vertices, denoted $1$ and $2$, and two oriented edges, one from $1$ to $2$ and
the other from $2$ to $1$. We denote by $Z_{2}$ the quotient of this path
algebra by the two-sided ideal generated by all paths of length 3. We regard
$Z_{2}$ as a differential graded algebra with grading by path length, and zero
differential. Let ${\>\mathsf{DGM}}_{2}$ be the category of differential
graded modules over $Z_{2}$, and let $\mathcal{D}_{2}$ be the derived category
of differential graded modules over $Z_{2}$ (obtained from
${\>\mathsf{DGM}}_{2}$ by inverting quasi-isomorphisms; note that this is not
the same as the standard derived category of the abelian category
${\>\mathsf{DGM}}_{2}$). For $i=1,2$, let $P_{i}$ denote the differential
graded module $Z_{2}(i)$, where $(i)$ is the trivial path at the vertex $i$.
The objects $\\{P_{1},P_{2}\\}$ form an $A_{2}$ configuration of spherical
objects in $\mathcal{D}_{2}$, in the sense of [ST01, Definition 1.1]. Let
$\mathcal{C}_{2}$ denote the full triangulated subcategory of
$\mathcal{D}_{2}$ generated by $P_{1},P_{2}$ under extensions. Then
$\mathcal{C}_{2}$ is a 2-Calabi–Yau category.

Especially important for our purposes are the following structural features of
$\mathcal{C}_{2}$.

  1. (1)

There is a unique morphism (up to scaling) from $P_{1}$ to $P_{2}$, and also
from $P_{2}$ to $P_{1}$. We denote the cones of these morphisms by $P_{12}$
and $P_{21}$ respectively. They are indecomposable spherical objects of
$\mathcal{C}_{2}$.

  2. (2)

The extension closure of $P_{1},P_{2}$ is the heart of a bounded $t$-structure
on $\mathcal{C}_{2}$. We call this $t$-structure the standard $t$-structure on
$\mathcal{C}_{2}$, and denote its heart by $\heartsuit_{\operatorname{std}}$.
Objects in the standard heart are direct sums of the indecomposables $P_{1}$,
$P_{2}$, $P_{12}$ and $P_{21}$.

  3. (3)

Every spherical object $X$ in $\mathcal{C}_{2}$ gives rise to an
autoequivalence $\sigma_{X}$ on $\mathcal{C}_{2}$ known as the spherical twist
along $X$. These autoequivalences form a group, generated by
$\sigma_{P_{1}},\sigma_{P_{2}}$ (henceforth simply $\sigma_{1},\sigma_{2}$).
The group of spherical twists is isomorphic to the 3-strand braid group
$B_{3}=B_{A_{2}}$, inducing a (weak) braid group action on $\mathcal{C}_{2}$
(see [KS00] and [ST01] for details). Let $\mathbb{S}$ be the set of isomophism
classes of spherical objects of $\mathcal{C}_{2}$. It is known that $B_{3}$
acts transitively on this set; see e.g. [BDL21, Theorem 1.1].

###  3.2. Standard stability conditions on $\mathcal{C}_{2}$

The proofs in the following section use a choice of a stability condition on
$\mathcal{C}_{2}$. Any choice will work, but for convenience we choose a
degenerate standard stability condition, defined as follows.

######  Definition 3.1.

A stability condition $\tau=(\mathcal{P},Z)$ is _standard_ if
$\mathcal{P}([\phi,\phi+1))=\heartsuit_{\operatorname{std}}$ for some
$\phi\in\mathbb{R}$. A standard stability condition $\tau$ is _degenerate_ the
phase of $P_{1}$ is equal to the phase of $P_{2}$.

Since $P_{1}$ and $P_{2}$ are simple in $\heartsuit_{\operatorname{std}}$,
they are automatically stable in any standard stability condition. In a
degenerate standard stability condition, the objects $P_{12}$ and $P_{21}$ are
also semistable of the same phase.

We will use the following proposition. It is a serious result, and
corresponding versions of this proposition are true for any 2-Calabi–Yau
category of a finite ADE type quiver. Several proofs of versions of this
proposition are known, in various levels of generality: see, e.g., [IUU10,
Theorem 1], [Ike14, Proposition 4.13], [AMY19, Corollary 9.5], [BDL21, Theorem
1.2].

######  Proposition 3.2.

  1. (1)

The space $\operatorname{Stab}(\mathcal{C}_{2})$ is connected.

  2. (2)

For any stability condition $\tau\in\operatorname{Stab}(\mathcal{C}_{2})$,
there exists a braid $\beta\in B_{3}$ such that $\beta\tau$ is a standard
stability condition.

From now on, we fix a degenerate standard stability condition $\tau$. The
spherical objects in $\mathcal{C}_{2}$ admit a simple classification, based on
their $\tau$-HN filtration factors. This classification will be a key tool in
later proofs. Given a set $S\subset\operatorname{Ob}\mathcal{C}_{2}$, let
$[s\mid s\in S]$ denote the set of all objects of $\mathcal{C}_{2}$ whose
$\tau$-HN filtration factors consist of shifted copies of elements of $S$. For
example, $[P_{1},P_{12}]$ consists of all objects whose $\tau$-HN filtration
factors are shifts of either $P_{1}$ or $P_{12}$.

The following proposition is similar to [BDL20, Proposition 5.8].

######  Proposition 3.3.

For each spherical object $X\in\mathbb{S}$, one of the following is true:

  1. (1)

$X\in[P_{2},P_{21}],$

  2. (2)

$X\in[P_{21},P_{1}],$

  3. (3)

$X\in[P_{1},P_{12}],$

  4. (4)

$X\in[P_{12},P_{2}].$

###  3.3. An automaton for the category $\mathcal{C}_{2}$

In this section we write down an explicit HN automaton for $\mathcal{C}_{2}$
(see Definition B.4). In the next section we discuss its implications on the
dynamics of the braid group action on spherical objects of $\mathcal{C}_{2}$.
Once again, fix $\tau$ to be a degenerate standard stability condition. To
construct a $\tau$-HN automaton for $\mathcal{C}_{2}$, we specify the
following pieces of data.

  1. (1)

Fig. 6 simultaneously depicts a $B_{3}$-labelled graph $\Theta$ and a
$\Theta$-set $\underline{S}\subset\underline{\operatorname{Ob}C_{2}}$.

${{[P_{1},P_{12}]}}$${{[P_{21},P_{1}]}}$${{[P_{12},P_{2}]}}$${{[P_{2},P_{21}]}}$${\sigma_{2}}$${\sigma_{2}^{-1}}$${\sigma_{1}}$${\sigma_{2}^{-1}}$${\sigma_{2}}$${\sigma_{1}^{-1}}$${\sigma_{1}}$${\sigma_{1}^{-1}}$${\sigma_{2}^{-1}}$${\sigma_{1}^{-1}}$${\sigma_{1}}$${\sigma_{2}}$
Figure 6. A $B_{3}$-labelled graph $\Theta$, together with an assignment of a
subset of $\operatorname{Ob}\mathcal{C}_{2}$ to each vertex.

  2. (2)

Fig. 7 depicts a $\Theta$-representation $\underline{M}$.

${{\mathbb{Z}[q^{\pm}]^{2}}}$${{\mathbb{Z}[q^{\pm}]^{2}}}$${{\mathbb{Z}[q^{\pm}]^{2}}}$${{\mathbb{Z}[q^{\pm}]^{2}}}$${\operatorname{Id}}$${\begin{pmatrix}q^{-1}&0\\\
1&1\end{pmatrix}}$${\begin{pmatrix}q^{-1}&q^{-1}\\\
0&1\end{pmatrix}}$${\operatorname{Id}}$${\begin{pmatrix}1&1\\\
0&q\end{pmatrix}}$${\begin{pmatrix}1&0\\\
q&q\end{pmatrix}}$${\begin{pmatrix}1&1\\\
0&q\end{pmatrix}}$${\operatorname{Id}}$${\begin{pmatrix}1&0\\\
q&q\end{pmatrix}}$${\begin{pmatrix}q^{-1}&0\\\
1&1\end{pmatrix}}$${\operatorname{Id}}$${\begin{pmatrix}q^{-1}&q^{-1}\\\
0&1\end{pmatrix}}$ Figure 7. A $\Theta$-representation $\underline{M}$.

As shown in Fig. 6, the set at every vertex has the form $[A,B]$, where
$A,B\in\Sigma_{\tau}$. The summands in the module
$\mathbb{Z}[q^{\pm}]^{2}=\mathbb{Z}[q^{\pm}]\oplus\mathbb{Z}[q^{\pm}]$ at that
vertex should be thought of as being indexed by $A$ and $B$ respectively.

  3. (3)

The morphism $i\colon\underline{S}\to\underline{M}$ is as follows. At each
vertex, the map $[A,B]\to\mathbb{Z}[q^{\pm}]^{2}$ takes an object $X\in[A,B]$
to the $A$ and $B$ coordinates of $\operatorname{HN}_{\tau}(X)$. The proof
that $i$ is $\Theta$-equivariant is a calculation that proceeds exactly as in
the proof of [BDL20, Proposition 5.1].

  4. (4)

Finally, $h_{v}\colon M_{v}\to\mathbb{Z}[q^{\pm}]^{\Sigma_{\tau}}$ is simply
the inclusion map. We need to check that for any vertex $v$ and any $X\in
S_{v}$, we have

$\operatorname{HN}_{\tau}(X)=h_{v}(i_{v}(X)).$

In other words, that the $\tau$-HN multiplicity vectors transform according to
matrices in Fig. 7. This check, which is involved, is essentially identical to
the one done in [BDL20, Proposition 5.1]. We omit the details here.

###  3.4. The dynamics of the braid group action on spherical objects

We now define two kinds of functionals
$\mathbb{S}\times\mathbb{S}\to\mathbb{Z}[q^{\pm}]$. The first of these is an
extension of a functional
$\\{P_{1},P_{2}\\}\times\mathbb{S}\to\mathbb{Z}[q^{\pm}]$ that counts the
‘occurences’ of the building block complexes $P_{2}$ and $P_{1}$,
respectively, in a complex $X\in\mathbb{S}$. We continue to denote by $\tau$ a
degenerate standard stability condition on $\mathcal{C}_{2}$.

We know that in this case, the set of indecomposable semistable objects in the
heart is just

$\Sigma_{\tau}=\\{P_{1},P_{2},P_{12},P_{21}\\}.$

With these coordinates, for any $X\in\mathcal{T}$, we write
$\operatorname{HN}_{\tau}(X)$ as

$\operatorname{HN}_{\tau}(X)=(\pi_{1}(X),\pi_{2}(X),\pi_{12}(X),\pi_{21}(X)).$

######  Definition 3.4.

We define functionals $\mathbb{S}\times\mathbb{S}\to\mathbb{Z}[q^{\pm}]$ as
follows. For any $X\in\mathbb{S}$,

$\displaystyle\operatorname{occ}_{q}(P_{1},X)$
$\displaystyle:=\pi_{2}(X)+\pi_{12}(X)+\pi_{21}(X),$
$\displaystyle\operatorname{occ}_{q}(P_{2},X)$
$\displaystyle:=\pi_{1}(X)+\pi_{12}(X)+\pi_{21}(X).$

For any $X,Y\in\mathbb{S}$, let $X=\beta P_{1}$, then

$\operatorname{occ}_{q}(X,Y)=\operatorname{occ}_{q}(P_{1},\beta^{-1}Y).$

######  Remark 3.5.

Note that $\operatorname{occ}_{q}(P_{1},X)$ counts the ‘occurrences’ of
$P_{2}$ in $X$ and vice versa, which is notationally confusing but somewhat
unavoidable. Every occurence of $P_{2}$ in $X$ must correspond to an HN
filtration factor of the form $P_{2}$, $P_{12}$ or $P_{21}$, up to shift.
Thus, summing the number of these factors, with degree, gives the number of
$P_{2}$’s, with degree.

######  Definition 3.6.

The functional $\operatorname{occ}_{q}$ has a close cousin,
$\overline{\hom}_{q}$:

$\overline{\hom}_{q}(X,Y):=\begin{cases}q^{n}(q^{-2}-q^{-1}),&Y\cong X[n],\\\
\sum_{n\in\mathbb{Z}}\dim_{\Bbbk}\operatorname{Hom}(X,Y[n])q^{-n},&\text{otherwise.}\end{cases}$

The following theorems relate these functionals to the $q$-deformed rational
numbers. For reasons that will soon be obvious, for any $X\in\mathbb{S}$ we
write $X\geq 0$ to mean $X\in[P_{2},P_{21}]\cup[P_{21},P_{1}],$ and $X\leq 0$
to mean $X\in[P_{1},P_{12}]\cup[P_{12},P_{2}].$

######  Theorem 3.7.

For every $X\in\mathbb{S}$,

$(-q)^{-\epsilon}\frac{\operatorname{occ}_{q}(P_{2},X)}{\operatorname{occ}_{q}(P_{1},X)},$

is a right $q$-deformed rational number, where

$\epsilon=\begin{cases}0,&X\geq 0,\\\ 1,&X\leq 0.\end{cases}$

Moreover, the induced map from $\mathbb{S}$ to the set of $q$-deformed
rationals is $B_{3}$-equivariant, and bijective (up to shift).

######  Theorem 3.8.

For every $X\in\mathbb{S}$,

$(-1)^{\epsilon}q^{\epsilon-1}\frac{\overline{\hom}_{q}(X,P_{2})}{\overline{\hom}_{q}(X,P_{1})},$

is a left $q$-deformed rational number, where

$\epsilon=\begin{cases}0,&X\geq 0,X\not\cong P_{2}[n],\\\ 1,&X\leq
0,X\not\cong P_{1}[n].\end{cases}$

Moreover, the induced map from $\mathbb{S}$ to the set of left $q$-deformed
rationals is $B_{3}$-equivariant, and bijective (up to shift).

These theorems were first proven in the case $q=1$, where they are identical,
by Rouquier and Zimmermann [RZ03, Proposition 4.8]. The proofs of these
theorems for arbitrary $q$ will occupy the rest of this section. In [BDL20,
Proposition 5.1], a proof of the original Rouquier–Zimmermann bijection is
established using the theory of Harder–Narasimhan automata; it is this proof
technique that we adapt to prove the above theorems. More precisely, to prove
Theorem 3.7, we will use the automaton introduced in Section 3.3 to analyse
the dynamics of spherical objects in $\mathcal{C}_{2}$ under the braid group
action.

The proof of Theorem 3.7 will proceed by an induction on the length of
elements of $B_{3}$. To facilitate this induction, we introduce a normal form
for the three-strand braid group $B_{3}$, which is (mostly) recognised by the
automaton (that is, any braid in this form that we actually need corresponds
to a path along the edges of the automaton). Let
$\omega:=\sigma_{2}\sigma_{1}\sigma_{2}\sigma_{1}\sigma_{2}\sigma_{1}$ be the
generator of the centre of $B_{3}$,

######  Lemma 3.9 (Continued normal form).

Consider any braid $\beta\in B_{3}$. We can write $\beta$ uniquely in one of
four forms, determined by the image of $P_{1}$:

(3) $\displaystyle\beta=$
$\displaystyle\sigma_{1}^{-a_{1}}\sigma_{2}^{a_{2}}...\sigma_{1}^{-a_{2n-1}}\sigma_{2}^{a_{2n}}\sigma_{1}^{M}\omega^{N},$
$\beta P_{1}\geq 0,\beta P_{1}\neq P_{1}[m],P_{2}[m],$ (4)
$\displaystyle\beta=$
$\displaystyle\sigma_{1}^{a_{1}}\sigma_{2}^{-a_{2}}...\sigma_{1}^{a_{2n-1}}\sigma_{2}^{-a_{2n}}\sigma_{1}^{M}\omega^{N},$
$\beta P_{1}\leq 0,\beta P_{1}\neq P_{1}[m],P_{2}[m],$ (5)
$\displaystyle\beta=$ $\displaystyle\sigma_{1}^{M}\omega^{N},$ $\beta
P_{1}=P_{1}[m],$ (6) $\displaystyle\beta=$
$\displaystyle\sigma_{1}\sigma_{2}\sigma_{1}^{M}\omega^{N},$ $\beta
P_{1}=P_{2}[m],$

where $a_{1}\in\mathbb{N}$, $a_{2},...,a_{2n}\in\mathbb{N}\setminus 0$,
$M,N\in\mathbb{Z}$.

Taking $P_{1}$ as the counterpart of $\infty$, and $P_{2}$ as the counterpart
of 0, Lemma 3.9 corresponds exactly to the continued fraction expansion in
Section 2.1.

###### Proof.

Consider any braid $\beta\in B_{3}$. Let $\overline{\beta}$ denote the
corresponding element of $\operatorname{PSL}_{2}(\mathbb{Z})$ (under the
standard surjection $B_{3}\to\operatorname{PSL}_{2}(\mathbb{Z})$.) Let
$\overline{\beta}\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]}=\big{[}\begin{smallmatrix}r\\\
s\end{smallmatrix}\big{]}$. Then by Proposition 2.2, there exists a matrix
$\widehat{\beta^{\prime}}$ of the form

$\overline{\beta^{\prime}}=\overline{\sigma}_{1}^{-a_{1}}\overline{\sigma}_{2}^{a_{2}}...\overline{\sigma}_{1}^{-a_{2n-1}}\overline{\sigma}_{2}^{a_{2n}},$

such that $\overline{\beta^{\prime}}\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]}=\big{[}\begin{smallmatrix}r\\\
s\end{smallmatrix}\big{]}$. Since $\overline{\beta^{\prime}}$ and
$\overline{\beta}$ act identically on $\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]}$, we have that
$\overline{\beta^{\prime}}=\overline{\beta}\overline{\sigma}_{1}^{M}$ for some
$M\in\mathbb{Z}$ (this can be deduced from the fact that every matrix in
$\operatorname{PSL}_{2}(\mathbb{Z})$ has determinant 1). Lemma 3.9 then
follows from the fact that the kernel of
$B_{3}\to\operatorname{PSL}_{2}(\mathbb{Z})$ is the centre $Z(B_{3})$. ∎

If $\beta$ is given by the form in Lemma 3.9, we say that $\beta$ is written
in (even) continued form. We say that $\beta$ is written in strict continued
form if any of the following hold:

  1. (1)

$\beta$ is written in form (3), $M\leq 0$;

  2. (2)

$\beta$ is written in form (4), $M\geq 0$;

  3. (3)

$\beta$ is written in form (5);

  4. (4)

$\beta$ is written in form (6), $M\leq 0$.

The following corollary of Lemma 3.9 follows from the calculation:

$\sigma_{1}^{M}\omega^{N}P_{1}=P_{1}[-2N-M].$

######  Corollary 3.10 (Strict continued normal form).

For any $X\in\mathbb{S}$, there exists a braid $\beta$ in strict continued
form such that $\beta P_{1}=X$.

Theorem 3.7 is an immediate corollary of the matrix formula for right
$q$-rationals (Proposition 2.10), and the following Propositions 3.11 and
3.12:

######  Proposition 3.11.

Consider any $\beta\in B_{3}$ which can be written in strict continued form,
and let $\overline{\beta}$ be the corresponding $q$-deformed matrix. Up to
multiplication by $-1$ and by powers of $q$, we have an equality:

$\overline{\beta}=\begin{bmatrix}\operatorname{occ}_{q}(P_{2},\beta
P_{1})&(-q)^{-\epsilon}\operatorname{occ}_{q}(P_{2},\beta P_{2})\\\
(-q)^{\epsilon}\operatorname{occ}_{q}(P_{1},\beta
P_{1})&\operatorname{occ}_{q}(P_{1},\beta P_{2})\end{bmatrix},$

where

$\epsilon=\begin{cases}0,&\beta P_{2}\geq 0,\\\ 1,&\beta P_{2}\leq
0.\end{cases}$

######  Proposition 3.12.

Consider any $\beta\in B_{3}$, which cannot be written in strict contined
form, then, up to multiplication by $-1$ and by powers of $q$, we have:

$\overline{\beta}=\begin{bmatrix}\operatorname{occ}_{q}(P_{2},\beta
P_{1})&(-q)^{\epsilon-1}\operatorname{occ}_{q}(P_{2},\beta P_{2})\\\
\operatorname{occ}_{q}(P_{1},\beta
P_{1})&(-q)^{2\epsilon-1}\operatorname{occ}_{q}(P_{1},\beta
P_{2})\end{bmatrix}$

where

$\epsilon=\begin{cases}0,&\beta P_{2}\geq 0,\\\ 1,&\beta P_{2}\leq
0.\end{cases}$

###### Proof of Proposition 3.11.

Consider any $\beta\in B_{3}$ which can be written in strict continued form.
We induct on the length of $\beta$. The base case $\beta=\operatorname{Id}$ is
trivial. Since multiplication by $\omega$ corresponds to scalar multiplication
by a power of $q$, we can assume without loss of generality that the
$\omega^{N}$-term in $\beta$ is trivial (i.e. $N=0$).

For the induction argument, note that $\beta$, read from right to left,
defines a path in $\Theta$ (Fig. 6). We will ignore the case that $\beta$ is
in the forms (5) and (6) (in the notation of Lemma 3.9), since this can be
easily done by hand.

Let $\beta_{l}$ be the braid consisting of the last $l$ elementary braids in
$\beta$, and suppose that

$\overline{\beta_{l}}=\begin{bmatrix}\operatorname{occ}_{q}(P_{2},\beta_{l}P_{1})&(-q^{-1})^{\epsilon}\operatorname{occ}_{q}(P_{2},\beta_{l}P_{2})\\\
(-q)^{\epsilon}\operatorname{occ}_{q}(P_{1},\beta_{l}P_{1})&\operatorname{occ}_{q}(P_{1},\beta_{l}P_{2}).\end{bmatrix}$

There are eight cases, depending on whether $\beta$ has the form (3) or (4) in
the notation of Lemma 3.9, and on the value of $\beta_{l+1}\beta_{l}^{-1}$.
Note that $\beta_{l}P_{1}\in[P_{i},P_{j}]$ if and only if
$\beta_{l}P_{2}\in[P_{i},P_{j}]$. For illustrative purposes, we will assume
that $\beta$ has the form (3) (and thus $\epsilon=0$), that
$\beta_{l}P_{1},\beta_{l}P_{2}\in[P_{21},P_{1}]$, and
$\beta_{l+1}=\sigma_{2}\beta_{l}$. First we calculate

$\displaystyle\overline{\beta_{l+1}}$
$\displaystyle=\overline{\sigma_{2}\beta_{l}}$
$\displaystyle=\begin{bmatrix}1&0\\\
1&q^{-1}\end{bmatrix}\begin{bmatrix}\operatorname{occ}_{q}(P_{2},\beta_{l}P_{1})&\operatorname{occ}_{q}(P_{2},\beta_{l}P_{2})\\\
\operatorname{occ}_{q}(P_{1},\beta_{l}P_{1})&\operatorname{occ}_{q}(P_{1},\beta_{l}P_{2}).\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\operatorname{occ}_{q}(P_{2},\beta_{l}P_{1})&\operatorname{occ}_{q}(P_{2},\beta_{l}P_{2})\\\
\operatorname{occ}_{q}(P_{2},\beta_{l}P_{1})+q^{-1}\operatorname{occ}_{q}(P_{1},\beta_{l}P_{1})&\operatorname{occ}_{q}(P_{2},\beta_{l}P_{2})+q^{-1}\operatorname{occ}_{q}(P_{1},\beta_{l}P_{2}).\end{bmatrix}$

On the other hand, by definition we have equalities

$\displaystyle\begin{bmatrix}\pi_{21}(\beta_{l}P_{1})&\pi_{21}(\beta_{l}P_{2})\\\
\pi_{1}(\beta_{l}P_{1})&\pi_{1}(\beta_{l}P_{2})\end{bmatrix}=\begin{bmatrix}\operatorname{occ}_{q}(P_{1},\beta_{l}P_{1})&\operatorname{occ}_{q}(P_{1},\beta_{l}P_{2})\\\
\operatorname{occ}_{q}(P_{2},\beta_{l}P_{1})-\operatorname{occ}_{q}(P_{1},\beta_{l}P_{1})&\operatorname{occ}_{q}(P_{2},\beta_{l}P_{2})-\operatorname{occ}_{q}(P_{1},\beta_{l}P_{2})\end{bmatrix}$

and

(7)
$\displaystyle\begin{bmatrix}\pi_{2}(\beta_{l+1}P_{1})&\pi_{2}(\beta_{l+1}P_{2})\\\
\pi_{21}(\beta_{l+1}P_{1})&\pi_{21}(\beta_{l+1}P_{2})\end{bmatrix}$ (8)
$\displaystyle=\begin{bmatrix}\operatorname{occ}_{q}(P_{1},\beta_{l+1}P_{1})-\operatorname{occ}_{q}(P_{2},\beta_{l+1}P_{1})&\operatorname{occ}_{q}(P_{1},\beta_{l+1}P_{2})-\operatorname{occ}_{q}(P_{2},\beta_{l+1}P_{2})\\\
\operatorname{occ}_{q}(P_{2},\beta_{l+1}P_{1})&\operatorname{occ}_{q}(P_{2},\beta_{l+1}P_{2})\end{bmatrix}$

Applying the linear representation embedded in $\Theta$,

(9)
$\displaystyle\begin{bmatrix}\pi_{2}(\beta_{l+1}P_{1})&\pi_{2}(\beta_{l+1}P_{2})\\\
\pi_{21}(\beta_{l+1}P_{1})&\pi_{21}(\beta_{l+1}P_{2})\end{bmatrix}$
$\displaystyle=\begin{bmatrix}q^{-1}&0\\\
1&1\end{bmatrix}\begin{bmatrix}\pi_{21}(\beta_{l}P_{1})&\pi_{21}(\beta_{l}P_{2})\\\
\pi_{1}(\beta_{l}P_{1})&\pi_{1}(\beta_{l}P_{2})\end{bmatrix}$ (10)
$\displaystyle=\begin{bmatrix}q^{-1}\operatorname{occ}_{q}(P_{1},\beta_{l}P_{1})&q^{-1}\operatorname{occ}_{q}(P_{1},\beta_{l}P_{2})\\\
\operatorname{occ}_{q}(P_{2},\beta_{l}P_{1})&\operatorname{occ}_{q}(P_{2},\beta_{l}P_{2})\end{bmatrix}$

Comparing Eq. 7 and Eq. 9, gives us that for $i\in\\{1,2\\}$,

$\displaystyle\operatorname{occ}_{q}(P_{2},\beta_{l+1}P_{i})$
$\displaystyle=\operatorname{occ}_{q}(P_{2},\beta_{l}P_{i}),$
$\displaystyle\operatorname{occ}_{q}(P_{1},\beta_{l+1}P_{i})$
$\displaystyle=q^{-1}\operatorname{occ}_{q}(P_{1},\beta_{l}P_{i})+\operatorname{occ}_{q}(P_{2},\beta_{l}P_{i}),$

which proves that

$\overline{\beta_{l+1}}=\begin{bmatrix}\operatorname{occ}_{q}(P_{2},\beta_{l+1}P_{1})&\operatorname{occ}_{q}(P_{2},\beta_{l+1}P_{2})\\\
\operatorname{occ}_{q}(P_{1},\beta_{l+1}P_{1})&\operatorname{occ}_{q}(P_{1},\beta_{l+1}P_{2}).\end{bmatrix}.$

Verifying the other seven cases (using the exact same procedure) completes the
induction argument. ∎

The proof of Proposition 3.12 is virtually the same as the proof of
Proposition 3.11. Any braid which cannot be written in strictly normal form is
a right multiple of a braid which can be written in strictly normal form by
$\sigma_{1}^{M}$, thus one can prove this case using the same method as in
part 1, but changing the base case appropriately.

This completes the proof of Theorem 3.7.

A proof of Theorem 3.8 can be constructed independently, analogously to the
above proof. In lieu of doing that, we will bootstrap Theorem 3.8 to Theorem
3.7, by way of the following relationship between $\operatorname{occ}_{q}$ and
$\overline{\hom}_{q}$:

######  Lemma 3.13 ($\hom$-values).

The functional $\overline{\hom}_{q}(P_{1},-)$ takes the following values:

$\overline{\hom}_{q}(P_{1},X)=\begin{cases}(1-q^{-1})\operatorname{occ}_{q}(P_{2},X)+q^{-1}\operatorname{occ}_{q}(P_{1},X),&X\geq
0\\\
(q^{-2}-q^{-1})\operatorname{occ}_{q}(P_{2},X)+q^{-1}\operatorname{occ}_{q}(P_{1},X),&X\leq
0.\end{cases}$

Symmetrically, the functional $\overline{\hom}_{q}(P_{2},-)$ takes the
following values:

$\overline{\hom}_{q}(P_{2},X)=\begin{cases}(q^{-2}-q^{-1})\operatorname{occ}_{q}(P_{1},X)+q^{-1}\operatorname{occ}_{q}(P_{2},X),&X\geq
0,\\\
(1-q^{-1})\operatorname{occ}_{q}(P_{1},X)+q^{-1}\operatorname{occ}_{q}(P_{2},X),&X\leq
0.\end{cases}$

###### Proof of Theorem 3.8.

Consider any $X\in\mathbb{S}$. Assume that $X\neq P_{1},P_{2}$. By 3.10, we
can write $X=\beta P_{1}$ for some $\beta$ in strict continued form. It is
easy to check that $\beta P_{1}\geq 0$ implies $\beta^{-1}P_{1}\leq 0$ (and
vice versa), and thus by Proposition 3.11 we have the following equality (up
to scaling):

$\displaystyle\overline{\beta}$
$\displaystyle=\begin{bmatrix}\operatorname{occ}_{q}(P_{2},\beta^{-1}P_{1})&(-q)^{\epsilon-1}\operatorname{occ}_{q}(P_{2},\beta^{-1}P_{2})\\\
(-q)^{1-\epsilon}\operatorname{occ}_{q}(P_{1},\beta^{-1}P_{1})&\operatorname{occ}_{q}(P_{1},\beta^{-1}P_{2})\end{bmatrix}^{-1}$
$\displaystyle=\begin{bmatrix}\operatorname{occ}_{q}(P_{1},\beta^{-1}P_{2})&(-1)^{\epsilon}q^{\epsilon-1}\operatorname{occ}_{q}(P_{2},\beta^{-1}P_{2})\\\
(-1)^{\epsilon}q^{1-\epsilon}\operatorname{occ}_{q}(P_{1},\beta^{-1}P_{1})&\operatorname{occ}_{q}(P_{2},\beta^{-1}P_{1})\end{bmatrix}.$

Let $\mathcal{R}^{\flat}(q)/\mathcal{S}^{\flat}(q)$ be the rational function
representation of $[\beta(\infty)]_{q}^{\flat}$, using the notation in 2.8.
Combining the matrix formula for right $q$-rationals (Proposition 2.10) and
the $\hom$-values lemma (Lemma 3.13), we have (again up to scaling):

$\displaystyle\begin{bmatrix}\mathcal{R}^{\flat}(q)\\\
\mathcal{S}^{\flat}(q)\end{bmatrix}$
$\displaystyle=\overline{\beta}\begin{bmatrix}q^{-1}\\\ q^{-1}-1\end{bmatrix}$
$\displaystyle=\begin{bmatrix}q^{-1}\operatorname{occ}_{q}(P_{1},\beta^{-1}P_{2})+(-1)^{\epsilon}(q^{\epsilon-2}-q^{\epsilon-1})\operatorname{occ}_{q}(P_{2},\beta^{-1}P_{2})\\\
(-1)^{\epsilon}q^{-\epsilon}\operatorname{occ}_{q}(P_{1},\beta^{-1}P_{1})+(q^{-1}-1)\operatorname{occ}_{s}{q}(P_{2},\beta^{-1}P_{1})\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\overline{\hom}_{q}(P_{1},\beta^{-1}P_{2})\\\
(-1)^{\epsilon}q^{1-\epsilon}\overline{\hom}_{q}(P_{1},\beta^{-1}P_{1})\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\overline{\hom}_{q}(X,P_{2})\\\
(-1)^{\epsilon}q^{1-\epsilon}\overline{\hom}_{q}(X,P_{1})\end{bmatrix}$

This shows that

$(-1)^{\epsilon}q^{\epsilon-1}\frac{\overline{\hom}_{q}(X,P_{2})}{\overline{\hom}_{q}(X,P_{1})}$

is indeed a left $q$-deformed rational number, and that the map sending $\beta
P_{1}$ to $[\beta(\infty)]_{q}^{\flat}$ is a bijection.

We now prove $B_{3}$-equivariance. By construction, we have that for every
$X\in\mathbb{S}$, there is some $\beta\in B_{3}$, such that $X=\beta P_{1}$
and

$(-1)^{\epsilon}q^{\epsilon-1}\frac{\overline{\hom}_{q}(\beta
P_{1},P_{2})}{\overline{\hom}_{q}(\beta
P_{1},P_{1})}=\beta\frac{\overline{\hom}_{q}(P_{1},P_{2})}{\overline{\hom}_{q}(P_{1},P_{1})}=\beta[\infty]_{q}^{\flat}.$

Consider any $\beta_{1},\beta_{2}\in B_{3}$ such that
$\beta_{1}P_{1}=\beta_{2}P_{1}$. To prove $B_{3}$-equivariance, it will be
sufficient to show that $\beta_{1}$ and $\beta_{2}$ act identically on
$[\infty]_{q}^{\flat}$. By elementary braid group arithmetic,
$\beta_{1}=\beta_{2}\sigma_{1}^{n}$ for some $n\in\mathbb{Z}$. Since
$\sigma_{1}$ fixes $[\infty]_{q}^{\flat}$, we have that

$\beta_{1}[\infty]_{q}^{\flat}=\beta_{2}[\infty]_{q}^{\flat}.$

This completes the proof of $B_{3}$-equivariance. ∎

##  4\. A family of compactifications of the space of Bridgeland stability
conditions

The aim of this section is to propose a family of compactifications of the
space of Bridgeland stability conditions of a triangulated category indexed by
$q>0$. This construction generalises results of [BDL20]. In particular, for
each real number $q>0$, the corresponding compactification will be the closure
of an embedding (via $q$-masses) of
$\operatorname{Stab}(\mathcal{T})/\mathbb{C}$ into a fixed infinite projective
space. We prove and conjecture several properties of this construction for
2-CY categories associated to connected quivers, and analyse the construction
in detail for the category $\mathcal{C}_{2}$.

###  4.1. The deformed mass embedding

Suppose that $\mathcal{T}$ is the 2-CY category associated to a connected
quiver (we will specialise to $\mathcal{C}_{2}$ again shortly). Let
$\mathbb{S}$ be the set of spherical objects of $\mathcal{T}$. Let
$\mathbb{R}^{\mathbb{S}}$ be the space of real-valued functions on
$\mathbb{S}$. Let $\mathbb{P}(\mathbb{R}^{\mathbb{S}})$ denote the
projectivisation of this space; that is,
$\mathbb{P}(\mathbb{R}^{\mathbb{S}})=(\mathbb{R}^{\mathbb{S}}\setminus
0)/\mathbb{R}^{\times}$. Recall the definition of the $q$-mass of an object
$X$, given a stability condition $\tau$, denoted $m_{q,\tau}(X)$ (Definition
B.3). For any $q>0$, we have a map

$m_{q}\colon\operatorname{Stab}(\mathcal{T})/\mathbb{C}\to\mathbb{P}(\mathbb{R}^{\mathbb{S}}),$

defined as

$m_{q}(\tau)=[m_{q,\tau}(X)]_{X\in\mathbb{S}}.$

The following proposition states that a stability condition can be recovered
uniquely up to $\mathbb{C}$-action from its $q$-mass vector.

######  Proposition 4.1 (Injectivity).

Let $\mathcal{T}$ be the 2-CY category associated to a connected quiver. Let
$\mathbb{S}$ be the set of spherical objects of $\mathcal{T}$. Then the map
$m_{q}\colon\operatorname{Stab}(\mathcal{T})/\mathbb{C}\to\mathbb{P}(\mathbb{R}^{\mathbb{S}})$
is injective.

Injectivity in the case that $q=1$ is proven in [BDL20, Proposition 4.1]. This
proof works in general, given $q$-deformed versions of two trigonometric
relations: Ikeda’s ‘$q$-deformed triangle inequality’ [Ike20, Lemma 3.6], and
a ‘$q$-deformed SSS-triangle congruence theorem’, which we describe presently.

For any point $z\in\mathbb{H}\cup\mathbb{R}^{>0}$, let $\phi(z)$ denote the
phase of $z$. For any pair of points
$z_{1},z_{2}\in\mathbb{H}\cup\mathbb{R}^{>0}$, say that the triangle
associated to $(z_{1},z_{2})$ is the Euclidean planar triangle with vertices
$(0,z_{1},z_{1}+z_{2})$.

Classically, the lengths of the sides of the trangle associated to
$(z_{1},z_{2})$ would be $|z_{1}|$, $|z_{2}|$ and $|z_{1}+z_{2}|$. For our
purposes however, it is natural to ‘$q$-deform’ this measure. The $q$-deformed
length of a side of this triangle will involve both the classical length, and
the phase of the corresponding vector. Precisely, for
$v\in\\{z_{1},z_{2},z_{1}+z_{2}\\}$, the $q$-deformed length of the side with
classical length $|v|$ is $q^{\phi(v)}|v|$. Ikeda’s $q$-deformed triangle
inequality ([Ike20, Lemma 3.6]) says that the $q$-deformed side-length
satisfies the triangle inequality:

$q^{\phi(z_{1}+z_{2})}|z_{1}+z_{2}|\leq
q^{\phi(z_{1})}|z_{1}|+q^{\phi(z_{2})}|z_{2}|.$

The next lemma says that the classical SSS-property is also preserved under
$q$-deformation: the $q$-deformed side-lengths uniquely determine the triangle
(up to congruence).

######  Lemma 4.2 ($q$-SSS).

Let
$z_{1},z_{2},z_{1}^{\prime},z_{2}^{\prime}\in\mathbb{H}\cup\mathbb{R}^{>0}$,
and suppose that

$\displaystyle q^{\phi(z_{1})}|z_{1}|$
$\displaystyle=q^{\phi(z_{1}^{\prime})}|z_{1}^{\prime}|,$ $\displaystyle
q^{\phi(z_{2})}|z_{2}|$
$\displaystyle=q^{\phi(z_{2}^{\prime})}|z_{2}^{\prime}|,$ $\displaystyle
q^{\phi(z_{1}+z_{2})}|z_{1}+z_{2}|$
$\displaystyle=q^{\phi(z_{1}^{\prime}+z_{2}^{\prime})}|z_{1}^{\prime}+z_{2}^{\prime}|.$

Then the triangle associated to $(z_{1},z_{2})$ is congruent to the triangle
associated to $(z_{1}^{\prime},z_{2}^{\prime})$.

###### Proof.

Assume that $z_{1}=z_{1}^{\prime}=1$ (by rotating and scaling appropriately).
For convenience, let

$\displaystyle\phi_{2}:=\phi(z_{2}),$ $\displaystyle m_{2}:=|z_{2}|,$
$\displaystyle z_{12}=z_{1}+z_{2},$ $\displaystyle\phi_{12}=\phi(z_{12}),$
$\displaystyle m_{12}=|z_{1}+z_{2}|,$ $\displaystyle c=q^{\phi_{2}}m_{2}.$

To prove the proposition, it is enough to show that given the assumption, the
vector $z_{2}$ is uniquely determined by the values $q^{\phi_{2}}m_{2}$ and
$q^{\phi_{12}}m_{12}$. For any $t\in[0,1]$, let $w_{t}$ be the vector with
phase $t$ and length $q^{-t}c$. We explicitly construct a function
$T_{q,c}:[0,1]\to\mathbb{R}$, which maps $t\in[0,1]$ to
$q^{\phi(1+w_{t})}|1+w_{t}|$. If $T_{q,c}$ is injective, then since
$T_{q,c}(\phi_{2})=q^{\phi_{12}}m_{12}$, we can derive $\phi_{2}$ from
$q^{\phi_{12}}m_{12}$. Thus given $q^{\phi_{12}}m_{12}$ and
$q^{\phi_{2}}m_{2}$, we can derive $\phi_{2}$ and $m_{2}$, which determine
$z_{2}$.

By the law of cosines,

$\phi(1+w_{t})=\frac{1}{\pi}\cos^{-1}\Big{(}\frac{1+cq^{-t}\cos(\pi
t)}{\sqrt{1+(cq^{-t})^{2}+2cq^{-t}\cos(\pi t)}}\Big{)},$

and

$|1+w_{t}|=\sqrt{1+(cq^{-t})^{2}+2cq^{-t}\cos(\pi t)}.$

Thus,

$T_{q,c}(t)=q^{\frac{1}{\pi}\cos^{-1}\big{(}\frac{1+cq^{-t}\cos(\pi
t)}{\sqrt{1+(cq^{-t})^{2}+2cq^{-t}\cos(\pi
t)}}\big{)}}\sqrt{1+(cq^{-t})^{2}+2cq^{-t}\cos(\pi t)}.$

The derivative of $T_{q,c}$ on the interval $[0,1]$ simplifies to

$T_{q,c}^{\prime}(t)=-\frac{q^{f(t)}}{\pi\sqrt{c^{2}+q^{2t}+2cq^{t}\cos(\pi
t)}}(cq^{t}\ln^{2}(q)\sin(\pi t)+\pi^{2}cq^{t}\sin(\pi t)),$

where

$f(t)=-t+\frac{\cos^{-1}\Big{(}\frac{1+cq^{t}\cos(\pi
t)}{\sqrt{1+(cq^{-t})^{2}+2cq^{-t}\cos(\pi t)}}\Big{)}}{\pi}.$

Since $T^{\prime}_{q,c}$ is evidently negative, $T_{q,c}$ is injective on
$[0,1]$, completing the proof. ∎

Further, we conjecture that the mass map is a homeomorphism onto its image.

######  Conjecture 4.3.

Let $\mathcal{T}$ be the 2-CY category associated to a connected quiver and
let $\mathbb{S}$ be the set of spherical objects of $\mathcal{T}$. Let $m_{q}$
denote the mass map for a $q\in(0,\infty)$. Then

  1. (1)

The map $m_{q}$ is a homeomorphism onto its image.

  2. (2)

The closure of the image of $m_{q}$ is a compactification of
$\operatorname{Stab}(\mathcal{T})/\mathbb{C}$ and is a real manifold with
boundary.

  3. (3)

When $\mathcal{T}$ is a 2-CY category of type ADE, the closure of the image of
$m_{q}$ is homeomorphic to a closed Euclidean ball.

In the remainder of this section we prove a weaker implication of the above
conjecture in type $A_{2}$; we show that the map $m_{q}$ is a homeomorphism
onto its image, that the image of $m_{q}$ is homeomorphic to an _open_
Euclidean disk, and that the boundary of the closure is homeomorphic to a
circle.

The analogue of this conjecture for the 2-CY category of the $A_{2}$-quiver at
$q=1$ is proved in [BDL20, Proposition 5.6]. The proof of the (following)
$q$-deformed proposition is essentially the same (again switching the triangle
inequality for the $q$-deformed analogue from [Ike20, Lemma 3.6]).

######  Proposition 4.4 (Homeomorphism).

The map
$m_{q}\colon\operatorname{Stab}(\mathcal{C}_{2})/\mathbb{C}\to\mathbb{P}^{\mathbb{S}}(\mathbb{R})$
is a homeomorphism onto its image. Moreover, the closure of
$m_{q}(\operatorname{Stab}(\mathcal{C}_{2})/\mathbb{C})$ is compact in
$\mathbb{P}^{\mathbb{S}}(\mathbb{R})$.

###  4.2. Analogy with the $q$-Farey tessellation

In the remainder of this section, we will only work with the category
$\mathcal{C}_{2}$, and with $q\in(0,1)$ (although everything we state is true
for $q\in(1,\infty)$ up to sign/orientation). We will use $M_{q}$ to denote

$M_{q}:=m_{q}(\operatorname{Stab}(\mathcal{C}_{2})/\mathbb{C})\subset\mathbb{P}^{\mathbb{S}}(\mathbb{R}).$

Our goal is to classify the points in the ‘boundary’
$\partial\overline{M}_{q}:=\overline{M}_{q}\setminus M_{q}$. We will show that
the $\operatorname{occ}_{q}$ and $\overline{\hom}_{q}$ functionals lie in
$\partial\overline{M}_{q}$, and then use Theorem 3.7 and Theorem 3.8 to
completely describe $\partial\overline{M}_{q}$.

In Section 2.2, we described ‘$q$-deformed rational curves’ as limits of
sequences of triangles in the $q$-deformed Farey tessellation. These curves
satisfy two notable properties:

  1. (1)

they are in natural, $B_{3}$-equivariant bijection with the rational numbers;

  2. (2)

their union is a dense subset of the boundary of the $q$-Farey tessellation.

The same phenomenon occurs in the embedding of
$\operatorname{Stab}(\mathcal{C}_{2})/\mathbb{C}$ in projective space. Let
$\overline{\hom}_{q}(X),\operatorname{occ}_{q}(X)$ denote the functionals
$\overline{\hom}_{q}(X,\cdotp),\operatorname{occ}_{q}(X,\cdotp)$ respectively,
viewed as ellements of $\mathbb{P}^{\mathbb{S}}(\mathbb{R})$. That is,

$\overline{\hom}_{q}(X):=[\overline{\hom}_{q}(X,Y)]_{Y\in\mathbb{S}},\quad\operatorname{occ}_{q}(X):=[\operatorname{occ}_{q}(X,Y)]_{Y\in\mathbb{S}}.$

Set

$I_{q,X}=[\overline{\hom}_{q}(X),\operatorname{occ}_{q}(X)]:=\\{t\operatorname{occ}_{q}(X)+(1-t)\overline{\hom}_{q}(X)\mid
t\in[0,1]\\}$

as indicated in the introduction to the paper. We will show shortly that the
convex sets $I_{q,X}$ for $X\in\mathbb{S}$ arise as limits of stability
conditions. The functional $\overline{\hom}_{q}(X)$ should be thought of as
the correspondent of the left $q$-deformed rational number naturally
associated to $X$, and $\operatorname{occ}_{q}(X)$ as the correspondent of the
right $q$-deformed rational; together with their convex combinations, we
obtain a perfect analogue of the $q$-deformed rational curve.

Recall the definition of standard stability conditions from 3.1. The subset of
the space of stability conditions that corresponds to the first two Farey
triangles is given by the following definition:

######  Definition 4.5.

[BDL20, §5.3] A standard stability condition $\tau$ is type-A if $P_{21}$ is
semistable with respect to $\tau$, and type-B if $P_{12}$ is semistable with
respect to $\tau$. A type-A (resp, type-B) standard stability condition is
strictly type-A (resp. strictly type-B) if it is not both type-A and type-B.

Let $\Lambda$ denote the set of standard stability conditions, let
$\Lambda_{A}$ (resp. $\Lambda_{B}$) denote the set of type-A (resp. type-B)
standard stability conditions.

Think of $\Lambda_{A}$ as the Farey triangle with vertices $0,\infty,1$, and
$\Lambda_{B}$ as the Farey triangle with vertices $0,\infty,-1$. So the
initial edge $E$ is the intersection $\Lambda_{A}\cap\Lambda_{B}$. Its image
under the $q$-mass map is the interval between $\operatorname{occ}_{q}(P_{1})$
and $\operatorname{occ}_{q}(P_{2})$:

######  Lemma 4.6.

We have an equality:

$m_{q}(\Lambda_{A}\cap\Lambda_{B})=\\{t\operatorname{occ}_{q}(P_{1})+(1-t)\operatorname{occ}_{q}(P_{2})\mid
t\in(0,1)\\}.$

We can now state how the ‘intervals’ $I_{q,X}$ arise in
$\partial\overline{M}_{q}$.

######  Proposition 4.7.

Consider any $X\in\mathbb{S}$, and suppose that $X=\beta P_{1}[n]$, then

$\lim_{m\to\infty}\overline{m_{q}(\beta\sigma_{1}^{-m}(\Lambda_{A}\cap\Lambda_{B}))}=I_{q,X}.$

###### Proof.

We first prove this statement for $\beta=\operatorname{id}$, and then deduce
it for all other $\beta$ by equivariance. Thus we have to show that

$\lim_{m\to\infty}\overline{m_{q}(\sigma_{1}^{-m}(\Lambda_{A}\cap\Lambda_{B}))}=I_{q,P_{1}}=[\overline{\hom}_{q}(P_{1}),\operatorname{occ}_{q}(P_{1})].$

By Lemma 4.6,

$\overline{m_{q}(\Lambda_{A}\cap\Lambda_{B})}=\\{t\operatorname{occ}_{q}(P_{1})+(1-t)\operatorname{occ}_{q}(P_{2})\mid
t\in[0,1]\\}.$

Since $\sigma_{1}^{-1}P_{1}=P_{1}[1]$, we have that

$\lim_{m\to\infty}\sigma_{1}^{-m}\operatorname{occ}_{q}(P_{1})=\operatorname{occ}_{q}(P_{1}).$

Thus, all we need to show is that

$\lim_{m\to\infty}\sigma_{1}^{-m}\operatorname{occ}_{q}(P_{2})=\overline{\hom}_{q}(P_{1}).$

Let $X,Y\in\mathbb{S}$ be such that $X\geq 0$, $Y\leq 0$. Consider the
following abbreviated vector in $\mathbb{P}^{\mathbb{S}}(\mathbb{R})$,
representing $\sigma_{1}^{-m}\operatorname{occ}_{q}(P_{1})$

$\displaystyle\sigma_{1}^{-m}\begin{bmatrix}\operatorname{occ}_{q}(P_{2},P_{1})\\\
\operatorname{occ}_{q}(P_{2},P_{2})\\\ \operatorname{occ}_{q}(P_{2},X)\\\
\operatorname{occ}_{q}(P_{2},Y)\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\operatorname{occ}_{q}(P_{2},\sigma_{1}^{m}P_{1})\\\
\operatorname{occ}_{q}(P_{2},\sigma_{1}^{m}P_{2})\\\
\operatorname{occ}_{q}(P_{2},\sigma_{1}^{m}X)\\\
\operatorname{occ}_{q}(P_{2},\sigma_{1}^{m}Y)\end{bmatrix}$
$\displaystyle=\begin{bmatrix}q^{-m}\\\ [m]_{q^{-1}}\\\
[m-1]_{q^{-1}}(1-q^{-1})\operatorname{occ}_{q}(P_{2},X)+[m]_{q^{-1}}\operatorname{occ}_{q}(P_{1},X)\\\
\operatorname{occ}_{q}(P_{2},Y)+(q^{-1}-1)[m]_{q^{-1}}\operatorname{occ}_{q}(P_{2},Y)+[m]_{q^{-1}}\operatorname{occ}_{q}(P_{1},Y)\end{bmatrix}.$

We multiply all the coordinates by $\frac{q^{-1}}{[m]_{q^{-1}}}$ to obtain:

$\displaystyle\begin{bmatrix}\frac{q^{-m-1}}{[m]_{q^{-1}}}\\\ q^{-1}\\\
\frac{[m-1]_{q^{-1}}}{[m]_{q^{-1}}}(q^{-1}-q^{-2})\operatorname{occ}_{q}(P_{2},X)+q^{-1}\operatorname{occ}_{q}(P_{1},X)\\\
\frac{q^{-1}}{[m]_{q^{-1}}}\operatorname{occ}_{q}(P_{2},Y)+(q^{-2}-q^{-1})\operatorname{occ}_{q}(P_{2},Y)+q^{-1}\operatorname{occ}_{q}(P_{1},Y)\end{bmatrix}.$

When we send $m$ to infinity, we get:

$\displaystyle\begin{bmatrix}q^{-2}-q^{-1}\\\ q^{-1}\\\
(1-q^{-1})\operatorname{occ}_{q}(P_{2},X)+q^{-1}\operatorname{occ}_{q}(P_{1},X)\\\
(q^{-2}-q^{-1})\operatorname{occ}_{q}(P_{2},Y)+q^{-1}\operatorname{occ}_{q}(P_{1},Y)\end{bmatrix}.$

Comparing to the $\hom$-values lemma (Lemma 3.13), we see that this is indeed
equal to $\overline{\hom}_{q}(P_{1})$. ∎

We will check that the appearance of these functionals really is a non-trivial
result of taking the closure.

######  Lemma 4.8 (Disjointness 1).

For any $X\in\mathbb{S}$ and all $t\in[0,1]$,

$t\operatorname{occ}_{q}(X)+(1-t)\overline{\hom}_{q}(X)\notin M_{q}.$

To prove this proposition, we need to give a more thorough account of the
dynamics of the $q$-deformed triangle inequality. Consider any stability
condition $\tau\in\operatorname{Stab}(\mathcal{C}_{2})/\mathbb{C}$. By the
triangle inequality for the $q$-deformed mass [Ike20, Proposition 3.3], we
have the inequalities

(11) $\displaystyle m_{q,\tau}(P_{21})$ $\displaystyle\leq
m_{q,\tau}(P_{1})+m_{q,\tau}(P_{2}),$ (12) $\displaystyle m_{q,\tau}(P_{1})$
$\displaystyle\leq q^{-1}m_{q,\tau}(P_{2})+m_{q,\tau}(P_{21}),$ (13)
$\displaystyle m_{q,\tau}(P_{2})$ $\displaystyle\leq
m_{q,\tau}(P_{21})+qm_{q,\tau}(P_{1}).$

The next lemma describes when these inequalities degenerate into equalities;
it is the $q$-analogue of the phenomenon noted at [BDL20, Remark 5.14].

######  Lemma 4.9 ($q$-Degeneracy).

Consider any stability condition $\tau$. If $\tau$ is strictly type-A
standard, then the triangle inequalities (11), (12), (13) are strict
inequalities. If $\tau$ is not strictly standard, then one of the triangle
inequalities (11), (12), (13) is an equality.

Interchanging the indices 1 and 2 gives corresponding inequalities for type-B
stability conditions, for which Lemma 4.9 holds.

Now we are ready to prove the disjointness of $I_{q,X}$ and $M_{q}$.

###### Proof of Lemma 4.8.

Consider any $X\in\mathbb{S}$. That $\operatorname{occ}_{q}(X)\notin M_{q}$
follows immediately from the fact that $\operatorname{occ}_{q}(X,X)=0$, but
$m_{q,\tau}(X)>0$ for all
$\tau\in\operatorname{Stab}(\mathcal{C}_{2})/\mathbb{C}$. Nevertheless, in
order to obtain disjointness for all the convex combinations, we must give a
more involved proof for both $\operatorname{occ}_{q}(X)$ and
$\overline{\hom}_{q}(X)$.

Suppose to generate a contradiction that
$\operatorname{occ}_{q}(X),\overline{\hom}_{q}(X)\in M_{q}$. Since every
stability condition is in the braid group orbit of $\Lambda$, we can assume
that

$\operatorname{occ}_{q}(X)=m_{q}(\tau)\quad\text{and}\quad\overline{\hom}_{q}(X)=m_{q}(\tau^{\prime}),$

for some $\tau,\tau^{\prime}\in\Lambda$. Let $X=\beta P_{1}$ for some braid
$\beta\in B_{3}$ which can be written in strict continued form. Assume for
convenience that $\beta$ has form (3) in the notation of Lemma 3.9, and
therefore $\beta^{-1}\sigma_{1}$ has form (4), that is,

(14)
$\beta^{-1}\sigma_{1}=\sigma_{1}^{a_{1}}\sigma_{2}^{-a_{2}}...\sigma_{1}^{a_{2n-1}}\sigma_{2}^{-a_{2n}}\sigma_{1}^{M}\omega^{N}.$

Thus by Proposition 3.11,

$\displaystyle\overline{\beta^{-1}\sigma_{1}}$
$\displaystyle=\begin{bmatrix}\operatorname{occ}_{q}(P_{2},\beta^{-1}P_{1}[-1])&-q^{-1}\operatorname{occ}_{q}(P_{2},\beta^{-1}P_{12})\\\
-q\operatorname{occ}_{q}(P_{1},\beta^{-1}P_{1}[-1])&\operatorname{occ}_{q}(P_{1},\beta^{-1}P_{12})\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\operatorname{occ}_{q}(P_{2},\beta^{-1}P_{1})&-q^{-1}\operatorname{occ}_{q}(P_{2},\beta^{-1}P_{2})\\\
-q\operatorname{occ}_{q}(P_{1},\beta^{-1}P_{1})&\operatorname{occ}_{q}(P_{1},\beta^{-1}P_{2})\end{bmatrix}\begin{bmatrix}q^{-1}&-q^{-1}\\\
0&1\end{bmatrix}.$

Therefore, for $i\in\\{1,2\\}$,

(15)
$\operatorname{occ}_{q}(P_{i},\beta^{-1}P_{12})=\operatorname{occ}_{q}(P_{i},\beta^{-1}P_{1})+\operatorname{occ}_{q}(P_{i},\beta^{-1}P_{2}).$

Now consider the two cases $M=0$ and $M>0$, where $M$ is the exponent of
$\sigma_{1}$ in Eq. 14. If $M>0$, then $\beta\sigma_{1}^{-1}$ has form (4),
and a similar matrix calculation gives

(16)
$\operatorname{occ}_{q}(P_{i},\beta^{-1}P_{21})=\operatorname{occ}_{q}(P_{i},\beta^{-1}P_{2})-q\operatorname{occ}_{q}(P_{i},\beta^{-1}P_{1}).$

If $M=0$, then $\beta\sigma_{2}$ has form (4), and another similar matrix
calculation gives

(17)
$\operatorname{occ}_{q}(P_{i},\beta^{-1}P_{21})=\operatorname{occ}_{q}(P_{i},\beta^{-1}P_{1})-q^{-1}\operatorname{occ}_{q}(P_{i},\beta^{-1}P_{2}).$

Assume $i=1$ (the statements for $i=2$ are needed for $\overline{\hom}_{q}$).
Then Eqs. 15, 16 and 17, along with our assumption that
$m_{q}(\tau)=\operatorname{occ}_{q}(C)$, imply that:

(18) $m_{q,\tau}(P_{12})=m_{q,\tau}(P_{1})+m_{q,\tau}(P_{2}),$

and either

(19) $m_{q,\tau}(P_{2})=m_{q,\tau}(P_{21})+qm_{q,\tau}(P_{1}),$

or

(20) $m_{q,\tau}(P_{1})=q^{-1}m_{q,\tau}(P_{2})+m_{q,\tau}(P_{21}).$

Thus the degeneracy lemma (Lemma 4.9) implies that $\tau$ is neither strictly
type-A, nor strictly type-B. Since we assumed that $\tau$ is standard, this
implies that $\tau$ is both type-A and type-B. However, any stability
condition which is both type-A and type-B satisfies

(21) $m_{q,\tau}(P_{21})=m_{q,\tau}(P_{1})+m_{q,\tau}(P_{2}).$

If we were to combine Eq. 21 with either Eq. 19 or Eq. 20, we would get that
$m_{q,\tau}(P_{i})=0$ for either $i=1$ or $i=2$, giving the desired
contradiction (as the $q$-mass associated to any stability condition is
strictly positive on all complexes). A contradiction can be generated in the
cases that $\beta$ has the form (4), (5), (6), in the same way. This shows
that $\operatorname{occ}_{q}(X)\notin M_{q}$.

Recall our supposition that $\overline{\hom}_{q}=m_{q}(\tau^{\prime})$. By our
computation of $\overline{\hom}_{q}$ in terms of $\operatorname{occ}_{q}$
(Lemma 3.13), for $j\in\\{1,2,12,21\\}$ we have

(22)
$\overline{\hom}_{q}(P_{1},\beta^{-1}P_{j})=(q^{-2}-q^{-1})\operatorname{occ}_{q}(P_{2},\beta^{-1}P_{j})+q^{-1}\operatorname{occ}_{q}(P_{1},\beta^{-1}P_{j}).$

If we combine Eq. 22 with Eqs. 15, 16 and 17, we re-derive Eqs. 18, 19 and 20,
replacing $\tau$ with $\tau^{\prime}$. This delivers the same contradiction as
in the case of $\tau$, proving that $\overline{\hom}_{q}(X)\notin M_{q}$.

The proof for the rest of $I_{q,X}$ follows from the argument above. The
contradiction we obtained from the degenerating inequalities carries through
when we pass to convex combinations. ∎

###  4.3. Embedding the boundary of $\overline{M_{q}}$ into
$\mathbb{Q}_{q}\cup\mathbb{I}_{q}$

We will now use Theorems 3.7 and 3.8 (the Rouquier–Zimmermann Theorems) to
define a homeomorphism (essentially a projection map) from the sets
$I_{q,X}=[\overline{\hom}_{q}(X),\operatorname{occ}_{q}(X)]$, lying in
$\mathbb{P}^{\mathbb{S}}(\mathbb{R})$, onto the corresponding $q$-deformed
curves from Section 2.5.

Let $\mathbb{O}_{q}$ denote the set

$\mathbb{O}_{q}:=\bigsqcup_{X\in\mathbb{S}}I_{q,X}\subset\mathbb{P}^{\mathbb{S}}(\mathbb{R}).$

Recall that $\mathbb{Q}_{q}$ denotes the union of all the $q$-deformed
rational curves, together with their endpoints. For $X=P_{1}$, fix a
homeomorphism from the interval
$I_{q,P_{1}}=[\overline{\hom}_{q}(P_{1}),\operatorname{occ}_{q}(P_{1})]$ to
the complete geodesic between $1/(1-q)$ and $\infty$, together with its
endpoints. Extend this by the $B_{3}$ action to a $B_{3}$-equivariant
$\rho_{q}:\mathbb{O}_{q}\to\mathbb{Q}_{q}$, so that $\rho_{q}$ maps
$\overline{\hom}_{q}(X)$ and $\operatorname{occ}_{q}(X)$ onto
$[r/s]_{q}^{\flat}$ and $[r/s]_{q}^{\sharp}$ respectively, where $r/s$ is the
rational number corresponding to $X$. In particular, this implies that
$\rho_{q}$ is a bijection.

The closure of $\mathbb{O}_{q}$ is contained in $\overline{M}_{q}$, so we need
to incorporate limits of points in $\mathbb{O}_{q}$ into our classification of
$\overline{M}_{q}$. Let $\overline{\mathbb{O}}_{q}$ denote the union

$\overline{\mathbb{O}}_{q}:=\mathbb{O}_{q}\cup\left\\{\lim_{n\to\infty}\operatorname{occ}_{q}(X_{n})\mid(X_{n})_{n\in\mathbb{N}}\in\mathbb{S}^{\mathbb{N}}\right\\}.$

As the notation suggests, $\overline{\mathbb{O}}_{q}$ is in fact the closure
of $\mathbb{O}_{q}$ (see Lemma 4.11 below). We now extend $\rho_{q}$ to a
$B_{3}$-equivariant homeomorphism between $\overline{\mathbb{O}}_{q}$ and
$\mathbb{Q}_{q}\cup\mathbb{I}_{q}$, using the following lemma.

######  Lemma 4.10.

A sequence $(\operatorname{occ}_{q}(X_{n}))_{n\in\mathbb{N}}$ converges in
$\mathbb{P}^{\mathbb{S}}(\mathbb{R})$ if and only if
$(\rho_{q}(\operatorname{occ}_{q}(X_{n})))_{n\in\mathbb{N}}$ converges in
$\mathbb{Q}_{q}\cup\mathbb{I}_{q}$. Two sequences
$(\operatorname{occ}_{q}(X_{n}))_{n\in\mathbb{N}}$ and
$(\operatorname{occ}_{q}(Y_{n}))_{n\in\mathbb{N}}$ converge to the same limit
if and only if $(\rho_{q}(\operatorname{occ}_{q}(X_{n})))_{n\in\mathbb{N}}$
and $(\rho_{q}(\operatorname{occ}_{q}(Y_{n})))_{n\in\mathbb{N}}$ converge to
the same limit.

###### Proof.

For each statement, the forwards direction is trivial, by Theorem 3.7. The
backwards direction follows from the fact that, by Proposition 3.11, for any
$X,Y\in\mathbb{S}$, we have

$\operatorname{occ}_{q}(X,Y)=\operatorname{occ}_{q}(X,P_{1})\operatorname{occ}_{q}(P_{2},Y)+\operatorname{occ}_{q}(X,P_{2})\operatorname{occ}_{q}(P_{1},Y).$

So for any $(X_{n})_{n\in\mathbb{N}}$ and any $Y\in\mathbb{S}$, if the
sequence of fractions
$\big{(}\frac{\operatorname{occ}_{q}(X,P_{2})}{\operatorname{occ}_{q}(X,P_{1})}\big{)}_{n\in\mathbb{N}}$
converges, then so does the sequence of points in projective space

$\begin{bmatrix}\operatorname{occ}_{q}(X_{n},P_{2})\\\
\operatorname{occ}_{q}(X_{n},P_{1})\\\
\operatorname{occ}_{q}(X_{n},Y)\end{bmatrix}=\begin{bmatrix}\operatorname{occ}_{q}(X_{n},P_{2})\\\
\operatorname{occ}_{q}(X_{n},P_{1})\\\
\operatorname{occ}_{q}(X_{n},P_{1})\operatorname{occ}_{q}(P_{2},Y)+\operatorname{occ}_{q}(X_{n},P_{2})\operatorname{occ}_{q}(P_{1},Y)\end{bmatrix}.$

∎

######  Lemma 4.11.

The set $\overline{\mathbb{O}}_{q}$, as defined above, is the closure of
$\mathbb{O}_{q}$.

###### Proof.

Consider any sequence in $\mathbb{O}_{q}$, with the form

$(t_{n}\operatorname{occ}_{q}(X_{n})+(1-t_{n})\overline{\hom}_{q}(X_{n}))_{n\in\mathbb{N}}.$

Assuming that the sequence $(X_{n})_{n\in\mathbb{N}}$ contains infinitely many
distinct elements of $\mathbb{S}$, using elementary properties of limits of
right $q$-deformed rational numbers, it can be shown that

$\lim_{n\to\infty}\rho_{q}(\operatorname{occ}_{q}(X_{n}))=\lim_{n\to\infty}\rho_{q}(\overline{\hom}_{q}(X_{n})).$

On the other hand, by the proof of Proposition 4.7, there exists a sequence
$(Y_{n})_{n\in\mathbb{N}}\in\mathbb{S}^{\mathbb{N}}$ such that

$\lim_{n\to\infty}\operatorname{occ}_{q}(Y_{n})=\lim_{n\to\infty}\overline{\hom}_{q}(X_{n}).$

Thus,

$\lim_{n\to\infty}\rho_{q}(\operatorname{occ}_{q}(X_{n}))=\lim_{n\to\infty}\rho_{q}(\operatorname{occ}_{q}(Y_{n})),$

and so by Lemma 4.10,

$\lim_{n\to\infty}\operatorname{occ}_{q}(X_{n})=\lim_{n\to\infty}\operatorname{occ}_{q}(Y_{n})=\lim_{n\to\infty}\overline{\hom}_{q}(X_{n}).$

Thus, under the assumption that the sequence $(X_{n})_{n\in\mathbb{N}}$
contains infinitely many distinct elements of $\mathbb{S}^{\mathbb{N}}$, we
have

$\lim_{n\to\infty}(t_{n}\operatorname{occ}_{q}(X_{n})+(1-t_{n})\overline{\hom}_{q}(X_{n}))_{n\in\mathbb{N}}=\lim_{n\to\infty}\operatorname{occ}_{q}(X_{n}).$

∎

The new points we have picked up by taking limits of $\operatorname{occ}_{q}$
functionals are also non-trivial members of the closure $\overline{M}_{q}$.

######  Lemma 4.12 (Disjointness 2).

For any $p\in\overline{\mathbb{O}}_{q}$, $p\notin M_{q}$.

###### Proof.

Consider any $p\in\overline{\mathbb{O}}_{q}\setminus\mathbb{O}_{q}$, and
suppose to generate a contradiction that $p=m_{q}(\tau)$ for some
$\tau\in\Lambda$. There exists a sequence
$(X_{k})_{k\in\mathbb{N}}\in\mathbb{S}^{\mathbb{N}}$ such that

$\lim_{k\to\infty}\operatorname{occ}_{q}(X_{k})=p.$

Let $X_{k}=\beta_{k}P_{1}$, where $\beta_{k}$ is in strict continued form.
Passing to a subsequence if necessary, we can assume for convenience that
$\beta_{k}$ either has form (3) for all $k\in\mathbb{N}$ or form (4) for all
$k\in\mathbb{N}$, in the notation of Lemma 3.9 (we can rule out the cases (5)
and (6), since in these cases the limit is trivial). Let’s assume that
$\beta_{k}$ has form (3) for all $k\in\mathbb{N}$. Then by the proof of the
first disjointness lemma (Lemma 4.8) we have that for all $k\in\mathbb{N}$,
$i\in\\{1,2\\}$,

(23)
$\operatorname{occ}_{q}(P_{1},\beta^{-1}_{k}P_{12})=\operatorname{occ}_{q}(P_{1},\beta^{-1}_{k}P_{1})+\operatorname{occ}_{q}(P_{1},\beta^{-1}_{k}P_{2}).$

And either

(24)
$\operatorname{occ}_{q}(P_{1},\beta^{-1}_{k}P_{21})=\operatorname{occ}_{q}(P_{1},\beta^{-1}_{k}P_{2})-q\operatorname{occ}_{q}(P_{1},\beta^{-1}_{k}P_{1}),$

or

(25)
$\operatorname{occ}_{q}(P_{1},\beta^{-1}_{k}P_{21})=\operatorname{occ}_{q}(P_{1},\beta^{-1}_{k}P_{1})-q^{-1}\operatorname{occ}_{q}(P_{1},\beta^{-1}_{k}P_{2}).$

Pass to a subsequence so that we have Eq. 24 for all $k\in\mathbb{N}$, or Eq.
25 for all $k\in\mathbb{N}$. Taking the limit, we get that

$m_{q,\tau}(P_{12})=m_{q,\tau}(P_{1})+m_{q,\tau}(P_{2}),$

and either

$m_{q,\tau}(P_{2})=m_{q,\tau}(P_{21})+qm_{q,\tau}(P_{1}),$

or

$m_{q,\tau}(P_{1})=q^{-1}m_{q,\tau}(P_{2})+m_{q,\tau}(P_{21}),$

As in the proof of the disjointness lemma part 1, this implies a
contradiction. ∎

###  4.4. Classification of the compactification of $\mathcal{C}_{2}$

The main theorem of this section is that we have now given a full description
of the closure.

######  Theorem 4.13.

We have an equality

$\overline{M}_{q}=M_{q}\sqcup\overline{\mathbb{O}}_{q}$

in $\mathbb{P}^{\mathbb{S}}(\mathbb{R})$.

We have a homeomorphism
$\rho_{q}:\overline{\mathbb{O}}_{q}\to\mathbb{Q}_{q}\cup\mathbb{I}_{q}$. So,
this theorem implies that
$\partial\overline{M}_{q}=\overline{\mathbb{O}}_{q}\cong\mathbb{R}\cup\\{\infty\\}$,
that is, the boundary of the $q$-deformed Thurston compactification of
$\operatorname{Stab}(\mathcal{C}_{2})$ is homeomorphic to a circle (see Part 3
of Proposition 2.13).

######  Remark 4.14.

We stated without proof in Proposition 2.13 that
$\mathbb{Q}_{q}\cup\mathbb{I}_{q}$ is equal to the boundary of the
$q$-deformed Farey tesselation, so in fact $\rho_{q}$ maps
$\overline{\mathbb{O}}_{q}$ onto $\partial\overline{L}_{q}$. Thus, $\rho_{q}$
is a $B_{3}$-equivariant homeomorphism from the boundary of the $q$-deformed
compactification of $\operatorname{Stab}(\mathcal{C}_{2})/\mathbb{C}$ onto the
boundary of the $q$-deformed Farey tessellation.

Theorem 4.13 can be bootstrapped onto the proof of the corresponding [BDL20,
Proposition 5.17] (which covers the case $q=1$), using ‘$q$-deformed Gromov
coordinates’. Consider any standard stability condition
$\tau\in\operatorname{Stab}(\mathcal{C}_{2})/\mathbb{C}$. By the triangle
inequality for the $q$-deformed mass function [Ike20, Proposition 3.3], there
exist $a_{\tau},b_{\tau},c_{\tau}\in\mathbb{R}^{\geq 0}$ such that

$m_{q,\tau}(P_{1})=b_{\tau}+c_{\tau},\;\;m_{q,\tau}(P_{2})=a_{\tau}+qc_{\tau},\;\;m_{q,\tau}(P_{21})=a_{\tau}+b_{\tau}.$

We call $(a_{\tau},b_{\tau},c_{\tau})$ the $q$-Gromov coordinates associated
to $\tau$. The following is the $q$-deformation of [BDL20, Proposition 5.14].
The proof is essentially identical.

######  Proposition 4.15 ($q$-Linearity).

Consider any standard stability condition
$\tau\in\operatorname{Stab}(\mathcal{C}_{2})/\mathbb{C}$, then

$m_{q,\tau}=a_{\tau}\operatorname{occ}_{q}(P_{1})+b_{\tau}\operatorname{occ}_{q}(P_{2})+c_{\tau}\operatorname{occ}_{q}(P_{21}).$

###### Proof of Theorem 4.13.

We just need to show that

$M_{q}\cup\overline{\mathbb{O}}_{q}$

is closed in $\mathbb{P}^{\mathbb{S}}(\mathbb{R})$ (since we have already
shown that $\overline{\mathbb{O}}_{q}\subset\partial\overline{M}_{q}$).
Consider any sequence $(m_{q}(\tau_{n}))_{n\in\mathbb{N}}$ which converges to
a limit in $\mathbb{P}^{\mathbb{S}}(\mathbb{R})$. Let
$\tau_{n}=\beta_{n}\tau_{n}^{\operatorname{std}}$, where each
$\tau_{n}^{\operatorname{std}}$ is a standard stability condition.

Suppose $\\{\beta_{n}\mid n\in\mathbb{N}\\}$ is a finite set. The proof that
$\lim_{n\to\infty}m_{q}(\tau_{n})$ lies in
$M_{q}\cup\overline{\mathbb{O}}_{q}$ is easy, and identical to that in the
proof of [BDL20, Proposition 5.17]. Briefly,
$\lim_{n\to\infty}m_{q}(\tau_{n})$ lies in the union of
$\beta_{n}(\overline{\Lambda})$ for finitely many $\beta_{n}$, and each of
these sets lies in $M_{q}\cup\overline{\mathbb{O}}_{q}$ by [BDL20, Proposition
5.15].

Now suppose that $\\{\beta_{n}\mid n\in\mathbb{N}\\}$ is an infinite set.
Consider the two corresponding sets of sequences

$O_{1}=\big{\\{}(\operatorname{occ}_{1}(\beta_{n}P_{1}))_{n\in\mathbb{N}},(\operatorname{occ}_{1}(\beta_{n}P_{2}))_{n\in\mathbb{N}},(\operatorname{occ}_{1}(\beta_{n}P_{21}))_{n\in\mathbb{N}}\big{\\}},$
$O_{q}=\big{\\{}(\operatorname{occ}_{q}(\beta_{n}P_{1}))_{n\in\mathbb{N}},(\operatorname{occ}_{q}(\beta_{n}P_{2}))_{n\in\mathbb{N}},(\operatorname{occ}_{q}(\beta_{n}P_{21}))_{n\in\mathbb{N}}\big{\\}}.$

Since $\overline{\mathbb{O}}_{q}$ is sequentially compact, we can assume
(simultaneously passing to subsequences if necessary) that all these sequences
converge. Let us denote the limits $p_{1},p_{2},p_{21}$ and
$p_{1,q},p_{2,q},p_{21,q}$ respectively. It is shown that $p_{1}=p_{2}=p_{21}$
in the proof of [BDL20, Proposition 5.17]. The limit $p_{1}=p_{2}=p_{21}$
corresponds to a real number, namely
$\rho_{q=1}(p_{1})=\rho_{q=1}(p_{2})=\rho_{q=1}(p_{21})$. We split the
remainder of the proof into two cases, depending on whether
$\rho_{q=1}(p_{1})=\rho_{q=1}(p_{2})=\rho_{q=1}(p_{21})$ is rational or
irrational.

Suppose that $\rho_{q=1}(p_{1})=\rho_{q=1}(p_{2})=\rho_{q=1}(p_{21})$ is
irrational. Then by the characterisation of limits of right $q$-rationals
above (Theorem 2.11), we have that
$\rho_{q}(p_{1,q})=\rho_{q}(p_{2,q})=\rho_{q}(p_{21,q})$ is the unique
‘$q$-irrational’ corresponding to
$\rho_{q}(p_{1})=\rho_{q}(p_{2})=\rho_{q}(p_{21})$. Since $\rho_{q}$ is a
homeomorphism, this implies that $p_{1,q}=p_{2,q}=p_{21,q}$.

By the $q$-linearity proposition (Proposition 4.15), we have sequences
$(a_{n})_{n\in\mathbb{N}}$, $(b_{n})_{n\in\mathbb{N}}$, and
$(c_{n})_{n\in\mathbb{N}}$ in $(\mathbb{R}^{\geq 0})^{\mathbb{N}}$ such that

$\displaystyle\lim_{n\to\infty}m_{q}(\tau_{n})$
$\displaystyle=\lim_{n\to\infty}m_{q}(\beta_{n}\tau_{n}^{\operatorname{std}})$
$\displaystyle=\lim_{n\to\infty}\big{(}a_{n}\operatorname{occ}_{q}(\beta_{n}P_{1})+b_{n}\operatorname{occ}_{q}(\beta_{n}P_{2})+c_{n}\operatorname{occ}_{q}(\beta_{n}P_{21})\big{)}$
$\displaystyle=p_{1,q}=p_{2,q}=p_{21,q}.$

Since $p_{1,q}=p_{2,q}=p_{21,q}\in\overline{\mathbb{O}}_{q}$, we have shown
that the limit lies in $\overline{\mathbb{O}}_{q}$.

Suppose that $\rho_{q=1}(p_{1})=\rho_{q=1}(p_{2})=\rho_{q=1}(p_{21})$ is
rational. By simultaneously passing to subsequences, we can assume that the
image under $\rho_{q}$ of each of the sequences in $O_{1}$ is either an
increasing sequence or a decreasing sequence. Let $X\in\mathbb{S}$ correspond
to the rational number $\rho_{q}(p_{1})=\rho_{q}(p_{2})=\rho_{q}(p_{21})$, so
by Theorem 2.11, each of
$\rho_{q}(p_{1,q}),\rho_{q}(p_{2,q}),\rho_{q}(p_{21,q})$ is either the left or
right $q$-deformation of $\rho_{q}(p_{1})$. Thus, since $\rho_{q}$ is a
homeomorphism,
$p_{1,q},p_{2,q},p_{21,q}\in\\{\operatorname{occ}_{q}(X),\overline{\hom}_{q}(X)\\}$.
Again, by Proposition 4.15, we have sequences
$(a_{n})_{n\in\mathbb{N}},(b_{n})_{n\in\mathbb{N}},(c_{n})_{n\in\mathbb{N}}\in(\mathbb{R}^{\geq
0})^{\mathbb{N}}$ such that

$\lim_{n\to\infty}m_{q}(\tau_{n})=\lim_{n\to\infty}\big{(}a_{n}\operatorname{occ}_{q}(\beta_{n}P_{1})+b_{n}\operatorname{occ}_{q}(\beta_{n}P_{2})+c_{n}\operatorname{occ}_{q}(\beta_{n}P_{21})\big{)}.$

Since we are in projective space, we can normalise so that for all
$n\in\mathbb{N}$, $a_{n}+b_{n}+c_{n}=1$. Passing to subsequences if necessary,
we can assume that each of
$(a_{n})_{n\in\mathbb{N}},(b_{n})_{n\in\mathbb{N}},(c_{n})_{n\in\mathbb{N}}$
converges. Then since
$\operatorname{occ}_{q}(\beta_{n}P_{1}),\operatorname{occ}_{q}(\beta_{n}P_{2}),\operatorname{occ}_{q}(\beta_{n}P_{21})$
converge to $p_{1,q},p_{2,q},p_{21,q}$ respectively, we must have that

$a_{n}\operatorname{occ}_{q}(\beta_{n}P_{1})+b_{n}\operatorname{occ}_{q}(\beta_{n}P_{2})+c_{n}\operatorname{occ}_{q}(\beta_{n}P_{21})$

converges to a convex combination of $p_{1,q},p_{2,q},p_{21,q}$. That is,
there exists $t\in[0,1]$ such that

$\lim_{n\to\infty}\big{(}a_{n}\operatorname{occ}_{q}(\beta_{n}P_{1})+b_{n}\operatorname{occ}_{q}(\beta_{n}P_{2})+c_{n}\operatorname{occ}_{q}(\beta_{n}P_{21})\big{)}=t\overline{\hom}_{q}(X)+(1-t)\operatorname{occ}_{q}(X).$

Thus we have shown that $\lim_{n\to\infty}m_{q}(\tau_{n})$ lies in
$[\overline{\hom}_{q}(X),\operatorname{occ}_{q}(X)]$. This completes the
proof. ∎

##  Appendix A Combinatorial properties of left $q$-deformed rational numbers

###  A.1. Counting closures in quivers

We return to thinking about right $q$-deformed rationals and left $q$-deformed
rationals as formal rational functions in $q$. The following theorem of
Morier-Genoud and Ovsienko shows that the coefficients of
$\mathcal{R}^{\sharp}(q)$ and $\mathcal{S}^{\sharp}(q)$ have combinatorial
significance. For any oriented quiver $\mathcal{G}$, a _closure_ in
$\mathcal{G}$ is a subquiver $\mathcal{H}$ such that there do not exist edges
in $\mathcal{G}$ from vertices in $\mathcal{H}$ to vertices outside of
$\mathcal{H}$. For $n\in\mathbb{N}$, an _$n$ -closure_ is a closure with $n$
vertices. For any quiver $\mathcal{G}$, let $C_{n}(\mathcal{G})$ denote the
number of $n$-closures in $\mathcal{G}$.

Consider any rational number $r/s\in(1,\infty)$ with continued fraction
expansion $[a_{1},...,a_{2n}]$. Let $\mathcal{G}_{r/s}^{\sharp}$ denote the
following quiver.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$a_{1}-1$$a_{2}$$a_{3}$$a_{(2n-1)}$$a_{2n}-1$

Let $\widehat{\mathcal{G}}_{r/s}^{\sharp}$ denote the following truncated copy
of $\mathcal{G}_{r/s}$.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$a_{2}-1$$a_{3}$$a_{(2n-1)}$$a_{2n}-1$

######  Theorem A.1 ([MGO20, Theorem 4]).

Consider any rational number $r/s\in(1,\infty)$, with continued fraction
expansion $[a_{1},...,a_{n}]$ and $q$-deformation

$\Big{[}\frac{r}{s}\Big{]}_{q}^{\sharp}=\frac{\alpha^{\sharp}_{0}+\alpha_{1}^{\sharp}q+...+\alpha_{N-1}^{\sharp}q^{N-1}+\alpha_{N}^{\sharp}q^{N}}{\beta_{0}^{\sharp}+\beta_{1}^{\sharp}q+...+\beta_{M-1}^{\sharp}q^{M-1}+\beta_{M}^{\sharp}q^{M}}.$

For all $i\in\mathbb{N}$,

$\alpha_{i}^{\sharp}=C_{i}\big{(}\mathcal{G}_{r/s}\big{)},\quad\beta_{i}^{\sharp}=C_{i}\big{(}\widehat{\mathcal{G}}_{r/s}\big{)}.$

The following corollary (which may have independent interest) will be used
later on. The left $q$-deformed rationals, which we motivated in Section 2 on
geometric grounds, and in Section 3 on homological grounds, also admit a
combinatorial interpretation directly analogous to that given for the right
$q$-deformed rationals by Theorem A.1. The left $q$-rational counts closures
in a slightly augmented version of $\mathcal{G}_{r/s}^{\sharp}$. Let
$\mathcal{G}^{\flat}_{r/s}$ denote the following quiver.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$a_{1}-1$$a_{2}$$a_{3}$$a_{(2n-1)}$$a_{2n}-1$

And let $\widehat{\mathcal{G}}^{\flat}_{r/s}$ denote the following truncation.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$a_{2}-1$$a_{3}$$a_{(2n-1)}$$a_{2n}-1$

######  Corollary A.2.

Consider any rational number $r/s\in(1,\infty)$, with left $q$-deformation

$\Big{[}\frac{r}{s}\Big{]}_{q}^{\flat}=\frac{\alpha^{\flat}_{0}+\alpha^{\flat}_{1}q+...+\alpha^{\flat}_{N-1}q^{N-1}+\alpha^{\flat}_{N}q^{N}}{\beta^{\flat}_{0}+\beta^{\flat}_{1}q+...+\beta^{\flat}_{M-1}q^{M-1}+\beta^{\flat}_{M}q^{M}}.$

Then for all $i\in\mathbb{N}$,

$\displaystyle\alpha_{i}^{\flat}=C_{i}\big{(}\mathcal{G}_{r/s}^{\flat}\big{)},$
$\displaystyle\beta_{i}^{\flat}=C_{i}\big{(}\widehat{\mathcal{G}}_{r/s}^{\flat}\big{)}.$

We omit the proof, which is a straightforward application of Theorem A.1 by
way of the following formula for the left $q$-deformed rational (using the
notation of Proposition 2.10):

$\Big{[}\frac{r}{s}\Big{]}_{q}^{\flat}=\frac{q\mathcal{R}^{\sharp}(q)+\mathcal{R}^{\sharp\prime}(q)-q\mathcal{R}^{\sharp\prime}(q)}{q\mathcal{S}^{\sharp}(q)+\mathcal{S}^{\sharp\prime}(q)-q\mathcal{S}^{\sharp\prime}(q)}.$

###  A.2. Left $q$-deformed rational numbers and the Jones polynomial

The next theorem links left $q$-deformed rationals to the Jones polynomial of
rational (two-bridge) knots. The proof of the theorem uses Theorem 3.8,
although we suspect there exists a purely combinatorial proof.

Due to work by Lee and Schiffler [LS19, Theorem 1.2] and by Morier-Genoud and
Ovsienko [MGO20, Proposition A.3], the proof of this theorem is not difficult,
and does not involve any knot theory. For any rational number
$r/s\in(1,\infty)$, let $V_{r/s}(q)$ denote the Jones polynomial associated
with the class of rational knots parametrised by $r/s$, and let $|V_{r/s}(q)|$
denote the polynomial obtained by making each coefficient positive.

######  Theorem A.3.

Consider any rational number $r/s\in(1,\infty)$ with left $q$-deformation

$\Big{[}\frac{r}{s}\Big{]}_{q}^{\flat}=\frac{\mathcal{R}^{\flat}(q)}{\mathcal{S}^{\flat}(q)}.$

Then $\mathcal{R}^{\flat}(q)=|V_{r/s}(q)|$.

To prove this theorem, we will need the following lemma. For any rational
number $r/s$, let $\mathcal{H}_{r/s}$ denote the following quiver (note that
the orientation of the edges is opposite to that of
$\mathcal{G}_{r/s}^{\flat}$).

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$a_{1}-1$$a_{2}$$a_{3}$$a_{(2n-1)}$$a_{2n}-1$

######  Lemma A.4.

Consider any rational number $r/s\in(1,\infty)$. Let
$|V_{r/s}|=\gamma_{0}+\gamma_{1}q+...+\gamma_{N-1}q^{N-1}+\gamma_{N}q^{N}$.
Then for all $i\in\mathbb{N}$, $\gamma_{i}=C_{i}(\mathcal{H}_{r/s}).$

Lemma A.4 is an easy corollary of [MGO20, Proposition A.3], since the sequence
of coefficients of the normalised Jones polynomial $J_{r/s}(q)$ in their paper
is just the reverse of the sequence of coefficients of the Jones polynomial
$V_{r/s}$, and the number of $i$-closures of $\mathcal{H}_{r/s}$ is equal to
the number of $(N-i)$-closures of $\mathcal{H}_{r/s}$ with opposite
orientation.

###### Proof of Theorem A.3.

Let $[a_{1},...,a_{2n}]$ be the continued fraction expansion of $r/s$. Let
$\overline{\mathcal{R}}^{\flat}(q)/\overline{\mathcal{S}}^{\flat}(q)$ be the
left $q$-deformed rational associated to $[a_{2n},...,a_{1}]$. By A.2,
$\overline{\mathcal{R}}^{\flat}(q)$ counts the number of closures in
$\mathcal{H}_{r/s}$ with its orientation reversed. It is easy to check that
the number of $k$-closures in an $n$-vertex quiver $\mathcal{G}$ equals the
number of $(n-k)$-closures in $\mathcal{G}$ with its orientation reversed.
Thus, by Lemma A.4, the sequence of coefficients of
$\overline{\mathcal{R}}^{\flat}(q)$ is equal to the reverese of the sequence
of coefficients of $|V_{r/s}(q)|$. So it is sufficient to show that the
sequence of coefficients of $\mathcal{R}^{\flat}(q)$ is the reverse of the
sequence of coefficients of $\overline{\mathcal{R}}^{\flat}(q)$.

By Theorem 3.8, we have that

$\mathcal{R}^{\flat}(q)=\overline{\hom}_{q}(\sigma_{1}^{-a_{1}}\sigma_{2}^{a_{2}}...\sigma_{1}^{-a_{2n-1}}\sigma_{2}^{a_{2n}}P_{1},P_{2}),$

and that

$\displaystyle\overline{\mathcal{R}}^{\flat}(q)$
$\displaystyle=\overline{\hom}_{q}(\sigma_{1}^{-a_{2n}}\sigma_{2}^{a_{2n-1}}...\sigma_{1}^{-a_{2}}\sigma_{2}^{a_{1}}P_{1},P_{2})$
$\displaystyle=\overline{\hom}_{q}(P_{1},\sigma_{2}^{-a_{1}}\sigma_{1}^{a_{2}}...\sigma_{2}^{-a_{2n-1}}\sigma_{1}^{a_{2n}}P_{2})$
$\displaystyle=\overline{\hom}_{q}(P_{2},\sigma_{1}^{-a_{1}}\sigma_{2}^{a_{2}}...\sigma_{1}^{-a_{2n-1}}\sigma_{2}^{a_{2n}}P_{1}).$

Let

$X=\sigma_{1}^{-a_{1}}\sigma_{2}^{a_{2}}...\sigma_{1}^{-a_{2n-1}}\sigma_{2}^{a_{2n}}P_{1}.$

By the $\hom$-values lemma (Lemma 3.13), we know that

$\hom_{q}(P_{2},X)=(q^{-2}-q^{-1})\operatorname{occ}_{q}(P_{1},X)+q^{-1}\operatorname{occ}_{q}(P_{2},X),$

and an analogous result for $\overline{\hom}_{q}(\cdotp,P_{2})$ gives that

$\hom_{q}(X,P_{2})=(1-q^{-1})\operatorname{occ}_{q^{-1}}(P_{1},X)+q^{-1}\operatorname{occ}_{q^{-1}}(P_{2},X).$

Thus, the sequence of coefficients of $\overline{\hom}_{q}(X,P_{2})$ is the
reverse of the sequence of coefficients of $\overline{\hom}_{q}(P_{2},X),$
completing the proof. ∎

##  Appendix B Bridgeland stability conditions and Harder–Narasimhan automata

###  B.1. Background on Bridgeland stability conditions

We assume the reader is familiar with the theory of Bridgeland stability
conditions [Bri07]. To briefly recall, a stability condition on a triangulated
category is specified as $\tau=(\mathcal{P}_{\tau},Z_{\tau})$, where
$\mathcal{P}_{\tau}$ is a slicing and $Z_{\tau}$ is a compatible central
charge. An object of the category is called _$\tau$ -semistable_ if it belongs
to one of the full (abelian) subcategories $\mathcal{P}_{\tau}(\phi)$ for some
$\phi\in\mathbb{R}$. In this case, the number $\phi$ is called the _phase_ of
that semistable object. An object is called _$\tau$ -stable_ if it is
semistable (of some phase $\phi$), and moreover if it is simple in the abelian
category $\mathcal{P}_{\tau}(\phi)$.

Recall also that every object of the category has a unique _Harder–Narasimhan_
filtration with respect to $\tau$.

######  Definition B.1.

Fix a stability condition $\tau$ on a triangulated category. Let $X$ be any
object of the category. The $\tau$-Harder–Narasimhan filtration of $X$ is the
unique filtration

${0}$${X_{1}}$${X_{2}}$${\cdots}$${X_{n}=X}$${A_{1}}$${A_{2}}$${A_{3}}$${\cdots}$${A_{n}}$

with the following properties.

  1. (1)

Each filtration factor $A_{i}$ is semistable of some phase $\phi_{i}$, and

  2. (2)

The numbers $\phi_{i}$ appear in decreasing order:
$\phi_{1}>\phi_{2}\cdots>\phi_{n}$.

Let $\mathcal{T}$ denote an arbitrary triangulated category. Recall that for
any subset $I\subset\mathbb{R}$, we define $\mathcal{P}_{\tau}(I)$ to be the
full subcategory of $\mathcal{T}$ consisting of objects whose
Harder–Narasimhan filtration factors have phases in $I$. The subcategory
$\mathcal{P}_{\tau}([0,1))$ corresponding to the half-open interval
$[0,1)\subset\mathbb{R}$ is a full abelian subcategory of $\mathcal{T}$, which
is the heart of a bounded $t$-structure on $\mathcal{T}$ (see [Bri07, Section
3]).

Let $\Sigma_{\tau}$ denote the set of indecomposable semistable objects of
$\mathcal{T}$ that lie in $\mathcal{P}_{\tau}([0,1))$. Then every
indecomposable semistable object lies in $\Sigma_{\tau}$ up to an integer
shift. We can count the (graded) multiplicity of any object of $\mathcal{T}$
in a given object of $\Sigma_{\tau}$, as follows.

######  Definition B.2.

Fix some $q>0$. The $\tau$-Harder–Narasimhan multiplicity vector (or simply
the HN multiplicity vector) of a semistable object $A\in\mathcal{T}$ is an
element of $\mathbb{Z}[q^{\pm}]^{\Sigma_{\tau}}$, defined as follows. Let $k$
be such that $A\in\mathcal{P}_{\tau}([k,k+1))$. For any $B\in\Sigma_{\tau}$,
let $n_{B}$ be the number of indecomposable summands of $A$ that are
isomorphic to $B$. Then the coordinate of the HN multiplicity vector of $A$ at
$B\in\Sigma_{\tau}$ is $n_{B}q^{k}$.

The $\tau$-Harder–Narasimhan multiplicity vector of a general object of
$\mathcal{T}$ is defined as the sum of the $\tau$-Harder–Narasimhan
multiplicity vectors of its $\tau$-Harder–Narasimhan filtration factors.

Next we introduce a graded version of Harder–Narasimhan mass (see [DHKK14,
Section 4.5], [Ike20, Section 3]).

######  Definition B.3.

Let $\tau$ be a stability condition on a triangulated category $\mathcal{T}$.
Let $q$ be a fixed positive real number. Let $X$ be an object of $\mathcal{T}$
and consider its Harder–Narasimhan filtration:

${0}$${X_{1}}$${X_{2}}$${\cdots}$${X_{n}=X.}$${A_{1}}$${A_{2}}$${A_{3}}$${\cdots}$${A_{n}}$

Let $\phi_{i}$ denote the phase of the semistable object $A_{i}$. The _$q$
-Harder–Narasimhan mass_, or _$q$ -mass_ of the object $X$ is defined to be

$m_{q,\tau}(X)=\sum_{i=1}^{k}q^{\phi_{i}}|Z_{\tau}(A_{i})|.$

Note that for $q=1$, the $q$-Harder–Narasimhan mass is simply the classical
Harder–Narasimhan mass as defined in [Bri07, Section 5]. The $q$-mass function
is due to [DHKK14, Section 4.4].

Finally, we denote the space of stability conditions on a triangulated
category $\mathcal{T}$ by $\operatorname{Stab}(\mathcal{T})$. There is a
natural $\mathbb{C}$-action on $\operatorname{Stab}(\mathcal{T})$ defined as
follows.

For $z=a+i\pi b\in\mathbb{C}$ and a stability condition
$\tau=(\mathcal{P}_{\tau},Z_{\tau})$, define
$z\tau=(\mathcal{P}_{z\tau},Z_{z\tau})$ by

$\mathcal{P}_{z\tau}(\phi)=\mathcal{P}_{\tau}(\phi-b),\quad
Z_{z\tau}(X):=e^{z}Z_{\tau}(X).$

So for any $X\in\mathcal{T}$, the action of $z=a+i\pi b$ rotates the vector
$Z(X)$ by the angle $\pi b$, scales it by the quantity $e^{a}$, and
simultaneously shifts the slicing by the quantity $b$. We will always consider
the space $\operatorname{Stab}(\mathcal{T})/\mathbb{C}$, which is the space of
stability conditions up to this natural $\mathbb{C}$ action.

###  B.2. Harder–Narasimhan and mass automata

Fix throughout this section a stability condition $\tau$ on $\mathcal{T}$.
Recall that $\Sigma_{\tau}$ is the set of indecomposable $\tau$-semistable
objects of $\mathcal{T}$ that lie in $\mathcal{P}_{\tau}([0,1))$. For any
$X\in\operatorname{Ob}\mathcal{T}$, set
$\operatorname{HN}_{\tau}(X)\in\mathbb{Z}[q^{\pm}]^{\Sigma_{\tau}}$ to be the
$\tau$-HN multiplicity vector of $X$. Suppose also that a group $G$ acts on
$\mathcal{T}$.

In this section we define Harder–Narasimhan automata on $\mathcal{T}$. Our
definition is just a graded version of the one from [BDL20, Section 3], but we
recall the key elements here. We begin with some notation.

Let $\Theta$ be an arbitrary quiver. Let $\Theta_{0}$ and $\Theta_{1}$ denote
the sets of vertices and edges of $\Theta$ respectively. Given
$\alpha\in\Theta_{1}$, we say that $s(\alpha)\in\Theta_{0}$ is its source, and
$t(\alpha)\in\Theta_{0}$ is its target.

A $G$-labelling of $\Theta$ is a function $\Theta_{1}\to G$. Then every path
(finite sequence of composable edges) in $\Theta$ automatically acquires a
label, namely the product of the corresponding group elements in reverse
order.

A $\Theta$-set $\underline{S}$ is a representation of $\Theta$ in the category
of sets. More precisely, it is specified by a collection of sets $\\{S_{v}\mid
v\in\Theta_{0}\\}$, together with edge functions $\\{f_{\alpha}\colon
S_{s(\alpha)}\to S_{t(\alpha)}\mid\alpha\in\Theta_{1}\\}$.

Let $A$ be any ring. A $\Theta$-representation $\underline{M}$ of $A$-modules
consists (as usual) of $A$-modules $\\{M_{v}\mid v\in\Theta_{0}\\}$, and
$A$-linear maps $\\{f_{\alpha}\colon M_{s(\alpha)}\to
M_{t(\alpha)}\mid\alpha\in\Theta_{1}\\}$. Any $\Theta$-representation is also
a $\Theta$-set.

A morphism of $\Theta$-sets (resp. $\Theta$-representations) consists of maps
between the sets (resp. module maps between the modules) at each vertex, so
that the square determined by every edge commutes.

A $G$-set $S$, together with a $G$-labelling on $\ell\colon\Theta_{1}\to G$,
naturally produces a $\Theta$-set, as follows. We declare each $S_{v}$ to be
$S$ for every $v\in\Theta_{0}$, and each $f_{\alpha}\colon S\to S$ to be the
action of $\ell(\alpha)$ on $S$. In particular, $\operatorname{Ob}\mathcal{T}$
is a $\Theta$-set, and we denote this as
$\underline{\operatorname{Ob}\mathcal{T}}$. Similarly, any $G$-representation,
together with a $G$-labelling of $\Theta$, naturally produces a
$\Theta$-representation.

######  Definition B.4.

Fix some $q>0$. A $\tau$-HN automaton for the $G$-action on $\mathcal{T}$
consists of the following data.

  1. (1)

A $G$-labelled quiver $\Theta$ (and hence the $\Theta$-set
$\underline{\operatorname{Ob}\mathcal{T}}$).

  2. (2)

A $\Theta$-subset
$\underline{S}\subset\underline{\operatorname{Ob}\mathcal{T}}$.

  3. (3)

A $\Theta$-representation $\underline{M}$ of $\mathbb{Z}[q^{\pm}]$-modules.

  4. (4)

A morphism $i\colon\underline{S}\to\underline{M}$.

  5. (5)

A $\mathbb{Z}[q^{\pm}]$-linear map $h_{v}\colon
M_{v}\to\mathbb{Z}[q^{\pm}]^{\Sigma_{\tau}}$ for every $v\in\Theta_{0}$.

This data is required to satisfy

$\operatorname{HN}_{\tau}(s)=h_{v}(i_{v}(s))$

for every $v\in\Theta_{0}$ and every $s\in S=S_{v}$.

######  Example B.5.

We illustrate the above definition with the following (somewhat trivial)
example. Let $\mathcal{T}$ be any triangulated category, and fix a stability
condition $\tau$ on it. Let $G$ be the group $\mathbb{Z}$, generated by the
triangulated shift $(-)[1]$. Then an example of an HN automaton for this setup
is given by the following data.

  1. (1)

An infinite $\mathbb{Z}$-labelled quiver $\Theta$.

${\cdots}$-101${\cdots}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{[-1]}$$\scriptstyle{[1]}$$\scriptstyle{[-1]}$$\scriptstyle{[1]}$$\scriptstyle{[-1]}$$\scriptstyle{[-1]}$

  2. (2)

The following $\Theta$-subset
$\underline{S}\subset\underline{\operatorname{Ob}\mathcal{T}}$.

${\cdots}$${\mathcal{P}_{\tau}([-1,0))}$${\mathcal{P}_{\tau}([0,1))}$${\mathcal{P}_{\tau}([1,2))}$${\cdots}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{[-1]}$$\scriptstyle{[1]}$$\scriptstyle{[-1]}$$\scriptstyle{[1]}$$\scriptstyle{[-1]}$$\scriptstyle{[-1]}$

  3. (3)

The following $\Theta$-representation $\underline{M}$.

${\cdots}$${\mathbb{Z}[q^{\pm}]^{\Sigma_{\tau}}}$${\mathbb{Z}[q^{\pm}]^{\Sigma_{\tau}}}$${\mathbb{Z}[q^{\pm}]^{\Sigma_{\tau}}}$${\cdots}$$\scriptstyle{q}$$\scriptstyle{q}$$\scriptstyle{q^{-1}}$$\scriptstyle{q}$$\scriptstyle{q^{-1}}$$\scriptstyle{q}$$\scriptstyle{q^{-1}}$$\scriptstyle{q^{-1}}$

  4. (4)

The morphism $i\colon\underline{S}\to\underline{M}$ is as follows. For any
$j\in\mathbb{Z}$, the map $i_{j}\colon S_{j}\to M_{j}$ is defined as

$i_{j}(X)=\operatorname{HN}_{\tau}(X).$

With this definition, $i$ is $\Theta$-equivariant (i.e., a morphism) because

$\operatorname{HN}_{\tau}(X[1])=q\cdot\operatorname{HN}_{\tau}(X).$

  5. (5)

Finally, for each $j\in\mathbb{Z}$, the map $h_{j}\colon
M_{j}\to\mathbb{Z}[q^{\pm}]^{\Sigma_{\tau}}$ is just the identity map. The
compatibility condition

$\operatorname{HN}_{\tau}(X)=h_{j}(i_{j}(X))$

is obvious.

######  Remark B.6.

The example above merely serves to illustrate the definition and does not
compute anything interesting. However, appropriately engineered
Harder–Narasimhan automata can be extremely useful for calculations,
particularly for results about dynamics of autoequivalences in the category.
An example of a computationally useful HN automaton is described in Section
3.1 of this paper. Other examples can be found in [BDL20].

We also have the analogue of a _mass automaton_ , as described in [BDL20,
Section 3].

######  Definition B.7.

Let $q$ be a fixed positive real number. A $\tau$-mass automaton is analogous
to a $\tau$-HN automaton, with the definition modified as follows. The maps
$h_{v}\colon M_{v}\to\mathbb{Z}[q^{\pm}]^{\Sigma_{\tau}}$ are replaced by
linear maps $m_{v}\colon M_{v}\to\mathbb{R}$, satisfying

$m_{q,\tau}(s)=m_{v}(i_{v}(s)).$

In particular, given an HN automaton for a stability condition $\tau$ and a
fixed $q>0$, we can produce a $\tau$-mass automaton by composing the HN vector
with the $q$-mass function.

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