# Mixed Dimer Configuration Model in Type $D$ Cluster Algebras II: Beyond the
Acyclic Case

Libby Farrell, Gregg Musiker, and Kayla Wright

###### Abstract

This is a sequel to the second and third author’s Mixed Dimer Configuration
Model in Type $D$ Cluster Algebras where we extend our model to work for
quivers that contain oriented cycles. Namely, we extend a combinatorial model
for $F$-polynomials for type $D_{n}$ using dimer and double dimer
configurations. In particular, we give a graph theoretic recipe that describes
which monomials appear in such $F$-polynomials, as well as a graph theoretic
way to determine the coefficients of each of these monomials. To prove this
formula, we provide an explicit bijection between mixed dimer configurations
and dimension vectors of submodules of an indecomposable Jacobian algebra
module.

##  1 Introduction

After the positivity conjecture for the coefficients of Laurent polynomials
for cluster variables was resolved in [CLS15, GHKK18], many researchers have
worked on trying to provide combinatorial interpretations for these
coefficient sequences. Many particular classes of cluster algebras have been
studied with this goal in mind. For example, cluster algebras coming from
surfaces, first defined by [FST06], have combinatorial interpretations for the
cluster variables [MS10, MSW11]. In their model, each cluster variable has an
associated graph called a snake graph and then the cluster expansion is given
by some weighted generating function indexed over matchings on the snake
graph. Since defining these snake graphs for cluster algebras from surfaces,
many others have studied when such a construction holds in other contexts. For
example, snake graphs have been defined for cluster-like algebras called
quasi-cluster algebras arising from non-orientable surfaces [Wil19], cluster
algebras from unpunctured orbifolds [BK+20] and super cluster algebras
[MOZ22].

In this paper, we return to a classical case - cluster algebras of type
$D_{n}$ or in other words, cluster algebras arising from a once-punctured
$n$-gon. There are a few reasons that this is of interest. Firstly, there are
some limitations to the snake graph formulation of cluster algebras from
surfaces. For example, oftentimes there are nontrivial coefficients in the
Laurent expansions for cluster variables that are given by the Euler
characteristic of the quiver Grassmannian that are not recorded in a single
snake graph. In addition to this, there are deep representation theoretic
connections with the lattice of perfect matchings of snake graphs and the
submodule lattice of a fixed indecomposable representation representation of
the quiver of a given dimension vector that could use further exploring. For
instance, a connection to between these lattices and the weak Bruhat order was
recently discovered in [CS21].

Another point of interest is that shedding light on the $D_{n}$ case can help
us to study cluster algebras associated to coordinate rings of the
Grassmannian. In these cluster algebras, a finite subset of cluster variables
given by Plücker coordinates, admit a graphical combinatorial interpretations.
More specifically, one can associate a planar bicolored graph embedded in a
disk known as a plabic graph and the Laurent expansion is given by (almost)
perfect matchings of this plabic graph as in [Lam15, MS16, MS17, BKM16].
However, combinatorial interpretations beginning with examples such as cluster
algebras associated to the Grassmannian of $3$-planes in $6$-space, which is
actually the type $D_{4}$ cluster algebra, still lack such interpretations.

With the above as our inspiration, we further explore the connection between
dimer configurations, representation theory and cluster algebras with the goal
of providing a combinatorial interpretation for Laurent expansions of cluster
variables that utilizes a mixture of dimer configurations and double dimer
configurations. We ramp up previous work on quivers of type $D_{n}$ to the
more complicated representation theoretic setting of allowing cycles in our
quiver. More specifically, we focus on a single and double dimer configuration
interpretation of the F-polynomial associated to a cluster variable or module
over the associated Jacobian algebra. We provide a weighted generating
function in terms of dimers and double dimers on a certain graph to give the
F-polynomial for a particular cluster variable. We obtain the exact monomials
by creating a bijection between these dimers and particular dimension vectors
of submodules of a fixed indecomposable Jacobian algebra module and the
coefficients are given by an Euler characteristic of the space of possible
submodules with this given dimension vector.

We begin this paper by reviewing cluster algebras from surfaces and
representations of quivers in type $D_{n}$ in Section 2. We then describe our
dimer theoretic interpretation of the F-polynomial in Section 3 and our main
result is Theorem 4.1 found in Section 4. Our results depend on a
classification of the possible crossing vectors that can appear in type
$D_{n}$ cluster algebras (which are notably no longer in bijection with
positive roots when the quiver contains an oriented cycle). For a full catalog
of such vectors, see Appendix A.

Acknowledgements: The authors would like to thank Aaron Chan for helpful
insights into the representation theoretic side of our paper as well as the
support of the NSF, grants DMS-1745638 and DMS-1854162.

##  2 Preliminaries

This section split into three subsections: Section 2.1 which discusses the
surface model for type $D_{n}$ cluster algebras i.e. tagged triangulations of
once-punctured $n$-gons and their associated type $D_{n}$ cluster algebras,
Section 2.2 which defines the $F$-polynomial associated to a cluster variables
and Section 2.3 which defines the relevant representation theoretic notions we
will need throughout the paper.

###  2.1 Cluster Algebras from Punctured Surfaces

In this section, we give a brief review of the cluster algebra structure on
triangulated surfaces as defined by Fomin, Shapiro and Thurston in [FST06].
Since our focus is type $D_{n}$ cluster algebras, our exposition in this
section will emphasize the role of punctures following both [FST06] and
[LF16]. More specifically, we will focus on the once-punctured disk; a helpful
exposition of material for the single puncture case is given in [DE21].

######  Definition 2.1.

A marked surface is a pair $(S,M)$ where $S$ is a connected oriented Riemann
surface and $M$ is a finite set of marked points such that there is at least
one marked point on every boundary component of $S$. If a marked point is on
the interior of $S$, we call it a puncture.

######  Definition 2.2.

An arc $\gamma$ is a curve in $S$, considered up to isotopy, such that

  1. 1.

its endpoints are in $M$;

  2. 2.

$\gamma$ is disjoint from $\partial S$, except for possibly its endpoints;

  3. 3.

$\gamma$ does not cut an unpunctured monogon or an unpunctured bigon.

We say that two arcs $\gamma,\delta$ are compatible if, up to isotopy, they do
not intersect. More formally, let $e(\gamma,\delta)$ denote the minimal number
of crossings between any isotopic representative of $\gamma$ and $\delta$.
Then $\gamma$ and $\delta$ are compatible if $e(\gamma,\delta)=0$. Maximal
sets of pairwise compatible arcs are called ideal triangulations of $(S,M)$.
For an example, see the right side of Figure 1 for an ideal triangulation of
the once-punctured pentagon. The elements of an ideal triangulation $T$,
called ideal triangles, may not always have three distinct sides. Namely, we
may have the case of triangles called self-folded triangles. We refer to the
inner arc of a self-folded triangle as a radius and the outer edge wrapping
around the puncture enclosing the radius as a loop. In Figure 1, the arcs 4
and 5 create a self-folded triangle, where 5 is the radius and 4 is the loop.

Ideal triangulations are connected by “quadrilateral flips;” where the
quadrilateral flip for any arc $\gamma\in T$ is given by the unique arc
$\gamma^{\prime}\neq\gamma$ that completes $T\setminus\\{\gamma\\}$ to a
triangulation. For example, the resulting triangulation on the right in Figure
2 is the mutation of arc 3 in Figure 1. This “quadrilateral flip” provides the
cluster structure on triangulations of surfaces. In order to define the
cluster structure, we associate a directed graph called a quiver to a
triangulation as follows.

Figure 1: The quiver associated to a triangulation of the once-punctured
pentagon.

######  Definition 2.3.

Let $T=\\{\tau_{1},\tau_{2},\dots,\tau_{n}\\}$ be an ideal triangulation of
$(S,M)$, we define a quiver $Q_{T}=(Q_{0},Q_{1})$ with the vertex set
$Q_{0}=\\{1,2,\dots,n\\}$ such that each vertex $i$ is in correspondence with
the arcs $\tau_{i}\in T$ and with arrow set $Q_{1}$ drawn in clockwise order
for each ideal triangle of $T$. Given a self-folded triangle with radius $r$
and loop $\ell$, draw arrows connecting $r$ and $\ell$ to the adjacent arc(s),
but not connecting $r$ to $\ell$.

To each arc $\tau_{i}$ in a triangulation $T$, we can associate an
indeterminate $x_{i}$ to arc $\tau_{i}\in T$. Let
$X_{T}=\\{x_{1},\dots,x_{n}\\}$ associated to a triangulation $T$, then we
call $(X,Q)=(X_{T},Q_{T})$ a seed.

Figure 2: The mutation in direction 3 of quiver and triangulation from Figure
1.

######  Example 2.1.

Consider the once-punctured pentagon triangulation on the right of Figure 1.
The associated quiver is given by the quiver on the left of Figure 1.

######  Definition 2.4.

For each $1\leq k\leq n$, define the mutation in direction $k$ of the seed
$(X,Q)$ by
$\mu_{k}(X,Q)=((X\setminus\\{x_{k}\\})\cup\\{x_{k^{\prime}}\\},\mu_{k}(Q))$
where $x_{k^{\prime}}$ is a rational function in $X\setminus\\{x_{k}\\}$ and
$\mu_{k}(Q)$ is another quiver on $n$ vertices. More concretely,

  * •

$x_{k^{\prime}}x_{k}=\displaystyle\prod_{i\to k\in Q_{1}}x_{i}+\prod_{k\to
j\in Q_{1}}x_{j}$;

  * •

$\mu_{k}(Q)$ is obtained from transforming $Q$ by the following three step
process:

    * –

For any arrow incident to $k$, reverse the orientation;

    * –

For any two path $i\to k\to j$, draw an arrow $i\to j$;

    * –

Delete any created 2-cycles.

######  Example 2.2.

For the quiver and triangulation from Example 2.1, if we mutate the quiver in
direction 3, we obtain the quiver shown in Figure 2. Moreover,
$x_{3^{\prime}}=\dfrac{x_{1}x_{4}x_{5}+x_{2}}{x_{3}}$.

######  Definition 2.5.

Let $\mathcal{X}$ denote the union of all clusters obtainable by a sequence of
mutations starting from a fixed initial seed $(X,Q)$. The cluster algebra
$\mathcal{A}(X,Q)$ is the algebra generated by $\mathcal{X}$ over some ground
ring $R$ i.e. $\mathcal{A}(X,Q)=R[\mathcal{X}]$.

One of the first remarkable properties proven about cluster algebras is the
Laurent Phenomenon.

######  Theorem 2.1.

[FZ02, Theorem 3.1] Any cluster variable $x\in\mathcal{A}$ can be expressed as

$x=\frac{N(x_{1},\dots,x_{n})}{x_{1}^{d_{1}}\cdots x_{n}^{d_{n}}}$

where $N(x_{1},\dots,x_{n})\in\mathbb{Z}[x_{n+1}^{\pm 1},\dots,x_{2n}^{\pm
1}][x_{1},\dots,x_{n}]$ and is not divisible by any $x_{i}$. The denominator
vector of $x$ is the vector $\underline{d}=(d_{1},\dots,d_{n})$.

Cluster algebras from surfaces have topological formulations that are helpful
for understanding the algebra. Let $(S,M)$ be an unpunctured marked surface
and let $T$ be a triangulation of $(S,M)$. Let $Q_{T}$ be the quiver
associated to the triangulation $T$ and let $X_{T}$ be the initial cluster
associated to $T$. In the cluster algebra
$\mathcal{A}=\mathcal{A}(X_{T},Q_{T})$, there are bijections

$\\{\text{cluster variables of }\mathcal{A}\\}\longleftrightarrow\\{\text{arcs
of }(S,M)\\}$ $\\{\text{clusters of
}\mathcal{A}\\}\longleftrightarrow\\{\text{triangulations of }(S,M)\\}.$

Moreover, let $\gamma\in T$ be an internal arc that is not in a self-folded
triangle and let $\gamma^{\prime}$ be the arc obtained by flipping $\gamma$ in
$T$. Then, cluster mutation $\mu_{\gamma}(X_{T})$ in direction $\gamma$ is
compatible with flip $\mu_{\gamma}(T)$ in a quadrilateral of the arc $\gamma$,
that is,

$\mu_{x_{\gamma}}(X_{T})=(X_{T}\setminus\\{x_{\gamma}\\})\cup\\{x_{\gamma^{\prime}}\\}$

corresponds to

$\mu_{\gamma}(T)=(T\setminus\\{\gamma\\})\cup\\{\gamma^{\prime}\\}.$

After observing the above correspondence, when $(S,M)$ is unpunctured, all
triangulations are connected by flips and arcs are in bijection with cluster
variables of the associated cluster algebra. However, the above correspondence
becomes more complicated if $(S,M)$ contains a puncture. For example, the
radii of self-folded triangles are not mutable and as a consequence, our set
of arcs is not in direct bijection with cluster variables in the associated
cluster algebra. To deal with these complications, Fomin, Shapiro, and
Thurston introduced a decoration one can place on arcs called a tag that
allows us to fix the problem of having immutable arcs. When one of the
endpoints of an arc is a puncture, we can choose to decorate it with a tag
$\Bowtie$ or leave it plain. We let $\gamma^{\Bowtie}$ denote the tagged
version of the arc $\gamma$ which gives the tagged versions of self-folded
triangles, see Figure 3. Since we are only considering a once-punctured
surface, the only arcs the can be tagged are arcs with one endpoint the unique
puncture; that is, an arc whose endpoints are not the puncture are equal to
their plain version.

Figure 3: A self-folded triangle (on the left) corresponds to a tagged
triangle (in the middle). A tagged arc can be interpreted as a “lollipop” as
shown on the right.

In order to define a tagged triangulation, we must define what it means for
two arcs to be compatible with this new decoration. We define this notion
specifically for the case of a once-punctured surface.

######  Definition 2.6.

Let $\gamma,\delta$ be two (possibly tagged) arcs. Then define the crossing
number $e^{\Bowtie}(\gamma,\delta)$ as follows:

  1. 1.

if both arcs are plain, then $e^{\Bowtie}(\gamma,\delta)=e(\gamma,\delta)$;

  2. 2.

if both are tagged, then $e^{\Bowtie}(\gamma,\delta)=0$;

  3. 3.

if exactly one is tagged, say $\gamma$, then
$e^{\Bowtie}(\gamma,\delta)=e(\ell,\delta)$ where $\ell$ is the loop around
the puncture.

With that, we say two (possibly tagged) arcs $\gamma,\delta$ are compatible if
$e^{\Bowtie}(\gamma,\delta)=0$. A maximal set of pairwise compatible tagged
arcs form a tagged triangulation.

With these definitions, we are now able to state the following result:

######  Theorem 2.2.

[FST06] Let $(S,M)$ be any marked surface except a once-punctured surface with
empty boundary. Let $T$ be a triangulation of $(S,M)$ and $Q_{T}$ its
associated quiver. Let $X_{T}$ be the initial cluster associated to $T$, in
$\mathcal{A}=\mathcal{A}(X_{T},Q_{T})$, there are bijections

$\\{\text{cluster variables of
}\mathcal{A}\\}\longleftrightarrow\\{\text{tagged arcs of }(S,M)\\}$
$\\{\text{clusters of }\mathcal{A}\\}\longleftrightarrow\\{\text{tagged
triangulations of }(S,M)\\}.$

Using this theorem, we can define a vector associated to each tagged arc of
$(S,M)$ that will index our cluster variables.

######  Definition 2.7.

Let $T=\\{\tau_{1},\dots,\tau_{n}\\}$ be a triangulation of a marked surface
$(S,M)$ and let $\gamma$ be an arc not in $T$. The crossing vector associated
to $\mathbf{\gamma}$, denoted $\underline{d}_{\gamma}\in\mathbb{Z}_{\geq
0}^{n}$, is the vector defined by $d_{i}=e^{\Bowtie}(\tau_{i},\gamma)$ for
$1\leq i\leq n$. That is, $\underline{d}_{\gamma}$ is the vector that records
the number of times $\gamma$ crosses the arcs of $T$.

######  Remark 2.1.

When $Q_{T}$ is acyclic i.e. there are no internal triangles in the
triangulation $T$, the crossing vector and denominator vector are the same.
However, when $Q_{T}$ contains an oriented cycle, these vectors may not
coincide. For example, see Figure 21 in [FST06].

In this paper, we will be use initial triangulations that are ideal - as it
makes the representation theory simpler. However, in light of the above
theorem, we make sure to examine the full set of cluster variables by also
considering tagged arcs. We will make sure these cases are always explicitly
stated.

In order to properly analyze all type $D_{n}$ cluster algebras, we heavily
rely on Vatne’s classification of all type $D_{n}$ quivers give in [Vat10].
Note that we will take these notations for the remainder of the paper.

######  Theorem 2.3.

[Vat10] When the underlying quiver is mutation-equivalent to the type $D_{n}$
Dynkin quiver, there are four forms the quiver can take. Namely, all type
$D_{n}$ quivers must be of types I, II, III and IV as shown in Figure 4, where
the subquivers labeled $Q,Q^{\prime},Q^{\prime\prime},\dots,Q^{(k)}$ are type
$A_{m}$ quivers (that need not by acyclic). In type I, the arrows between $a$
and $c$ and between $b$ and $c$ can be oriented in either direction.

We use this classification and the correspondence between cluster variables,
(possibly tagged) arcs and crossing vectors as initial data in our
combinatorial model. To this end, we cataloged all families of arcs that can
appear on the once-punctured disk in Appendix A. The catalog is an exhaustive
list of all the crossing vectors that can appear in each type, with respect to
Vatne’s classification, of any ideal initial triangulation of the once-
punctured disk. The Appendix boils down to understanding when crossing vectors
are fully supported on oriented cycles on the quiver and which vectors
degenerate to the acyclic case. Moreover, we emphasize which crossing vectors
are composed of 0’s and 1’s and which crossing vectors have 2’s as this
distinction complicates both the combinatorics and representation theory, as
we will be associating representations to each crossing vectors by insisting
that the dimension vector of the representation and the crossing vector
coincide.

Figure 4: Vatne’s classification of the four types of quivers in type $D_{n}$.

###  2.2 Principal Coefficients

Each crossing vector indexes a cluster variable associated to an arc $\gamma$.
In this paper, we will give a combinatorial interpretation of the
$F$-polynomial associated to a cluster variable $x_{\gamma}$ i.e. indexed by
some vector $\underline{d}_{\gamma}$. In order to do this, we need to
associate coefficients to our cluster algebra by adding what are called frozen
vertices to the associated quiver. These are additional vertices of the quiver
that record the dynamics of mutation without being mutable themselves. This
follows the definitions and results about cluster algebras with coefficients
as in [FZ07].

######  Definition 2.8.

A framed (principal extension) quiver $Q$ is a quiver on $2n$ vertices where
there are $n$ mutable vertices $\\{1,\dots,n\\}$ and $n$ frozen vertices
$\\{1^{\prime},\dots,n^{\prime}\\}$ such that each $i^{\prime}\to i$ and no
frozen vertices are connected in any other way.

Once we have the definition of a framed quiver, we associate an indeterminate
$x_{i}$ to each mutable vertex $i\in Q_{0}$ and another indeterminate $y_{j}$
to each frozen vertex $j^{\prime}\in Q_{0}$. Each set of $2n$ indeterminates
form a cluster $\tilde{X}=\\{x_{1},\dots,x_{n},y_{1},\dots,y_{n}\\}$. The pair
$(\tilde{X},Q)$ is called an extended seed.

######  Remark 2.2.

The choice of frozen vertices in the quiver gives rise to cluster algebras
with principal coefficients. We take this definition for our purposes in order
to most conveniently define the $F$-polynomial to an associated cluster
variable.

To define the $F$-polynomial, we need to define the variables the polynomial
is in. To this end, fix an initial seed $(X,Q)$, define the new variables for
$1\leq j\leq n$

$\hat{y_{j}}=x_{n+j}\prod_{i=1}^{n}x_{i}^{\\#\\{i\to j\in Q_{1}\\}}.$

Using these new variables, we are ready to define the $F$-polynomial and also
$g$-vectors. Although we state it as a definition, the following is also a
theorem that cluster variables with principle coefficients have this form.

######  Definition 2.9.

[FZ07, Definition 3.3/Proposition 7.8] Let $1\leq\ell\leq n$, there exists a
unique primitive polynomial $F_{\ell}\in\mathbb{Z}[u_{1},\dots,u_{n}]$ and a
unique vector $\underline{g_{\ell}}=(g_{1},\dots,g_{n})\in\mathbb{Z}^{n}$ such
that the cluster variable $x\in A(\tilde{X},Q)$ is given by

$x=F_{\ell}(\hat{y_{1}},\dots,\hat{y_{n}})x_{1}^{g_{1}}\cdots x_{n}^{g_{n}}.$

The polynomial $F_{\ell}$ is called an $\mathbf{F}$-polynomial and
$\underline{g_{\ell}}$ is called a $\mathbf{g}$-vector.

###  2.3 The Jacobian Algebra and Quivers with Potential Representations

In this section, we review some representation theory that will dictate the
behavior of our combinatorial model for the $F$-polynomial. Namely, we will
define the Jacobian algebra of a triangulation of a punctured surface, first
defined by Labardini-Fragoso in [LF09]. For this section, assume that $\Bbbk$
is an (algebraically closed) field.

######  Definition 2.10.

Recall that we denote $Q=(Q_{0},Q_{1})$ to be a quiver where $Q_{0}$ is the
set of vertices of $Q$ and $Q_{1}$ is the set of arrows of $Q$. For
$i\xrightarrow{\alpha}$ $j$ in $Q_{1}$, let $s(\alpha)=i$ denote the source of
the arrow $\alpha$ and let $t(\alpha)=j$ denote the target of $\alpha$. Let
$\alpha_{0},\dots,\alpha_{p}\in Q_{1}$. A path in $Q$ is a sequence
$\alpha_{0}\cdots\alpha_{p}$ where $t(\alpha_{i})=s(\alpha_{i+1})$ for all
$0\leq i<p$.

######  Definition 2.11.

Let $\epsilon_{i}$ denote the path of length 0 at vertex $i$. Let
$\alpha=\alpha_{0}\cdots\alpha_{p}$ denote a path of length $p$ and
$\beta=\beta_{0}\cdots\beta_{q}$ denote a path of length $q$. The
concatenation of paths $\alpha\epsilon_{i}$ is given by

$\alpha\epsilon_{i}=\begin{cases}\alpha&\text{if }t(\alpha_{p})=i\\\
0&\text{otherwise}\end{cases}$

Similarly, the concatenation of $\epsilon_{i}\alpha$ is given by

$\epsilon_{i}\alpha=\begin{cases}\alpha&\text{if }s(\alpha_{0})=i\\\
0&\text{otherwise}\end{cases}$

In general, the concatenation of $\alpha\beta$ is given by

$\alpha\beta=\begin{cases}\alpha_{0}\dots\alpha_{p}\beta_{0}\dots\beta_{q}&\text{if
}t(\alpha_{p})=s(\beta_{0})\\\ 0&\text{otherwise}\end{cases}$

######  Definition 2.12.

The path algebra of $Q$, denoted $\Bbbk Q$ is the $\Bbbk$-algebra with basis
paths in $Q$, including paths of length 0, and multiplication given by
concatenation of paths.

######  Example 2.3.

Consider the following quiver:

${1}$${2}$${4}$${3}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\delta}$$\scriptstyle{\gamma}$

Then the path algebra $\Bbbk Q$ will be an infinite-dimensional algebra
because of the oriented cycle $\alpha\beta\gamma\delta$. The basis will
consist of elements such as

$\\{\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4},\alpha,\beta,\gamma,\delta,\alpha\beta,\beta\gamma,\gamma\delta,\delta\alpha,\alpha\beta\gamma,\beta\gamma\delta,\gamma\delta\alpha,\delta\alpha\beta,$
$\alpha\beta\gamma\delta,\beta\gamma\delta\alpha,\gamma\delta\alpha\beta,\delta\alpha\beta\gamma,\alpha\beta\gamma\delta\alpha,\beta\gamma\delta\alpha\beta,\gamma\delta\alpha\beta\gamma,\delta\alpha\beta\gamma\delta,\dots\\}$

######  Definition 2.13.

Let $R_{Q}$ be the ideal of $\Bbbk Q$ generated by the arrows of $Q$. More
generally, denote by $R_{Q}^{m}$ the ideal of the path algebra $\Bbbk Q$
generated by paths of length $m$ in $Q$. A two-sided ideal $I$ of $A$ is
admissible if there exists $m\geq 2$ such that $R_{Q}^{m}\subseteq I\subseteq
R^{2}_{Q}$. If $Q$ is a quiver and $I$ is an admissible ideal of $\Bbbk Q$,
then the pair $(Q,I)$ is a bound quiver.

######  Example 2.4.

Let $(S,M)$ be an unpunctured marked surface and consider a triangulation $T$
of $(S,M)$. It is possible to associate an admissible ideal to the
triangulation $T$, [LF09, ABCJP10]. Let $I_{T}$ to be the ideal of $\Bbbk Q$
generated by all $2$-paths $\alpha\beta$ such that $s(\alpha)=t(\beta)$, but
that $\alpha$ and $\beta$ arise from two different marked points in $(S,M)$;
this ideal is admissible. On the level of the quiver $Q_{T}$, the ideal
consists of all $2$-paths in $Q_{T}$ coming from the same triangle in $T$.

The admissible ideal referred to in Example 2.4 comes from a potential
associated to a triangulation of a surface. We now define a potential of a
quiver that arises from a triangulation of a surface.

######  Definition 2.14.

Let $P=\alpha_{0}\cdots\alpha_{p}$ be a path in $Q$. We say that $P$ is a
cycle when $s(\alpha_{0})=t(\alpha_{p})$.

######  Definition 2.15.

A potential $W$ on $Q$ is a possibly infinite linear combination of cycles in
$\Bbbk Q$. We call the pair $(Q,W)$ a quiver with potential.

######  Definition 2.16.

Two potentials $W$ and $W^{\prime}$ on $Q$ are cyclically equivalent when
$W-W^{\prime}$ lies in the closure of the span of all elements of the form
$\alpha_{1}\cdots\alpha_{p}-\alpha_{2}\cdots\alpha_{p}\alpha_{1}$, where
$\alpha_{1}\cdots\alpha_{p}$ is a cycle.

######  Definition 2.17.

When a quiver comes from a triangulation $T$ of a once-punctured surface,
define the potential associated to $\mathbf{T}$ by

$W(T)=\sum_{\Delta}W^{\Delta}+W^{P},$

where the sum is taken over $\Delta$ \- a clockwise oriented triangle in the
quiver i.e. an internal triangle in $T$; $W^{\Delta}$ is the potential given
by the cycle created by the triangle $\Delta$; and $W^{P}$ is the potential
given by the counterclockwise cycle around the puncture P as in [LF09].

Figure 5: A triangulation with overlayed arc $\gamma$, quiver with potential
and an the indecomposable quiver with potential representation $M_{\gamma}$.

######  Example 2.5.

Consider the ideal triangulation $T$ of the punctured pentagon and its
associated quiver $Q_{T}$ in Figure 5. The potential is given by
$W(T)=bcd+fgh+eij+def$, where $W^{\Delta}=bcd+fgh+eij$ which corresponds to
the internal triangles formed by the arcs 2,4 and 3; 4,6 and 5; and 3,5 and 7
respectively. Additionally, $W^{P}=def$ corresponds to the counterclockwise
cycle 3,4 and 5 coming from the puncture.

######  Definition 2.18.

Let $C=\alpha_{0}\cdots\alpha_{p}$ be a cycle in $Q$. Then the formal cyclic
derivative of $C$ with respect to $\alpha_{i}$ is

$\displaystyle\partial_{\alpha_{i}}(C)=\sum_{\alpha_{j}=\alpha_{i}}\alpha_{j+1}\cdots\alpha_{p}\alpha_{1}\cdots\alpha_{j-1}.$

The cyclic derivative extends linearly on linear combinations of cycles in
$Q$. Note that the cyclic derivative of two cyclically equivalent potentials
$W$ and $W^{\prime}$ on $Q$ are equal.

We are now ready to define a representation of a quiver with potential.

######  Definition 2.19.

Let $(Q,W)$ be a quiver with potential. A representation $M$ of $(Q,W)$ is a
pair $M=(M_{a},M_{\varphi_{\alpha}})$ where

  * •

$M_{a}$ is an assignment of $\Bbbk$-vector space $M_{a}$ to each a vertex
$a\in Q_{0}$;

  * •

$M_{\varphi_{\alpha}}$ is an assignment of a $\Bbbk$-linear map
$\varphi_{\alpha}$ to each arrow $\alpha\in Q_{1}$ such that for each cyclic
derivative $\partial_{a_{i}}$ of $W$, we insist the composition of the maps in
the cyclic derivative is 0.

Now that we have defined representations of quivers with potential, we re-
contextualize this setting in terms of the Jacobian algebra.

######  Definition 2.20.

Let $W$ be a potential on $Q$. The Jacobian ideal of $W$ is the ideal
generated by all cyclic derivatives $\partial_{\alpha}(W)$ for $\alpha\in
Q_{1}$.

With that, we define a quotient of the path algebra by this ideal.

######  Definition 2.21.

Let $I$ be the Jacobian ideal of a potential $W$ in $Q$. The Jacobian algebra
is the quotient $\Bbbk Q/I$. If $W$ is a potential consisting of a sum of all
cycles in $Q$ (up to cyclic equivalence), we refer to the corresponding
Jacobian algebra as the Jacobian algebra of $Q$.

Both Definition 2.20 and 2.21 are well-defined up to cyclic equivalence. This
is because the cyclic derivative of two cyclically equivalent potentials $W$
and $W^{\prime}$ on $Q$ are equal; hence giving that the Jacobian ideals of
$W$ and $W^{\prime}$ are also equal. Moreover, since the Jacobian ideals of
cyclically equivalent potentials $W$ and $W^{\prime}$ on $Q$ are equal, the
Jacobian algebras of $W$ and $W^{\prime}$ are also equal.

######  Remark 2.3.

A quiver with potential $(Q,W)$ is a bound quiver $(Q,I)$ using the admissible
ideal $I$ generated by cyclic derivatives (with respect to each arrow of
$Q_{1}$) of the potential $W$. Note that this ideal is admissible since each
cycle in $W$ has positive length, and therefore, the ideal generated by a
combination of these cycles must have bounded length.

######  Example 2.6.

Consider the quiver from Example 2.3. Endow this quiver with the potential
$W=\alpha\beta\gamma\delta$. The Jacobian ideal is given by

$I=\langle\partial_{\alpha}(\alpha\beta\gamma\delta),\partial_{\beta}(\alpha\beta\gamma\delta),\partial_{\gamma}(\alpha\beta\gamma\delta),\partial_{\delta}(\alpha\beta\gamma\delta)\rangle=\langle\beta\gamma\delta,\gamma\delta\alpha,\delta\alpha\beta,\alpha\beta\gamma\rangle.$

The Jacobian algebra $\Bbbk Q/I$ is a finite-dimensional algebra with basis

$\\{\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4},\alpha,\beta,\gamma,\delta,\alpha\beta,\beta\gamma,\gamma\delta,\delta\alpha\\}.$

######  Definition 2.22.

Let $\Bbbk Q/I$ be the Jacobian algebra of $Q$. A module
$M=(M_{a},M_{\varphi_{\alpha}})$ over $\Bbbk Q/I$ is given by two pieces of
data

  * •

$M_{a}$ is an assignment of $\Bbbk$-vector space $M_{a}$ to each a vertex
$a\in Q_{0}$;

  * •

$M_{\varphi_{\alpha}}$ is an assignment of a $\Bbbk$-linear map
$\varphi_{\alpha}$ to each arrow $\alpha\in Q_{1}$ such that for each relation
$\alpha_{1}\cdots\alpha_{r}\in I$, we insist
$\varphi_{\alpha_{r}}\cdots\varphi_{\alpha_{1}}=0$.

We say such a module is indecomposable if it cannot be expressed as the direct
sum of proper submodules. Studying indecomposable Jacobian algebra modules is
equivalent to studying representations of the quiver with potential. More
precisely, we have the following equivalence of categories:

######  Theorem 2.4.

[ASS+06], Theorem 1.6: Let $A=\Bbbk Q/I$, where $Q$ is a finite connected
quiver and $I$ an admissible ideal of $\Bbbk Q$. There exists an equivalence
of categories between modules over $A$ and finite-dimensional representations
of the bound quiver $(Q,I)$.

######  Example 2.7.

Consider the Jacobian algebra computed in Example 2.6. An example of an
indecomposable Jacobian algebra module is given by

${\Bbbk}$${\Bbbk}$${0}$${\Bbbk}$$\scriptstyle{1}$$\scriptstyle{1}$$\scriptstyle{0}$$\scriptstyle{0}$

Note that the assignment of linear maps gives that a path given by the
composition of any three consecutive arrows is 0 which is forced by the
relations in the Jacobian ideal.

######  Example 2.8.

Consider another quiver $Q$ shown below:

${3}$${5}$${1}$${2}$${4}$$\scriptstyle{\delta}$$\scriptstyle{\beta}$$\scriptstyle{\nu}$$\scriptstyle{\epsilon}$$\scriptstyle{\alpha}$$\scriptstyle{\gamma}$

Then define the potential $W=\alpha\beta\delta+\epsilon\delta\gamma$. Taking
cyclic derivatives of $W$, we obtain the Jacobian ideal
$I=\langle\beta\delta,\delta\alpha,\delta\gamma,\epsilon\delta,\alpha\beta+\gamma\epsilon\rangle$.
An example of an indecomposable module over the Jacobian algebra $\Bbbk Q/I$,
or equivalently, a representation of $(Q,W)$ is given by

${\Bbbk}$${\Bbbk}$${\Bbbk}$${\Bbbk^{2}}$${\Bbbk}$$\scriptstyle{0}$$\scriptstyle{1}$$\scriptstyle{(1~{}0)^{T}}$$\scriptstyle{(0~{}1)}$$\scriptstyle{1}$$\scriptstyle{(0~{}1)^{T}}$

Note that in Example 2.8, we needed to make one of the arrows between non-zero
vector spaces the zero map in order for it to satisfy the relations in the
Jacobian ideal. Namely the arrow $3\xrightarrow{\delta}4$ was 0. We call such
an arrow in a Jacobian algebra module singular. Such arrows will become
relevant when defining our dimer model in Section 3.

We now explain how to uniquely associate a quiver with potential or Jacobian
algebra module to an arc in a triangulated surface following [Dom17]. In
particular, we emphasize that an arc overlayed on a triangulation of a surface
uniquely determines an indecomposable Jacobian algebra module.

Let $T$ be an ideal initial triangulation of a surface $(S,M)$ and let
$(Q_{T},W_{T})$ be its associate quiver with potential. Overlay a plain
oriented arc $\gamma\not\in T$ on $(S,M)$ and we explain how to produce a
quiver with potential representation $M_{\gamma}$. The dimension vector of
this representation is simply given by the crossing vector of $\gamma$; that
is,

$\dim(M_{\gamma})_{i}=\\#\\{\text{intersections with }\gamma\text{ with arc
}\tau_{i}\in T\\}/\sim$

where $\sim$ is up to isotopy. The maps are given by comparing the segments of
$\gamma$ between arrows of the quiver. Let $\alpha:i\to k\in Q_{1}$, and let
$r_{1},\dots,r_{s}$ be the intersections of $\gamma$ with $\tau_{i}\in T$ and
let $q_{1},\dots,q_{t}$ be the intersections of $\gamma$ with $\tau_{k}\in T$.
Let $[r_{a},q_{b}]$ denote the segment of $\gamma$ between the intersection
points $r_{a}$ and $q_{b}$. The map $\varphi_{\alpha}$ given by the matrix
$N_{\varphi_{\alpha}}=(n_{a,b})\in\text{Mat}_{s+t,s+t}(\mathbb{Z})$ is defined
by

$n_{b,a}=\begin{cases}1&\text{ if the interior of }[r_{a},q_{b}]\text{ is
contained in one triangle of }T\\\ 1&\text{ if there is a segment
}[q_{b},q_{b^{\prime}}]\text{ that surrounds the puncture clockwise }\\\
&\text{ and the interior of }[r_{a},q_{b^{\prime}}]\text{ is contained in one
triangle of }T\\\ 0&\text{ otherwise}.\end{cases}$

######  Example 2.9.

Consider the triangulation, and quiver with potential $W(T)=bcd+fgh+eij+def$
from Example 2.5 in Figure 5. Its Jacobian ideal is given by

$I_{W}=\langle cd,db,bc+ef,fd_{i}j,de+gh,hf,hg,je,ei\rangle.$

Consider the red arc $\gamma$ overlayed on $T$ with the specified orientation.
It crosses arcs, 2,4,5,3,2 of $T$ in order – giving that the crossing vector
of $\gamma$ or the dimension vector of $M_{\gamma}$ is
$\underline{d}=(0,2,1,1,1,0,0)$. The maps between the one-dimensional vector
spaces are all the identity except the arrow $4\to 3$ because the interior of
the segment $[r_{1},q_{1}]$ is contained in two triangles. This gives that
this arrow is singular.

The arrow $2\to 1$ is given by the matrix $(0~{}1)$ as the segment
$[q_{1},r_{1}]$ is not contained in a single triangle whereas the segment
$[q_{2},r_{1}]$ is. Similarly, the arrow $2\to 4$ is given by the matrix
$(1~{}0)$ as the segment $[q_{1},r_{1}]$ is contained in a single triangle
whereas the segment $[q_{2},r_{1}]$ is not. The arrow $3\to 2$ is given by the
matrix $(1~{}1)^{T}$ as $[r_{1},q_{2}]$ is contained in a single triangle and
$[q_{1},q_{2}]$ surrounds the puncture clockwise and the interior of
$[r_{1},q_{2}]$ is contained in a single triangle. The full representation is
shown on the right of Figure 5.

When $\gamma$ is tagged, the process for producing an indecomposable quiver
with potential representation from $\gamma^{\Bowtie}$ is quite similar to the
above. The dimension vector is given by the crossing vector as before as
defined in Definition 2.6. The maps are slightly modified, imagining replacing
the tagged arc with a lollipop as in the rightmost picture in Figure 3.

The one last piece of representation theory we will need is the notion of the
$F$-polynomial in terms of quiver representations.

######  Theorem 2.5.

[DWZ08] The $F$-polynomial as defined in Definition 2.9 can be expressed in
terms of finite dimensional modules over the Jacobian algebra. Namely, we have
that for any finitely generated module $M$ of the Jacobian algebra that

$F_{M}=\sum_{\underline{e}}\chi(Gr_{\underline{e}}(M))\underline{y}^{\underline{e}}$

where the sum is taken over all dimension vectors $\underline{e}$ indexing
submodules $N$ of $M$ and $\chi(Gr_{\underline{e}}(M))$ is the Euler
characteristic of the quiver Grassmannian of all submodules $N\subset M$ with
dimension vector $\underline{e}$.

##  3 Dimer Configurations

In this section, we describe a combinatorial model for how to obtain the
$F$-polynomial associated to any type $D_{n}$ cluster algebra. Our model
assigns a planar, bipartite graph to a type $D_{n}$ quiver. Each of the
vertices in the graph correspond to a $2k$-gon, we call a tile, where $k$ is
the degree of the vertex in the quiver and each arrow in the quiver gives an
assignment of how to attach the tiles together. We use the data of the quiver
$Q$ and a crossing vector $\underline{d}$ to assign a minimal mixed dimer
configuration to this graph. From this assignment, we create a poset of mixed
dimer configurations each of which corresponds to a monomial of the
$F$-polynomial associated to $\underline{d}$ and whose coefficient is
determined by the number of cycles appearing in the mixed dimer configuration.
Our methods extend our previous model for the acyclic case [MW20], where we
were able to utilize Thao Tran’s work [Tra09], to the non-acyclic case.

Let $Q=(Q_{0},Q_{1})$ be any type $D_{n}$ quiver. Recall there are four types
of quivers that are mutation equivalent to the type $D_{n}$ Dynkin diagram
categorized by Vatne in Figure 4. Let $\mathcal{A}(Q)$ be the associated
cluster algebra. We first define the base graph associated to $Q$.

###  3.1 Defining the Base Graph

######  Definition 3.1.

Let $i\in Q_{0}$ be a vertex of degree $k$ in our quiver. Associate a $2k$-gon
called tile $i$ to this vertex. Each tile in an even-sided polygon, hence it
admits a bipartite coloring. We attach the tiles based on this black and white
coloring with the following convention:

If $i,j\in Q_{0}$ such that $i\to j\in Q_{1}$, attach tiles $i$ and $j$ by the
convention that we “see white on the right.” Call the resulting graph
$\bar{G}$, the uncontracted base graph associated to $Q$.

######  Example 3.1.

Figure 6: A quiver, its uncontracted and contracted base graphs.

Consider the $D_{7}$ quiver on the left of Figure 6. The associated
uncontracted base graph is given by the middle of Figure 6.

######  Remark 3.1.

For an $n$-cycle in $Q$, there is a more efficient process to obtain the
uncontracted base graph. Namely, begin constructing the graph with an
“$n$-star” i.e. $n$ line segments attached at one of their endpoints and use
these edges to create the rest of the tiles for vertices in the $n$-cycle in
$Q$. If the $n$-cycle is clockwise, the vertex of the $n$-star is white. If
the $n$-cycle is counterclockwise, the vertex of the $n$-star is black.

From this graph $\bar{G}$, we create a refinement of this graph via a local
move called double-edge contraction that will result in a graph comprised of
only squares and hexagons.

######  Definition 3.2.

Double-edge contraction is a graph transformation that takes two edges in a
graph $G$ and contracts them to a point. Locally, the transformation is:

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

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

In $\bar{G}$, for any tile $i$ that is a $2k$-gon for $k\geq 3$, perform
double-edge contraction on any edges that are boundary edges i.e. edges that
only belong to tile $i$ and neighbors no other tiles. Repeat this process as
many times as possible. Call the resulting graph $G$ the base graph associated
to $Q$.

######  Example 3.2.

Taking the uncontracted base graph from Example 3.1, we perform double edge
contraction which turns the octagon tile 2 into a square, and the hexagons 3
and 5 into squares. The resultant base graph is illustrated on the right of
Figure 6.

######  Remark 3.2.

Although this refinement of the uncontracted base graph $\bar{G}$ to the base
graph $G$ does not affect the combinatorics of our model, it yields a graph
that is easier to work with in practice.

###  3.2 Minimal Mixed Dimer Configuration

We now associate a minimal mixed dimer configuration to the base graph $G$ of
a quiver $Q$. In order to define the minimal mixed dimer configurations, we
first define single dimer, double dimer, and mixed dimer configurations. We
tie this back to cluster algebras by using another piece of global data - the
crossing vector $\underline{d}$ associated to a cluster variable
$\underline{x}$ in $\mathcal{A}(Q)$. The complete catalog of all crossing
vectors in type $D_{n}$ cluster algebras can be found in Appendix A.

For the following definitions, suppose that $G$ is an arbitrary planar
bipartite graph.

######  Definition 3.3.

A dimer configuration, also known as a (perfect) matching, is a subset
$D\subset E(G)$ such that every vertex $v\in V(G)$ is contained in exactly one
edge $e\in D$.

For example, consider the red edges in the following graph:

######  Definition 3.4.

A double dimer configuration $D^{\prime}$ of $G$ is a multiset of the edges of
$G$ such that every vertex $v\in V(G)$ is contained in exactly two edges
$e,e^{\prime}\in D^{\prime}$.

For example, consider the red edges in the following graph:

######  Definition 3.5.

Let $\underline{d}$ be an $n$-tuple whose entries are each $0$, $1$, or $2$,
i.e. $\underline{d}\in\\{0,1,2\\}^{n}$. A mixed dimer configuration
$M=M(\underline{d)}$ of $G$ is a multiset of the edges of $G$ such that every
vertex $v\in V(G)$ is contained in zero, one, or two edges in $M$.
Furthermore, we say that $M$ satisfies the valence condition with respect to
$\underline{d}$ if

  * •

Each vertex incident to a tile labeled $i$ with $d_{i}=2$ is contained in two
edges in $M$.

  * •

Each vertex incident to a tile labeled $i$ with $d_{i}=1$ is contained in at
least one edge in $M$.

######  Example 3.3.

Let $\underline{d}=(d_{0},d_{1},d_{2},d_{3})=(2,2,1,0)$. An example of a mixed
dimer configuration can be seen by the red edges highlighted in the following
graph:

The crossing vector $\underline{d}$ associated to some arc $\gamma$ in the
once-punctured $n$-gon will be what determines the mixed dimer configurations
in our model. Note that $\underline{d}$ is also the dimension vector of a
unique representative of an isomorphism class of an indecomposable quiver with
potential representation, $M_{\gamma}$.

With this, in order to define the minimal matching, we present lemmas about
the structure of crossing vectors $\underline{d}$ that will imply the well-
definition of our construction. We postpone the proofs of Lemma 3.1 and Lemma
3.2 until Appendix A. For the following lemmas, let $T$ be a triangulation of
the once-punctured $n$-gon and $Q$ be the corresponding quiver.

######  Lemma 3.1.

Suppose $\gamma$ is some arc not in $T$ and let
$\underline{d}=\text{cross}(\gamma)$. $Q^{\text{supp}(\underline{d})}$, the
induced subquiver using vertices $i\in Q_{0}$ with $d_{i}>0$, is connected.

######  Lemma 3.2.

Suppose $\gamma$ is some arc not in $T$ such that there exists some arc
$\tau\in T$ that $\gamma$ crosses twice. Let
$\underline{d}=\text{cross}(\gamma)$. $Q^{\text{supp}_{2}(\underline{d})}$,
the induced subquiver using vertices $i\in Q_{0}$ with $d_{i}=2$, is a
connected tree.

Now, we aim to define a minimal mixed dimer configuration associated to
$(Q,\underline{d})$. Let $G(Q)=G$ be the base graph obtained by the process
described in Definitions 3.1 and 3.2. Let $\gamma$ be an arc such that
$cross(\gamma)=\underline{d}$ and let $M_{\gamma}$ be the corresponding
indecomposable representation.

######  Definition 3.6.

We begin by addressing the nuance in the cyclic case. As we saw in Example
2.8, some of the arrows in $M_{\gamma}$ may be 0 between nonzero vector
spaces. In order to correctly model the combinatorics, our model must consider
these singular arrows in the definition of the minimal matching. For any
singular arrow $i\to j$ in $M_{\gamma}$, enumerate the edge $e_{i,j}$
straddling tiles $i$ and $j$. Define the set of such edges we distinguish
$\displaystyle Z=\cup_{i\to j\text{singular}}e_{i,j}$.

Let $G_{1}$ be the induced subgraph of $G$ using tiles $i$ with $d_{i}\geq 1$.
Traversing clockwise along the boundary of the graph $G_{1}$, distinguish the
edges that go black to white clockwise, call this set of edges $M(G_{1})$. If
$\underline{d}$ contains no 2’s, then define $M(G_{1})\cup
Z=M_{-}(\underline{d})$.

If $\underline{d}$ contains at least one 2, then let $G_{2}$ be the induced
subgraph of $G$ using tiles $i$ with $d_{i}=2$. Traversing clockwise along the
boundary of the graph $G_{2}$, distinguish the edges that go black to white
clockwise, call this set of edges $M(G_{2})$. Define $M(G_{1})\sqcup
M(G_{2})\sqcup Z=M_{-}(\underline{d})$. We refer to $M_{-}(\underline{d})$ as
the minimal matching associated to $(Q,\underline{d}$)

Figure 7: Phases of enumerating the edges in the minimal matching for Example
3.2.

######  Example 3.4.

Taking the base graph from Example 3.2, and $\underline{d}=(1,1,2,1,1,1)$, the
minimal matching $M_{-}(\underline{d})$ is shown in Figure 7. The leftmost
graphic shows the set $Z=e_{5,3}\cup e_{1,0}$, reflecting the singular arrows
$5\to 3$ and $1\to 0$. The middle graphic represents then adding the edges in
$M(G_{1})$. The rightmost figure is the addition of the edges in $M(G_{2})$
i.e. is the minimal matching for this choice of $(Q,\underline{d})$.

Observe that the definition of the minimal matching $M_{-}(\underline{d})$ is
well-defined. By Lemma 3.1, we have that $G_{1}$ is connected; moreover, by
Lemma 3.2, we have that $G_{2}$ is connected. Hence, $M(G_{1}),M(G_{2})$ are
well-defined respectively.

###  3.3 Poset of Mixed Dimer Configurations

We now create the poset of mixed dimer configurations where each of the mixed
dimer configurations appearing in this poset corresponds to a monomial that
appears in the $F$-polynomial associated to $\underline{d}$. Namely, the
minimal element of this poset corresponds to the minimal mixed dimer
configuration defined in Section 3.2 and we define the poset relation as in
Section 3.4 of [MW20]. For more details and motivation, please refer to
[MW20].

######  Definition 3.7.

Fix a type $D_{n}$ quiver $Q$ and a base graph $G=G(Q)$. Let
$\underline{d}\in\\{0,1,2\\}^{n}$ and let $D,D^{\prime}$ be two mixed dimer
configurations on $G$ with valence condition given by $\underline{d}$. We say
$D$ covers $D^{\prime}$, $D\lessdot D^{\prime}$, if there exists a tile $i$ of
$G$ such that $D^{\prime}$ is obtained from $D$ by “flipping” tile $i$ of $D$
i.e. exchanging highlighted edges black to white clockwise on tile $i$ in $D$
to white to black clockwise as shown in Figure 8. Implicit in this definition
is that $d_{i}>0$.

Figure 8:

######  Definition 3.8.

Let $(\bar{P},\leq)$ be the poset of mixed dimer configurations that satisfy
the valence condition and are reachable via a sequence of flips from
$M_{-}(Q,\underline{d})$. For two such mixed dimer configurations $D$ and
$D^{\prime}$, we say that $D\leq D^{\prime}$ if there exists a sequence of
allowable flips from $D$ to obtain $D^{\prime}$.

The poset $\bar{P}$ turns out to have more elements than the number of
monomials in the $F$-polynomial associated to $\underline{d}$. As in [MW20],
we put another condition on mixed dimer configurations that accurately
reflects the cluster combinatorics by disallowing some mixed dimer
configurations. We will let $(P,\leq)\subseteq(\bar{P},\leq)$ be the subposet
of mixed dimer configurations that satisfy the valence condition, are
reachable via a sequence of allowable flips from $M_{-}$, and satisfy a
condition known as being “node monochromatic” we define in the next section.

####  3.3.1 Node Monochromatic Mixed Dimer Configurations

As in the acyclic case, we must define a special set of vertices of $G$ that
we call “nodes” to disallow certain mixed dimer configurations. These nodes
allow us to disallow configurations that connect nodes of different color in
order to correctly model the $F$-polynomial. In order to create paths in a
mixed dimer configuration, we must have valence 2 on some vertices in
$M_{-}(\underline{d})$. Hence, the definition of these nodes is only relevant
when there is at least one 2 present in $\underline{d}$.

Recall Vatne’s classification of type $D_{n}$ quivers, described in Theorem
2.3, which states that any type $D_{n}$ quiver consists of a type $D_{n}$ part
and at least one type $A_{m}$ part. We first define nodes on the type $D_{n}$
part of the quivers. In types I, II and III, we define both red and blue
nodes; whereas in type IV, we just define blue nodes. After this, we define
green nodes on the type $A_{m}$ part of the quiver, which applies to all
types.

######  Remark 3.3.

In the support of $\underline{d}$-vectors that contain a 2, types II and III
degenerate to type I quivers when considering the induced subgraph on the
quiver. In particular, the type $D_{n}$ parts of these quivers are acyclic on
the support of $\underline{d}$. For details of how these types reduce to type
I, refer to the classification of crossing vectors in these types found in
Figures 23 and 24 in Appendix A. So, in types I, II and III, nodes are defined
in the same way and are very similar to the 6 nodes of three colors: red, blue
and green as defined in [MW20]. Namely, two pairs of these nodes are defined
on the type $D_{n}$ part of the quiver, since tiles are disconnected in the
induced base graph of the support of $\underline{d}$.

However, in type IV quivers, the type $D_{n}$ part of the quiver that contains
the central cycle is always fully supported in $\underline{d}$ when there is a
2 in the vector. So, in this case, we must define the nodes in a different
way. We define 4 nodes of two colors: blue and green in this case because the
type $D_{n}$ part of the induced subgraph is connected in this case. Hence,
rather than having two pairs of nodes on the type $D_{n}$ part, we only have
one pair of nodes in the type $D_{n}$ part.

Let $\underline{d}$ be a crossing vector associated to a cluster variable
$\underline{x}$ in $\mathcal{A}(Q)$. Let $Q_{\text{supp}}$ be the induced
subquiver of $Q$ using vertices $i\in Q$ with $d_{i}>0$ i.e. supported on
$\underline{d}$. Suppose that $Q$ is type I, II or III. If $Q_{\text{supp}}$
contains an oriented cycle, then this cycle is necessarily in a type $A_{m}$
part of the model i.e. some $Q^{(\ell)}$ using Vatne’s notation. By the
surface model, the type $D_{n}$ part of the quiver in $Q_{\text{supp}}$
reduces to the case where we have a fork as in the classical Dynkin diagram.
This allows us to define two pairs of nodes: two red nodes
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}u,v}$ and
two blue nodes
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}w,x}$. Let
$a$ and $b$ be the two forking vertices and let $c$ be the vertex that is
connected to the rest of the type $A_{m}$ part of the quiver. Define the red
nodes ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}u,v}$
to be place on the two vertices of tile $a$ that are not shared with tile $c$.
Define the blue nodes
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}w,x}$ to be
place on the two vertices of tile $b$ that are not shared with tile $c$. See
Figure 9.

Figure 9: The blue and red nodes on the tiles $a$ and $b$ in types I, II and
III.

Now, we define the blue nodes in type IV quivers. In the base graph, we create
a $k$-star that corresponds to the vertex shared by all the tiles in the
$k$-cycle. Define the vertex in the $k$-star to be the node
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}w}$. In the
minimal matching, there is a unique vertex that is connected to
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}w}$ using
the edges in $M_{-}$. Namely, the arrow $a\to b$ connecting the central
$k$-cycle to the type $A_{m}$ spike must be a singular arrow - and we define
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}x}$ to be
the vertex on tile $a$ adjacent to
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}w}$ but not
shared by $b$.

Figure 10: The blue nodes on the tiles $a$ and $b$ in type IV.

Finally, to define the last pair of green nodes, we focus on the type $A_{m}$
part of the quiver in any type. Let $i$ be the vertex in $Q_{\text{supp}}$
connected to the maximal number of vertices with $d_{j}=0$ or $1$. If $i$ is
not part of an oriented cycle, we define the green nodes as in the acyclic
case [MW20]. If $i$ is part of an oriented cycle in $Q_{\text{supp}}$, then
define the green nodes
${\color[rgb]{0.5,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{.5}\pgfsys@color@cmyk@fill{0}{0}{1}{.5}y,z}$
as in Figure 11.

Figure 11: The green nodes based on various tiles in or not in
$Q_{\text{supp}}$. The leftmost column shows the case when $j\in
Q_{\text{supp}}$ and $k\not\in Q_{\text{supp}}$; the center column shows the
case when $j\not\in Q_{\text{supp}}$ and $k\in Q_{\text{supp}}$; and the
rightmost column shows the case when $j,k\in Q_{\text{supp}}$.

######  Definition 3.9.

A mixed dimer configuration $M$ of $G$ is node-monochromatic if any path
consisting of edges in $M$ between nodes connects nodes of the same color. If
there exists a path consisting of edges in $M$ between nodes of different
colors, we say $M$ is node-polychromatic. We define $(P,\leq)$ to be the
subposet of $(\bar{P},\leq)$ consisting of mixed dimer configurations that
satisfy the valence condition, are reachable via a sequence of allowable flips
from $M_{-}$, and are node-monochromatic.

######  Proposition 3.1.

$M_{-}$ is in $P$, i.e. is a mixed dimer configuration on the base graph that
satisfies the valence condition and is node-monochromatic.

###### Proof.

First note that $M_{-}$ trivially satisfies the condition that it is reachable
by a sequence of allowable flips from $M_{-}$ by taking the empty sequence.
Note that $M_{-}$ also satisfies the valence condition by construction. By
definition of $M_{-}$, the sets $M_{1},M_{2}$ will satisfy the valence
condition and for any $k$-cycle fully supported in $\underline{d}$, the vertex
corresponding to the $k$-star is matched by the edges in $Z$ corresponding to
the unique singular arrow in that $k$-cycle. So, it suffices to show that
$M_{-}$ is node-monochromatic by showing that there are no paths between nodes
of different colors. If the associated quiver is acyclic, $M_{-}$ is node-
monochromatic, see Proposition 3.4.1 of [MW20]. So, we need to show $M_{-}$ is
node-monochromatic if our induced subquiver with respect to $\underline{d}$
contains an oriented cycle.

If the oriented cycle appears in the type $A_{m}$ part of our induced
subquiver, then by definition of the green nodes in Figure 11, these green
nodes must be connected in $M_{-}$. In particular, these sets of nodes cannot
be connected to red or blue nodes as any such path would imply a vertex of
degree 3 in $M_{-}$. If the oriented cycle appears in the type $D_{n}$ part of
the subquiver, then this quiver must be type IV. In this case, the blue nodes
are connected in $M_{-}$ by definition, which again implies that they cannot
connect to the green nodes as any such path would imply a vertex of degree 3
in $M_{-}$. Therefore, $M_{-}$ must be node-monochromatic. ∎

In the acyclic case, refer to Example 3.4 and Figure 1 in [MW20] to see an
example of this poset $P$. We provide another example when $Q$ contains an
oriented cycle before proceeding stating our main result in the following
section.

######  Example 3.5.

In Vatne type IV where the central cycle is a 3-cycle, consider the $D_{6}$
quiver shown in Figure 6 with $\underline{d}=(1,1,2,1,1,1)$. The base graph
and minimal matching with the nodes is given by:

and the poset $P$ of node monochromatic mixed dimer configurations is given in
Figure 12.

Figure 12: Poset of mixed dimer configurations in Example 3.5. The tiles
enclosed by cycles are shaded grey for emphasis.

Now, with these definitions and this running example, we have the ingredients
to state our main result. Namely, we will show that the elements of this poset
$P$ give the non-zero monomials in the $F$-polynomial associated to
$\underline{d}$.

##  4 Main Theorem

In this section, we connect the representation theory and combinatorics
presented in the previous sections together in our main result. In particular,
we give the generating function for cluster variables in terms of mixed dimer
configurations indexed by certain dimension vectors of modules of the
associated Jacobian algebra.

######  Theorem 4.1.

Given any quiver $Q$ of type $D_{n}$ and crossing vector $\underline{d}$, we
let $F_{\underline{d}}$ denote the $F$-polynomial corresponding to the cluster
variable with crossing vector $\underline{d}$. This expression is based on the
appropriate cluster algebra of type $D_{n}$ and assuming an initial seed
defined by the choice of quiver $Q$ and the standard initial cluster of
$\\{x_{1},x_{2},\dots,x_{n}\\}$.

Furthermore, let $M_{-}=M_{-}(Q,\underline{d})$ be the minimal matching, as
defined in Section 3.2 and let $P$ be the poset of mixed dimer configurations
that satisfy the valence condition, are reachable via a sequence of allowable
flips from $M_{-}$, and satisfy the node monochromatic condition as defined in
Definition 3.9. Then the expansion of $F$-polynomial $F_{\underline{d}}$ can
be expressed as a weighted multi-variate rank-generating function on the poset
$(P,\leq)$ determined by $M_{-}$:

$F_{\underline{d}}=\sum_{D\in P}2^{c}u_{0}^{t_{0}}u_{1}^{t_{1}}\cdots
u_{n-1}^{t_{n-1}},$

where the sum is taken over mixed dimer configurations $D$ obtained by
flipping tile $i$ $t_{i}$-times (keeping track of multiplicities) and $c$ is
the number of cycles in $D$.

######  Example 4.1.

Consider the quiver and $\underline{d}$-vector from Example 3.5. Each element
of the poset $P$ in Figure 12 correspond to a monomial in the $F$-polynomial:

$\displaystyle F_{\underline{d}}$
$\displaystyle=1+u_{1}+u_{1}u_{5}+u_{4}u_{5}+u_{1}u_{2}+u_{2}u_{5}+u_{2}u_{4}u_{5}+u_{1}u_{4}u_{5}+2u_{1}u_{2}u_{5}+u_{0}u_{2}u_{5}$
$\displaystyle+2u_{1}u_{2}u_{4}u_{5}+u_{0}u_{2}u_{4}u_{5}+u_{1}u_{2}^{2}u_{5}+u_{0}u_{1}u_{2}u_{5}+u_{1}u_{2}u_{3}u_{4}u_{5}+u_{1}u_{2}^{2}u_{4}u_{5}$
$\displaystyle+u_{0}u_{1}u_{2}u_{4}u_{5}+u_{0}u_{1}u_{2}^{2}u_{5}+u_{1}u_{2}^{2}u_{3}u_{4}u_{5}+u_{0}u_{1}u_{2}^{2}u_{4}u_{5}+u_{0}u_{1}u_{2}^{2}u_{3}u_{4}u_{5}.$

To highlight one of the monomials in the poset, notice the term
$2u_{1}u_{2}u_{5}$ corresponds to traversing up the poset $P$ on the left by
flipping tiles 1,2 and then 5. There is one cycle in the mixed dimer
configuration enclosing the face 2 which gives the coefficient $2^{1}$.

In order to prove Theorem 4.1, we rely on representation theory. In the
acyclic case [MW20], we proved this result by creating a bijection between
mixed dimer configurations and vectors that parameterize subrepresentations of
a fixed indecomposable quiver representation of dimension vector
$\underline{d}$. In particular, we utilized a categorization of these vectors
created by Tran [Tra09]. The acyclic case is particularly nice because the
indecomposable quiver representations are indexed by positive roots of the
$D_{n}$ root system. As mentioned previously, when $Q$ contains an oriented
cycle, positive roots no longer are in bijection with the cluster variables.
We instead rely on the surface model with the crossing vectors cataloged in
Appendix A.

To an arc $\gamma$ in a triangulated surface, one can associate an
indecomposable Jacobian algebra module $M_{\gamma}$ whose dimension vector
$\underline{d}_{\gamma}$ is given by enumerating the crossings of $\gamma$
with the arcs of the triangulation. With this, we mimic the work of Tran to
parameterize $\underline{e}$ vectors that index sub-modules of $M_{\gamma}$ we
call submodule-indexing vectors. We then prove Theorem 4.1 by creating a
bijection between these submodule-indexing $\underline{e}$ vectors and mixed
dimer configurations. To state this theorem, we define a few properties on
arrows given by [Tra09] in Definition 4.1. Define a partial order $\geq$ on
$\mathbb{Z}^{n}$ via

$\underline{a}\geq\underline{a^{\prime}}\text{ if
}\underline{a}-\underline{a^{\prime}}\in\mathbb{Z}_{\geq 0}^{n}.$

######  Definition 4.1.

Let $\underline{d}\in\\{0,1,2\\}^{n}$ and $\underline{e}$ such that $0\leq
e_{i}\leq d_{i}$ for each $1\leq i\leq n$. An arrow $j\to k\in Q_{1}$ is
acceptable with respect to $(\underline{d},\underline{e})$ if
$e_{j}-e_{k}\leq[d_{j}-d_{k}]_{+}:=\max(d_{j}-d_{k},0)$. An arrow is called
critical with respect to $(\underline{d},\underline{e})$ if either

$(d_{j},e_{j})=(2,1)~{}~{}\text{ and }~{}~{}(d_{k},e_{k})=(1,0)$

or

$(d_{k},e_{k})=(2,1)~{}~{}\text{ and }~{}~{}(d_{j},e_{j})=(1,1)$

Let $C=\\{i\in Q_{0}~{}:~{}(d_{i},e_{i})=(2,1)\\}$ and let $\nu(C)$ be the
number of critical arrows that have a vertex in $C$.

With these definitions, we are going to define a set of vectors
$\underline{e}$ we call submodule-indexing that satisfy certain conditions. In
[Tra09], analogous conditions gave criterion for indexing dimension vectors of
subrepresentations of a given indecomposable acyclic quiver representation.
Note that in the new cases of non-acyclic quiver, singular arrows can play a
special role.

######  Definition 4.2.

Let $\underline{d}\in\\{0,1,2\\}^{n}$. We say that a vector
$\underline{e}\in\\{0,1,2\\}^{n}$ is submodule-indexing with respect to
$\underline{d}$ if

  1. 1.

$0\leq\underline{e}\leq\underline{d}$;

  2. 2.

Any arrow $j\to k\in Q_{1}$ that is not singular, is acceptable; and

  3. 3.

$\nu(C)\leq 1$.

######  Remark 4.1.

We refer to condition 2 from Definition 4.1 as the acceptability condition and
condition 3 as the criticality condition.

Now we are ready to state a bijection between mixed dimer configurations and
these submodule-indexing vectors.

######  Theorem 4.2.

Let $T$ be an ideal triangulation of a once-punctured $n$-gon. Let
$(Q_{T},W_{T})$ be the associated quiver and the $J_{T}=\Bbbk Q_{T}/I$ be its
Jacobian algebra. Let $M$ be an indecomposable $J_{T}$-module associated to an
arc $\gamma$ with dimension vector $\underline{d}$. Then there exists a
bijection between

$\\{\text{mixed dimer configurations }D\text{ in
}P\\}\mathrel{{\stackon[4pt]{\longleftrightarrow}{\scriptscriptstyle\sim}}}\\{\underline{e}\text{
submodule-indexing with respect to }\underline{d}\\}$

where $P$ is the poset defined in Definition 3.9 and is order-preserving where
the set on the right has a natural partial ordering defined right above
Definition 4.1.

We first work to prove Theorem 4.2 by demonstrating a map in both directions.
The map taking mixed dimer configurations to submodule-indexing
$\underline{e}$-vectors is given by taking the superimposition of the mixed
dimer configuration with $M_{-}$ and deleting cycles created by this
multigraph. The tiles that these cycles are formed on correspond to the
entries in $\underline{e}$. The map taking a submodule-indexing vector
$\underline{e}$ to a mixed dimer configuration is given by taking a sequence
of “weighted flips” from the minimal matching on tiles that are in
$\text{supp}(\underline{e})$. We first describe the ladder map by weighting
the edges of the base graph $G$ via

$w(e):=\\#\\{\text{ edges on }e\text{ in }M_{-}\\}$

for all $e\in E(G)$.

######  Definition 4.3.

The weights of the edges are transformed via the following prescription after
flipping a tile:

This is known as a weighted flip.

######  Remark 4.2.

Implicit in this definition is the fact that our weights will stay non-
negative when we perform the flips given in Definition 3.7. However, in the
context of this direction of the bijection, we will allow for weighted flips
at any tile; meaning that we now allow the weight of an edge to be negative.
Following the conventions placed in [MW20], we emphasize the deficits which
arise when we flip edges that are not in a given mixed dimer configuration, we
distinguish edges of negative weight in yellow and call them “antiedges.”
Allowing weights to be negative in intermediate steps of a sequence of flips
simplifies the proof as we do not need to prove a sequence of allowable flips
exists to obtain a mixed dimer configuration from a submodule-indexing vector
$\underline{e}$. Namely, we just show that after flipping in any order, the
mixed dimer configuration we obtain has all non-negative weighting and belongs
to the poset $P$.

######  Theorem 4.3.

Let $\gamma$ be an arc superimposed on a triangulated once-punctured $n$-gon
and let $Q$ be the quiver associated to this triangulation. Let
$\underline{d}$ be the crossing vector of the arc $\gamma$. Let $G$ be the
base graph constructed using the data of $Q$ and $\underline{d}$ as described
in Definition 3.1 and 3.2. Suppose that $\underline{e}$ is a submodule-
indexing vector $\underline{e}$. Then there exists a unique way to produce a
mixed dimer configuration $D$ in poset $P$ via the following procedure:

  1. 1.

Weight the edges of the base graph by $w(e)$ for $e\in M_{-}$. Take the
positive entry $e_{i_{1}}>0$ in $\underline{e}$ of minimal index, and flip
tile $i_{1}$ $e_{i_{1}}$ number of times. Transform its edge weights as
prescribed in Definition 4.3. Let $D_{1}$ be the mixed dimer configuration
obtained.

  2. 2.

From $D_{1}$, take the next positive entry $e_{i_{2}}>0$ in $\underline{e}$ of
minimal index, and flip tile $i_{2}$ $e_{i_{2}}$ number of times. Again,
transform its edge weights as prescribed in Definition 4.3 to arrive at the
mixed dimer configuration $D_{2}$.

  3. 3.

Iterate this process until we have exhausted all positive entries in
$\underline{e}$. The resulting mixed dimer configuration $D$ will only have
non-negative weights, i.e. no yellow edges will remain. Moreover, it will be
an element of $P$.

Before proving Theorem 4.3, let’s first see an example of this algorithm.

######  Example 4.2.

Suppose $\underline{d}=(0,1,2,1,2,2,1,1,0)$ and the submodule-indexing vector
$\underline{e}=(0,0,0,0,0,2,1,0,0)$. The minimal matching and steps of
flipping tiles $5$ twice, then $6$ once is shown in Figure 13.

Figure 13:

In order to prove Theorem 4.3 and eventually Theorem 4.2, we need to create a
dictionary between mixed dimer configurations and conditions on
$\underline{e}$ vectors. We have the following sequence of lemmas that allows
us to relate edges distinguished on a mixed dimer configuration with
coordinates of a vector. These lemmas mimic arguments found in [MW20] for the
acyclic case, but we note the special features that arise when there are
cycles in the quiver, and the special role of singular arrows in particular.

######  Lemma 4.1.

Let $Q$ be a type $D_{n}$ quiver and let $\underline{d}$ be the dimension
vector of an indecomposable Jacobian algebra module $M$. Let $\underline{e}$
such that $0\leq\underline{e}\leq\underline{d}$ and let $D$ be the mixed dimer
configuration obtained by flipping tile $k$ $e_{k}$ number of times from
$M_{-}$. For any non-singular arrow $i\to j\in Q_{1}$, let $m_{i,j}$ be the
number of edges distinguished in $D$ on the edge between tiles $i$ and $j$.
Then

$m_{i,j}=\max(d_{i}-d_{j},0)+(e_{j}-e_{i})=:n_{i,j}.$

###### Proof.

We proceed by induction on $|\underline{e}|:=\sum_{k=0}^{n-1}e_{k}$. When
$|\underline{e}|=0$, i.e. $\underline{e}=(0,0,\dots,0)$, the associated mixed
dimer configuration is $M_{-}$. By definition of $M_{-}$, we only distinguish
internal edges if $G_{1}\subsetneq G$ or $G_{2}\subsetneq G$. If $d_{i}>0$ and
$d_{j}=0$, then $i\in G_{1}$, but $j\notin G_{1}$. Since $i\to j$ is oriented
black to white with respect to $i$, then $m_{i,j}=d_{i}=\max(d_{i}-0,0)$. If
$j\to i$, then the edge straddling $i$ and $j$ is oriented white to black
clockwise with respect to tile $i$. By definition of $M_{-}$, we will not
distinguish that edge giving that $n_{j,i}=\max(d_{j}-d_{i},0)=0=m_{j,i}$.

Suppose up to $k\in\mathbb{N}$, our formula holds. Now, suppose
$|\underline{e}|=k+1$, and choose $\tilde{\underline{e}}$ with
$|\tilde{\underline{e}}|=k$, such that $\underline{e}$ can be obtained by
adding $1$ to the $i^{\text{th}}$ entry of $\tilde{\underline{e}}$. In other
words, the tile $i$ has been flipped from the associated mixed dimer
configuration associated to $\tilde{\underline{e}}$ to obtain the mixed dimer
configuration associated to $\underline{e}$.

If $i\to j$, then the edge straddling tiles $i$ and $j$ is oriented black to
white clockwise on tile $i$. As tile $i$ has been flipped,
$e_{i}=\tilde{e_{i}}-1$ and the number of edges distinguishing on the edges
straddling $i$ and $j$ decreases by 1 from the dimer configuration associated
to $\tilde{\underline{e}}$. Therefore,

$n_{i,j}=\max(d_{i}-d_{j},0)+(e_{j}-(e_{i}+1))=\max(d_{i}-d_{j},0)+(e_{j}-e_{i})-1.$

Therefore, performing a flip at tile $i$ decreased both $n_{i,j}$ and
$m_{i,j}$ by 1. If $j\to i$ in $Q$, then the edge straddling tiles $i$ and $j$
is oriented white to black clockwise on tile $i$. As tile $i$ has been
flipped, $e_{i}=\tilde{e_{i}}+1$ and the number of edges distinguishing on the
edges straddling $i$ and $j$ increases by 1 from the dimer configuration
associated to $\tilde{\underline{e}}$. Therefore,

$n_{j,i}=\max(d_{j}-d_{i},0)+(e_{i}+1-e_{j})=\max(d_{i}-d_{j},0)+(e_{i}-e_{j})+1.$

Therefore, performing a flip at tile $i$ increased both $n_{j,i}$ and
$m_{j,i}$ by 1. ∎

######  Lemma 4.2.

(Analogous to Lemma 4.1 for singular arrows). Following the notation of Lemma
4.1, we can make an analogous statement for singular arrows $i\to j\in Q_{1}$.
Namely,

$m_{i,j}-1=\max(d_{i}-d_{j},0)+(e_{j}-e_{i})=:n_{i,j}.$

###### Proof.

When an arrow $i\to j$ in singular, we add an extra distinguished edge
straddling tiles $i$ and $j$ in $M_{-}$ that is not a boundary edge going
black to white clockwise in $G_{1}$ or $G_{2}$. Hence, the formula from Lemma
4.1 can be modified by subtracting 1 to reflect this extra edge. ∎

######  Corollary 4.1.

The values $n_{i,j}:=\max(d_{i}-d_{j},0)+(e_{j}-e_{i})$ all satisfy
$n_{i,j}\geq 0$ if and only if all weights on interior edges on the mixed
dimer configuration $D$ associated to $e$ are nonnegative. In particular, all
arrows $i\to j$ are acceptable with respect to $(\underline{d},\underline{e})$
if and only if all weights on interior edges are nonnegative on the mixed
dimer configuration $D$ associated to $e$.

######  Lemma 4.3.

(Analogous formula to Lemma 4.1 with boundary edges). Let $Q$ be a type
$D_{n}$ quiver and let $\underline{d}$ be the dimension vector of an
indecomposable Jacobian algebra module $M$. Let $\underline{e}$ such that
$0\leq\underline{e}\leq\underline{d}$ and let $D$ be the mixed dimer
configuration obtained by flipping tile $k$ $e_{k}$ number of times from
$M_{-}$. Let the outer face of our graph $G$ be indexed by $\infty$. We assign
an arrow to each of the boundary edges of $G$ with the convention that we “see
white on the right.” To each of these boundary edges, assign a weight to the
edge $\alpha$ on tile $i$ as follows:

$n_{i,\infty}(\alpha)=\max(d_{i},0)-e_{i}~{}~{}~{}\text{if }i\to\infty\text{
about }\alpha$ $n_{\infty,i}(\alpha)=\max(-d_{i},0)+e_{i}~{}~{}~{}\text{if
}\infty\to i\text{ about }\alpha$

Let $m_{i,\infty}(\alpha)$ (respectively $m_{\infty,i}(\alpha)$) be the number
of edges distinguished on $\alpha$ on tile $i$ in $D$ where $i\to\infty$ about
$\alpha$ (respectively $\infty\to i$ about $\alpha$.) Then for any boundary
edge $\alpha$ on tile $i$,

$n_{i,\infty}(\alpha)=m_{i,\infty}(\alpha)~{}~{}\text{ and
}~{}~{}n_{\infty,i}(\alpha)=m_{\infty,i}(\alpha).$

###### Proof.

As in the proof of Lemma 4.1, we proceed by induction on $|e|$. Suppose
$|e|=0$, then the associated mixed dimer configuration is $M_{-}$. Let
$\alpha$ be a boundary edge on a tile $i$. If $i\to\infty$ across the edge
$\alpha$, then $m_{i,\infty}(\alpha)=\max(d_{i},0)=d_{i}.$ Moreover, the edge
$\alpha$ is oriented black to white clockwise giving that it is distinguished
$d_{i}$-times in $M_{-}$. Therefore,
$m_{i,\infty}(\alpha)=d_{i}=n_{i,\infty}(\alpha)$. If $\infty\to i$ across the
edge $\alpha$, we have $m_{\infty,i}(\alpha)=\max(-d_{i},0)=0.$ Moreover, the
edge $\alpha$ is oriented white to black clockwise giving that it is never
distinguished in $M_{-}$. Therefore,
$m_{i,\infty}(\alpha)=0=d_{i}=n_{i,\infty}(\alpha)$.

Suppose up to $k\in\mathbb{N}$, our formula holds. Now, suppose
$|\underline{e}|=k+1$, and choose $\tilde{\underline{e}}$ with
$|\tilde{\underline{e}}|=k$, such that $\underline{e}$ can be obtained by
adding $1$ to the $i^{\text{th}}$ entry of $\tilde{\underline{e}}$. In other
words, the tile $i$ has been flipped from the associated mixed dimer
configuration associated to $\tilde{\underline{e}}$ to obtain the mixed dimer
configuration associated to $\underline{e}$. Suppose that $\alpha,\beta$ are
boundary edges on tile $i$ where $\infty\to i$ across the edge $\alpha$ and
$i\to\infty$ across the edge $\beta$. After flipping tile $i$, the $w(\beta)$
decreases by 1 and $w(\alpha)$ increases by 1. Since $\tilde{e_{i}}=e_{i}+1$,
$m_{i,\infty}(\beta)$ is transformed by

$m_{i,\infty}(\beta)=\max(d_{i},0)-(e_{i}+1)=\max(d_{i},0)-e_{i}-1,$

and $m_{\infty,i}(\alpha)$ is transformed by

$m_{\infty,i}(\alpha)=\max(-d_{i},0)+e_{i}+1.$

Hence, we have that $n_{i,\infty},m_{i,\infty}$ and respectively
$n_{\infty,i},m_{\infty,i}$ are transformed in the same way after a flip at
tile $i$. ∎

We need one more technical lemma regarding the graphical structure of a quiver
$Q$ mutation-equivalent to a type $D_{n}$ Dynkin diagram and the possible
subquivers arising as the support of a submodule-indexing vector
$\underline{e}$.

######  Lemma 4.4.

Suppose $Q$ is a quiver mutation-equivalent to an orientations of a type
$D_{n}$ Dynkin diagram and $\underline{e}$ is a choice of submodule-indexing
vector with respect to $\underline{d}$ for some $d$-vectors $\underline{d}$
that is not fully supported by only 1’s on any cycle of $Q$. Then $Q$ is
guaranteed to contain a vertex $j$ with the property that $e_{j}>0$ and such
that for any arrow pointing $i\to j$, we have $e_{j}>e_{i}$.

###### Proof.

If the given submodule-indexing vector $\underline{e}$ contains a $2$, then
let $Q^{e}$ denote the subquiver of $Q$ containing exclusively the vertices
$i$ such that $e_{i}=2$. By Lemma 3.2, the subquiver $Q^{e}$ must be a tree
and therefore must contain a source. Letting $j$ denote the vertex of such a
source, we see that vertex $j$ vacuously satisfies $e_{j}>e_{i}$ for any arrow
pointing $i\to j$.

On the other hand, if the given submodule-indexing vector $\underline{e}$
contains no $2$’s, then we let $Q^{e}$ denote the subquiver of $Q$ given by
the support of $\underline{e}$, i.e. containing exclusively the vertices $i$
such that $e_{i}=1$. By assumption, the subquiver $Q^{e}$ again must be a tree
and therefore must contain a source, and we can choose vertex $j$ just as
above. ∎

With these lemmas, we are now ready to prove Theorem 4.3.

###### Proof.

Suppose $\underline{e}$ is submodule-indexing and let $D$ be the mixed dimer
configuration produced as prescribed in the statement of Theorem 4.3. In order
to show that $D\in P$, we must show that $D$ is a mixed dimer configuration
reachable via a sequence of flips from $M_{-}$ and is node-monochromatic. We
first show that $D$ is reachable via a sequence of flips from $M_{-}$ by
induction on $|e|$. For the base case, we agree to associate the submodule-
indexing vector $\underline{e}=(0,0,\dots,0)$ to the minimal matching $M_{-}$
which is reachable by an empty sequence of flips from $M_{-}$.

Suppose that for any nonzero submodule-indexing vector $\underline{e}$ with
$|e|=k$, the associated mixed dimer configuration $D$ is reachable via a
sequence of flips from $M_{-}$. Now suppose that $|e|=k+1$ and that we have
added 1 to the $i^{\text{th}}$ entry of $\underline{e}$. As $\underline{e}$ is
submodule-indexing, any arrow involving vertex $i$ must be acceptable with
respect to $(\underline{d},\underline{e})$ i.e.
$e_{i}-e_{j}\leq\max(d_{i}-d_{j},0)$. By Lemma 4.1, we have that
$n_{i,j}=m_{i,j}$ which implies that the number of edges straddling tiles $i$
and $j$ must be non-negative. Moreover, by Lemma 4.3, since
$0\leq\underline{e}\leq\underline{d}$, we have that for any boundary edge
$\alpha$ on tile $i$,
$n_{i,\infty}(\alpha)=\max(d_{i},0)-e_{i}=d_{i}-e_{i}\geq 0$ giving that
$m_{i,\infty}(\alpha)$, the number of edges on $\alpha$ in $D$ is non-
negative. Similarly, $n_{\infty,i}(\alpha)=\max(-d_{i},0)+e_{i}=0+e_{i}\geq 0$
giving that $m_{\infty,i}(\alpha)$, the number of edges on $\alpha$ in $D$ is
non-negative. If $i\to j$ is a singular arrow, the number of edges straddling
$i$ and $j$ will have only increased by 1, so $m_{i,j}\geq 0$ as well. Hence,
the resulting mixed dimer configuration $D$ has edges with all non-negative
weights.

We now show that there exists some order in which we can read the entries of
$\underline{e}$ which will yield a mixed dimer configuration with non-negative
weights at each flip along the way in the sequence. Even though we constructed
$\underline{e}$ inductively in a way such that the last entry increased by one
is the $i^{\text{th}}$ entry, it is not necessarily the case that the
corresponding mixed dimer configuration $D$ is reachable by a flip sequence
ending in a flip of tile $i$.

Instead, here we invoke Lemma 4.4 with $j$ denoting the label of the vertex
whose existence is posited by the lemma. If $\underline{d}$ is fully supported
by 1’s on a cycle of $Q$, then there exists some singular arrow $a\to b$ with
$d_{a}=d_{b}=1$. In this case, let $j$ be the vertex $b$. We let
$\underline{e^{\prime}}$ be the result of subtracting the $j^{\text{th}}$ unit
vector from $\underline{e}$. Then, for any non-singular arrow $i\to j$, we
have $n_{i,j}^{\prime}=\max(d_{i}-d_{j},0)+(e_{j}-1-e_{i})$ must be
nonnegative. If the arrow is the reverse orientation $j\to i$, then
$n_{j,i}^{\prime}=\max(d_{j}-d_{i},0)+(e_{i}-e_{j}+1)$ is also nonnegative.
Moreover, on the boundary, we also have $n_{i,\infty}^{\prime}=d_{i}-e_{i}$
and $n_{\infty,i}^{\prime}=e_{i}$ are nonnegative. In the case of a fully
supported cycle with only 1’s and singular arrow $a\to j$, by Lemma 4.2, we
have that $m_{a,j}-1=\max(d_{a}-d_{j},0)+e_{j}-e_{a}$. Since $e_{a}\leq 1$,
$e_{j}\leq 1$, their difference is at most -1 implying that
$n_{a,j}=e_{j}-e_{a}+1$ is nonnegative.

Therefore $\underline{e^{\prime}}$ corresponds to a mixed dimer configuration
$D^{\prime}$. Since $|e^{\prime}|=k$, by the inductive hypothesis,
$D^{\prime}$ is reachable by a sequence of allowable flips from $M_{-}$. This
gives that $D$ itself is reachable from $M_{-}$ by a sequence of allowable
flips, where we tack on a flip of tile $j$. This last flip is allowable since
both $D^{\prime}$ and $D$ are mixed dimer configurations, i.e. with
nonnegative weights on edges, and the difference
$\underline{e}-\underline{e^{\prime}}$ is the $j^{\text{th}}$ unit vector.

We now show $D$ is a node monochromatic mixed dimer configuration. We again
proceed by induction on $|e|$. By Proposition 3.1, we have that $M_{-}$ is
node-monochromatic, establishing the base case. So suppose that up to
$k\in\mathbb{N}$, we have that when $|e|=k$, the resulting mixed dimer
configuration $D$ is node-monochromatic. Now take $\underline{e}$ with
$|e|=k+1$ where we have added 1 to the $i^{\text{th}}$ entry of
$\underline{e}$. We may assume that $D$ was obtained via a sequence of $k+1$
allowable flips from $M_{-}$. We claim that the only way we could have
produced a path connecting nodes of different colors is if there is more than
one critical arrow in $Q$ with respect to $(\underline{d},\underline{e})$.

If $Q^{\text{supp}}$ is acyclic, then we employ the proof of Theorem 4.2.2
from [MW20]. Moreover, in type IV where we have full support on the central
$m$-cycle, we may also employ the argument in the acyclic case when the type
$A_{m}$ part of the quiver has no oriented cycles. This is because such a path
connecting blue to green nodes can only occur on an acyclic subquiver of
$Q^{\text{supp}}$. Namely, a path only occurs in a mixed dimer configuration
when there are valence 2 tiles and must end after a consecutive tile has
valence 1. In any $\underline{d}$-vector in type IV with a 2, the central
$k$-cycle is fully supported where the $\underline{d}$-vector entries are all
1. Hence, a path connecting the type $D_{n}$ part of the quiver to the type
$A_{m}$ part of the quiver will only use the tile with the blue nodes and the
consecutive tile whose $\underline{d}$-vector entry is 2 and no other tiles on
the $m$-cycle - reducing to the acyclic case.

So it suffices to show that $D$ is node-monochromatic when $Q^{\text{supp}}$
contains an oriented cycle on the type $A_{m}$ part of the quiver. First note
that in types I, II and III, the
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}red}$ or
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}blue}$
nodes cannot be connected without creating two critical arrows. Let $p,q,r$ be
the forking vertices in the type $D_{n}$ part of $Q^{\text{supp}}$. Then, for
any orientation of the fork, paths between
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}red}$ and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}blue}$
nodes create two critical arrows as shown in Figure 14.

Figure 14: No paths between
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}red}$ and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}blue}$
nodes can occur when $\underline{e}$ satisfies the criticality condition.

Suppose that the cycle is on the type $A_{m}$ part of the quiver and we aim to
show that the
${\color[rgb]{0.5,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{.5}\pgfsys@color<EMAIL_ADDRESS>nodes cannot attach to either the
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}red}$ or
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}blue}$
nodes without having more than one critical arrow. Note that it suffices to
show that there is no connection between the
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}blue}$
nodes and
${\color[rgb]{0.5,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{.5}\pgfsys@color<EMAIL_ADDRESS>nodes as in types I, II and III, the
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}blue}$
nodes and
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}red}$ nodes
are symmetric and type IV has no red nodes. Suppose that tile $a$ has the two
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}blue}$
nodes ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}w,x}$
and let $b$ be the vertex connecting the type $D_{n}$ part of the quiver to
the type $A_{m}$ part of the quiver. Suppose that the 3-cycle that has full
support occurs at the set vertices $i,j,k$ read in cyclic order in the type
$A_{m}$ part of the quiver. In order to connect the
${\color[rgb]{0.5,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{.5}\pgfsys@color<EMAIL_ADDRESS>nodes to the
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}blue}$
nodes, we must have flipped tiles connecting the 3-cycle $i,j,k$ with the
vertices in the type $A_{m}$ part of the quiver connecting to tile $b$, then
tile $b$. For example, if $b$ is directly attached to tile $i$, then we first
flip tile $i$, then $b$ as shown in Figure 15:

Figure 15:

With this, we see that this implied that the quiver must have had more than
one critical arrow with respect to $(\underline{d},\underline{e})$ as the
arrow $b\to a$ has $(d_{b},e_{b})=(2,1)$ and $(d_{a},e_{a})=(1,0)$ and the
arrow $i\to j$ has $(d_{i},e_{i})=(2,1)$ and $(d_{j},e_{j})=(1,0)$. Similarly,
if $a\to b$, we would flip tiles $b$, then $a$ giving that the arrow $a\to b$
is critical with $(d_{b},e_{b})=(2,1)$ and $(d_{a},e_{a})=(1,1)$ with $i\to j$
has $(d_{i},e_{i})=(2,1)$ and $(d_{j},e_{j})=(1,0)$ still critical. The same
argument holds if the 3-cycle $i,j,k$ was not directly adjacent to $b$ using a
flip sequence along the vertices connecting $b$ to the 3-cycle $i,j,k$.

Therefore, even when there is a cycle in $Q^{\text{supp}}$, we see that node
monochromatic paths imply that the quiver must have had more than one critical
arrow with respect to $(\underline{d},\underline{e})$. Hence, the dimer $D$
associated to a submodule-indexing vector must be node monochromatic implying
that $D\in P$ as desired. ∎

To complete the proof of Theorem 4.2, we now describe the other side of the
bijection i.e. the map taking mixed dimer configurations to submodule-indexing
$\underline{e}$-vectors.

######  Theorem 4.4.

Let $\gamma$ be an arc superimposed on a triangulated once-punctured $n$-gon
and let $Q$ be the quiver associated to this triangulation. Let
$\underline{d}$ be the crossing vector of the arc $\gamma$. Let $G$ be the
base graph constructed using the data of $Q$ and $\underline{d}$ as described
in Definition 3.2. Suppose $D$ is a mixed dimer configuration in $P$ i.e. is
node monochromatic. There exists a unique way to produce a submodule-indexing
vector $\underline{e}$. The process is given by the following procedure:

  1. 1.

Superimpose $D$ with $M_{-}$ on the base graph $G$ to obtain the multigraph
$D\sqcup M_{-}$. In this superimposition, if $D$ and $M_{-}$ have any edge
$e\in D\cap M_{-}$ in common on $G$, delete one copy of $e$ from $D\sqcup
M_{-}$. Call the resulting multigraph $M_{1}$.

  2. 2.

Using the edges $M_{1}$, create a cycle of maximal length $\ell_{1}>2$, call
it $C_{1}$. For all faces $i$ enclosed by $C_{1}$, add $+1$ to $v_{i}$ in
$\underline{v}$ and delete $C_{1}$ from $M_{1}$. Call the resulting multigraph
$M_{2}=M_{1}\setminus C_{1}$.

  3. 3.

Examine $M_{2}$ and if there are any cycles of length $\ell_{2}>2$, find a
cycle of maximal length and call it $C_{2}$. For all faces $i$ enclosed by
$C_{2}$, add +1 to $v_{i}$ and delete $C_{2}$ from $M_{2}$.

  4. 4.

Iterate this process of deleting cycles of of largest length and adding 1’s to
the vector $\underline{v}$ until nothing is left besides 2-cycles. The
resultant vector $\underline{v}$ corresponds to $\underline{e}$.

Before proving Theorem 4.4, we compute an example of this side of the
bijection.

######  Example 4.3.

In this example, let $\underline{d}=(0,1,2,1,2,2,1,1,0)$ and consider the base
graph shown in Figure 16. We consider a mixed dimer configuration $D$ and we
show that its associated $\underline{e}$-vector is given by
$\underline{e}=(0,0,0,0,0,2,1,0,0)$. When we superimpose $D$ with $M_{-}$ and
delete any paired sets of edges in their intersection, we obtain $M_{1}$. In
this case, $C_{1}$ is the cycle around the tiles labeled 5 and 6. This
indicates that $\underline{v}=(0,0,0,0,0,1,1,0,0)$ at this step. After we
delete the edges used to make this cycle, we obtain $M_{1}\setminus C_{1}$ and
see a cycle around the tile 5. This gives that
$\underline{v}=(0,0,0,0,0,2,1,0,0)$ at this step and we see are left with an
empty disjoint union of two cycles after this deletion. Hence,
$\underline{v}=(0,0,0,0,0,2,1,0,0)=\underline{e}$ is the submodule-indexing
vector associated to $D$ which is illustrated in Figure 17.

Figure 16: On the left, we have the mixed dimer configuration $D$ in blue and
on the right, we have the minimal matching $M_{-}$ in red. Figure 17: On the
left, we have the superimposition of $D$ with $M_{-}$ after deleting any pairs
of edges in the intersection, $M_{1}$. After deleting the cycle around tiles 5
and 6, we obtain $M_{2}$ illustrated on the right.

Now we prove Theorem 4.4.

###### Proof.

Suppose $D$ is a non-minimal mixed dimer configuration in the poset $P$ and
initialize $\underline{v}=(0,0,\dots,0)$. To begin the algorithm, we must find
a cycle of maximal length strictly larger than 2. In the superimposition of
$D\sqcup M_{-}$ we call $M_{1}$, we obtain that each vertex that was
previously valence 2 is now valence 4 in $M_{1}$ and each vertex that was
valence 1 is now valence 2 in $M_{1}$. Because $D\neq M_{-}$, there must exist
some cycle of length at least 4 in their superimposition as $D$ differs from
the minimal matching by at least one flip. Hence, there exists some cycle of
maximal length $\ell_{1}\geq 4$.

We claim that after each deletion of a maximal length cycle, the vector
$\underline{v}$ is a submodule-indexing vector with respect to
$\underline{d}$. To do this, we proceed by induction on $n$, the number of
deletions of cycles required so that $D\sqcup M_{-}$ is a (possible empty)
disjoint union of 2-cycles.

When $n=0$, we agree to associate $M_{-}$ to $\underline{e}=(0,0,\dots,0)$
which is submodule-indexing. Suppose that $D^{\prime}$ is a mixed dimer
configuration that requires $i-1$ iterations of our algorithm to obtain the
associated resultant vector $\underline{v}^{\prime}$. Suppose further that
this vector $\underline{v}^{\prime}$ is submodule-indexing. Now suppose that
$D$ is a mixed dimer configuration that requires one more application of our
algorithm than $D^{\prime}$ i.e. $D$ takes $i$ iterations and the resulting
vector is called $\underline{v}$. Namely, we have $\underline{v}$ is obtained
from adding 1’s to $\underline{v^{\prime}}$ to the tiles enclosed by cycle
$C_{i}$. After the $i^{\text{th}}$ iteration of the algorithm, we have $0\leq
v_{k}\leq d_{k}$ for every $k$, is satisfied as the only way a cycle can
enclose tile $j$ in the superimposition of $D$ and $M_{-}$ is if $D$ differs
from $M_{-}$ at a tile $j$. This precisely happens when a flip occurred at
tile $j$ and implicit in an occurrence of a flip is that $d_{j}>0$. In
$\underline{v}-\underline{v^{\prime}}$, the only new 1’s occur if $C_{i}$
enclosed those corresponding tiles, $v_{j}\leq d_{j}$ for all $j$ enclosed by
$C_{i}$. This gives that $0\leq\underline{v}\leq\underline{d}$.

To show that $\underline{v}$ is acceptable and satisfies the boundedness
condition on the number of critical arrows, we need to induct on the number of
tiles enclosed by $C_{i}$. We first show that $\underline{v}$ is acceptable.
If $C_{i}$ enclosed a single tile, call it $j$, then to verify that the
acceptability condition is satisfied, it suffices to check that any non-
singular arrows involving vertex $j$. Note that $v_{j}=1$ or $2$ because $j$
could have been enclosed in a previous cycle in an earlier iteration. Suppose
that $j$ is the tail of an arrow i.e. we have some arrow $k\to j\in Q_{1}$. If
$v_{k}\leq 1$, then acceptability is satisfied as $v_{k}-v_{j}\leq 0$. If
$v_{k}=2$, then $d_{k}=2$. Moreover, acceptability is satisfied as long as
$v_{j}\neq 1$ and $d_{j}=2$. However, by Lemma 4.1, this implies that
$m_{k,j}<0$ contradicting Corollary 4.1. Now suppose $j$ is the head of an
arrow i.e. there is an arrow $j\to k\in Q_{1}$. Then if $v_{k}=2$, then
acceptability is satisfied as $v_{j}-v_{k}\leq 0$. If $0\leq v_{k}\leq 1$,
then the only case that $j\to k$ fails acceptability is when $v_{k}=v_{j}-1$
and $d_{j}=d_{k}$. However, by Lemma 4.1, this implies that $m_{k,j}<0$
contradicting Corollary 4.1.

Now, assume that if $C_{i}$ enclosed $k$ cycles, then the resulting
$\underline{v}$ is submodule-indexing. Now, suppose that $C_{i}$ encloses
$k+1$ tiles, $t_{1},\dots,t_{k+1}$. We aim to show that
$\underline{v}+\underline{e_{t_{k+1}}}=\underline{y}$ is submodule-indexing.
It suffices to show that any non-singular arrow with vertex $t_{k+1}$
satisfies the acceptability condition. Since, $t_{k+1}$ is enclosed by $C_{i}$
and may have been enclosed by another cycle in a previous iteration, we have
that $y_{t_{k+1}}\geq 1$. Suppose that $t_{k+1}$ is the tail of an arrow i.e.
we have some arrow $j\to t_{k+1}\in Q_{1}$.

If $y_{j}\leq 1$, then acceptability is satisfied as $y_{j}-y_{t_{k+1}}\leq
0$. If $y_{j}=2$, then $d_{k}=2$. Moreover, acceptability is satisfied as long
as $y_{k}\neq 1$ and $d_{k}=2$. However, by Lemma 4.1, this implies that
$m_{j,t_{k+1}}<0$ contradicting Corollary 4.1. Now suppose $t_{k+1}$ is the
head of an arrow i.e. there is an arrow $t_{k+1}\to j\in Q_{1}$. Then if
$y_{j}=2$, then acceptability is satisfied as $y_{t_{k+1}}-y_{j}\leq 0$. If
$0\leq y_{j}\leq 1$, then the only case that $t_{k+1}\to j$ fails
acceptability is when $y_{j}=y_{t_{k+1}}-1$ and $d_{j}=d_{k}$. However, by
Lemma 4.1, this implies that $m_{k,j}<0$ contradicting Corollary 4.1.

Therefore, we have shown that if $C_{i}$ encloses $k+1$ cycles, the resulting
vector $\underline{y}$ is submodule-indexing. By induction, we have that after
the $i^{\text{th}}$ step of our algorithm, $\underline{v}$ is submodule-
indexing. Therefore, any vector obtained via this algorithm must satisfy the
acceptability condition. Hence, the resultant $\underline{e}$ must be
acceptable.

Now, we show that $\underline{v}$ satisfies the criticality condition. In
order to do this, we again induct on the number of steps needed to complete
our algorithm as well as induct on the number of tiles enclosed by cycle
$C_{i}$ at each step. Suppose that $C_{1}$ encloses a unique tile $j$ giving
that $\underline{v}=\underline{e_{j}}$, the unit vector with a 1 in the
$j^{\text{th}}$ position. Note that if $d_{j}\neq 2$, no critical arrows can
be formed as $C:=\\{i\in Q~{}:~{}(d_{i},e_{i})=(2,1)\\}=\emptyset$. So,
suppose that $d_{j}=2$. Then, as $(d_{j},v_{j})=(2,1)$, $j\in S$ and the only
way that the criticality condition could have failed is if this created two
critical arrows i.e. the quiver and associated pair
$(\underline{d},\underline{e})$ is

$k\longleftarrow j\longrightarrow\ell$ $(1,0)\leftarrow(2,1)\rightarrow(1,0)$

where we must have $v_{k}=0=v_{\ell}$ as the only tile enclosed by a cycle is
$j$. Note that since the subgraph $G_{2}$ associated to all tiles with
$\underline{d}$ entry 2 is connected which means that $j$ must be the unique
tile in $G_{2}$. Moreover, since both critical arrows would need to have
source $j$, it is impossible that $j,k,\ell$ form a 3-cycle in the sense of
the quiver. Hence, we are reduced locally to the acyclic case and can import
the argument for the base case found in Theorem 4.2.3 of [MW20].

Now, suppose that if $C_{1}$ encloses $k$ tiles, the resulting vector
$\underline{v}$ satisfies the criticality condition. We now aim to show that
if $C_{1}$ encloses $k+1$ tiles, then the resulting vector
$\underline{w}:=\underline{v}+\underline{e_{j}}$ satisfies the criticality
condition. Note that by our inductive assumption,
$(\underline{v},\underline{d})$ has at most one critical arrow. Suppose that
$(\underline{w},\underline{d})$ created 0 critical arrows. Since
$\underline{w}$ only differs from $\underline{v}$ in the $j^{\text{th}}$
entry, it suffices to only check any arrows involving $j$. Moreover, we need
to show that two critical arrows cannot be created by adding $+1$ to
$\underline{v}$ to obtain $\underline{w}$. The only ways that this could
happen is if $j$ is the unique vertex in $C$, i.e. $d_{j}$ is the unique 2 in
$\underline{d}$ and the quiver and associated pair
$(\underline{d},\underline{e})$ has one of the following local orientations:

$\displaystyle p~{}~{}\longleftarrow~{}~{}$ $\displaystyle
j~{}~{}\longleftarrow~{}~{}q$ (i) $\displaystyle(1,0)\leftarrow(2$
$\displaystyle,1)\leftarrow(1,1)$ $\displaystyle p~{}~{}\longrightarrow~{}~{}$
$\displaystyle j~{}~{}\longrightarrow~{}~{}q$ (ii)
$\displaystyle(1,1)\rightarrow(2$ $\displaystyle,1)\rightarrow(1,0)$
$\displaystyle p~{}~{}\longleftarrow~{}~{}$ $\displaystyle
j~{}~{}\longrightarrow~{}~{}q$ (iii) $\displaystyle(1,0)\leftarrow(2$
$\displaystyle,1)\rightarrow(1,0)$ $\displaystyle
p~{}~{}\longrightarrow~{}~{}$ $\displaystyle j~{}~{}\longleftarrow~{}~{}q$
(iv) $\displaystyle(1,1)\rightarrow(2$ $\displaystyle,1)\leftarrow(1,1)$

Note that case (iv) will not be possible as if $w_{p}=w_{q}=1$, this means
that both $p$ and $q$ were enclosed by $C_{1}$ in which case, since $j$ is in
between these vertices, this could not happen without having enclosed $j$ as
well by the connectedness of the tiles enclosed by $C_{1}$.

Moreover, the orientation in case (iii) occurred in the base case. Namely,
since $C_{1}$ enclosed a connected set of tiles, $p$ and $q$ cannot both be
terminal vertices. Therefore, it suffices to analyze cases (i) and (ii). Since
these cases are symmetric, we focus on case (i).

Note that if $j,p,q$ are not in a cycle in the quiver, then we are reduced to
the acyclic case. Therefore, we rely on the proof of Theorem 4.2.3 in [MW20].
If $j,p,q$ indeed are in a cycle in the quiver, then it is either a 3-cycle in
the type $D_{n}$ part of the type II the spike of a type IV surface, the
4-cycle in the type III surface or is in the type $A_{m}$ part of any type
$D_{n}$ quiver. Note that, $j,p,q$ cannot be in a larger central cycle in the
type IV surface, if there is a 2 in the $\underline{d}$-vector, it occurs at
the attaching spike of the quiver rather than in the central cycle.

If $j,p,q$ forms a 3-cycle in the type $D_{n}$ part of the type II surface or
a 4-cycle in the type $D_{n}$ part of the type III surface, then the nodes
$u,v$ and $w,x$ are on tiles $p,q$ and this degenerates into the acyclic case.
If the quiver is type IV, then the blue nodes are on tile $p$. Namely, the
node $w$ is on the white vertex on the edge straddling $p$ and $q$. In order
for $w_{q}=1$ and $w_{p}=0$, there must have existed a previous cycle
enclosing tile $q$ and not tile $p$. However, by definition of $M_{-}$, tile
$q$ would have only have been enclosed by a previous cycle if $j$ also was –
as the only edge enumerated in $M_{-}$ on tile $q$ is the straddling $p,q$
representing the singular arrow $p\to q$. Hence, this case is also impossible.
Similarly, if $j,p,q$ forms a 3-cycle in the type $A_{m}$ part of the quiver.
In order to flip the tile $q$, we must flip the tile $p$ giving that $w_{q}=1$
and $w_{p}=0$ is impossible. Therefore, $\underline{w}$ satisfies the
criticality condition if $(\underline{v},\underline{d})$ created 0 critical
arrows.

If $(\underline{v},\underline{d})$ created one critical arrow either
$k\to\ell$ with $(d_{k},v_{k})=(2,1)$ and $(d_{\ell},v_{\ell})=(1,0)$ or
$k^{\prime}\rightarrow\ell^{\prime}$ with
$(d_{k^{\prime}},v_{k^{\prime}})=(1,1)$ and
$(d_{\ell^{\prime}},v_{\ell^{\prime}})=(2,1)$, the only way for the
$\underline{w}$ to create another critical arrow is if we added 1 to the
source of an arrow whose sink is in $C$. Namely, $w_{j}=1$ because if
$w_{j}=2$, then no more critical arrows could be created. Therefore,
$\underline{w}$ satisfies the criticality condition if
$(\underline{v},\underline{d})$ created one critical arrow. Thus,
$\underline{v}$ is submodule-indexing as desired. ∎

Now that we have demonstrated the map in both directions, we show that the
maps described in Theorem 4.3 and Theorem 4.4 are inverses of one another to
complete the proof of Theorem 4.2.

###### Proof.

By definition of the algorithm in Theorem 4.4, we see that the number of steps
in the algorithm exactly equals $|e|$, where $\underline{e}$ is the resulting
vector associated to the dimer $D$. Therefore, we will induct on $|e|$, i.e.
the number of steps needed to perform the algorithm in Theorem 4.4, to show
that the maps described in Theorems 4.4 and 4.3 are inverses of one another.

First suppose that $|e|=0$. Then, by both the algorithm for Theorem 4.4 and
for Theorem 4.3, we have that $\underline{e}$ is associated to $M_{-}$ and
vice versa. Suppose that some $m\in\mathbb{N}$, when $|e|=k\leq m$, i.e. when
we perform the first $k$ steps of the algorithm in Theorem 4.4, the maps are
inverses of one another. Let $\underline{e}^{\prime}$ be such that
$|e^{\prime}|=m$. Then define $\underline{e}$ to be the resulting vector from
adding 1 to the $i^{\text{th}}$ entry of $\underline{e}^{\prime}$, so
$|e|=m+1$.

Applying Theorem 4.3 to $\underline{e}$ and $\underline{e}^{\prime}$, let
$D^{\prime}$ be the mixed dimer configuration corresponding to
$\underline{e}^{\prime}$ and $D$ be the mixed dimer configuration associated
to $\underline{e}$. Note that as $|e|=m+1$, we had to perform $m+1$ flips from
$M_{-}$ to obtain $D$ and by assumption, we had to perform one more flip at
tile $i$ to obtain $D$ from $D^{\prime}$.

We aim to show that using the algorithm in Theorem 4.4, the vector
$\underline{e}$ is the vector associated to $D$. If we take the
superimposition of $D\sqcup D^{\prime}$, note that will consist of all
2-cycles and exactly one 4-cycle enclosing tile $i$. This implies that
$D\sqcup M_{-}$ will also have at least one cycle containing $i$. Moreover,
$D\sqcup M_{-}$ will have exactly one more cycle containing $i$ than that of
$D^{\prime}\sqcup M_{-}$. Hence, the corresponding vector obtained by
performing the algorithm in Theorem 4.4 must be
$\underline{e}^{\prime}+\underline{e_{i}}$ which is exactly $\underline{e}$.

Next we show the reverse direction. Suppose that given a mixed dimer
configuration $D$, using the algorithm in Theorem 4.3, we obtain
$\underline{e}$ such that $|e|=m+1$. Let $D^{\prime}$ be the mixed dimer
configuration associated to $\underline{e}^{\prime}$, where
$\underline{e}^{\prime}$ is the result of subtracting 1 from the
$i^{\text{th}}$ entry of $\underline{e}$. By the algorithm in Theorem 4.4, we
must have enclosed the tile $i$ in dimer $D$ $e_{i}$ number of times and
$(e_{i}-1)$ number of times in dimer $D^{\prime}$. By our inductive
assumption, let $\underline{e}^{\prime}$ be the vector associated to
$D^{\prime}$. Note that the order in which we flip tiles in the algorithm in
Theorem 4.3 does not matter, so to obtain the mixed dimer configuration $D$
from $D^{\prime}$, it suffices to flip the tile $i$. Therefore, we see that
the mixed dimer configuration associated to $\underline{e}$ prescribed by the
algorithm in Theorem 4.3 is exactly the mixed dimer configuration $D$ we began
with. Therefore, we conclude that these maps are indeed inverses of each
other.

∎

######  Example 4.4.

Examples 4.2 and 4.3 demonstrate each side of the bijection are inverses of
one another.

Now that we have established the connection between mixed dimer configurations
and submodule-indexing vectors, we provide the connection between submodule-
indexing vectors and representation theory. Ultimately, this gives the
connection to the $F$-polynomial through representation theory. The following
theorem is adapted from [DWZ08] and [Tra09].

######  Theorem 4.5.

Let $T$ be an ideal triangulation of a once-punctured $n$-gon and $J_{T}$ be
its associated Jacobian algebra. Let $M$ be an indecomposable $J_{T}$-module
associated to an arc $\gamma$ with dimension vector $\underline{d}$. Let
$Gr_{\underline{e}}(M)$ be the variety of all submodules of $M$ with dimension
vector $\underline{e}$ known as the quiver Grassmannian. Let $\chi$ denote the
Euler characteristic. Then

$F_{\underline{d}}=\sum_{\underline{e}}\chi(Gr_{\underline{e}}(M))\prod_{i=1}^{n}u_{i}^{e_{i}}$

where the sum ranges over $\underline{e}\in\mathbb{Z}^{n}$ with
$0\leq\underline{e}\leq\underline{d}$.

The following lemma, proven in [Tra09], helps to determine when there is a
subrepresentation of a given dimension.

######  Lemma 4.5.

Let $M^{\prime}$ and $M^{\prime\prime}$ be vector spaces of dimensions
$d^{\prime}$ and $d^{\prime\prime}$, respectively, and
$\phi:M^{\prime}\rightarrow M^{\prime\prime}$ be a linear map of maximal
possible rank min$(d^{\prime},d^{\prime\prime})$. Let $e^{\prime}$ and
$e^{\prime\prime}$ be two integers such that such that $0\leq e^{\prime}\leq
d^{\prime}$ and $0\leq e^{\prime\prime}\leq d^{\prime\prime}$. Then the
following conditions are equivalent:

  1. 1.

There exist subspaces $N^{\prime}\subseteq M^{\prime}$ and
$N^{\prime\prime}\subseteq M^{\prime\prime}$ such that
dim$(N^{\prime})=e^{\prime}$, dim$(N^{\prime\prime})=e^{\prime\prime}$, and
$\phi(N^{\prime})\subseteq N^{\prime\prime}$.

  2. 2.

$e^{\prime}-e^{\prime\prime}\leq[d^{\prime}-d^{\prime\prime}]_{+}$.

For a representation of quiver $Q=(Q_{0},Q_{1})$ with dimension vector
$\underline{d}$, we determine when there is a subrepresentation with of a
given dimension vector $\underline{e}$. We do this by adapting an argument in
[Tra09] relying on Lemma 4.5 and Theorem 4.5.

######  Theorem 4.6.

For vectors $\underline{d}$ and $\underline{e}=(e_{1},\dots,e_{n})$, the
coefficient of the monomial $u_{1}^{e_{1}}u_{2}^{e_{2}}\cdots u_{n}^{e_{n}}$
in $F_{\underline{d}}$ is nonzero if and only if

  1. 1.

$0\leq\underline{e}\leq\underline{d}$,

  2. 2.

all arrows in $Q$ are acceptable, and

  3. 3.

$\nu(S)\leq 1$ for all connected components $S$ of $C$.

If all of the conditions above are satisfied, then the coefficient of
$u_{1}^{e_{1}}u_{2}^{e_{2}}\cdots u_{n}^{e_{n}}$ is $2^{c}$, where $c$ is the
number of connected components $S$ such that $\nu(S)=0$.

###### Proof.

We may restrict attention to $\underline{e}\in\mathbb{Z}^{n}$ which satisfy
the first two conditions of Definition 4.2. Otherwise, ignoring singular
arrows, Theorem 4.5 and Lemma 4.5 imply that the coefficient of
$u^{\underline{e}}$ in $F_{\underline{d}}$ is 0. If
$\underline{d}\in\\{0,1\\}^{n}$, then Theorem 4.5 follows from Theorem 7.4 of
[Tra09] and [DWZ08] in the acyclic case. The non-acyclic case reduces to the
acyclic proof because the support of quiver will be acyclic when
$\underline{d}\in\\{0,1\\}^{n}$.

If $d_{i}\geq 2$ for some $i$, then we show that $\underline{e}$ indexes a
subrepresentation if and only if $\underline{e}\leq\underline{d}$ satisfies
the criticality condition. Throughout this proof, we only consider arrows in
$Q_{1}$ which are not singular. Note that $Q_{1}$ can only contain singular
arrows when there are some $d_{i}\geq 2$. By the surface model, and more
specifically using Lemma 3.2, we can index $\underline{d}$ such that

$\displaystyle\underline{d}=\underline{e}_{p}+\cdots+\underline{e}_{r-1}+2\underline{e}_{r}+\cdots+2\underline{e}_{m-2}+\underline{e}_{m-1}+\underline{e}_{m},$

for some $1\leq p<r\leq m-2$, where $\underline{e}_{i}$ is the standard basis
vector with $i^{\text{th}}$ entry 1 and all other entries 0.

An indecomposable representation $M=(M_{i})$ with dimension vector
$\underline{d}$ can be selected in which we insist that the maps between
$M_{i}$ and $M_{j}$ where $d_{i}=d_{j}$ are identity maps. We denote maps in
the representation as $\varphi_{i,j}$ where $i$ denotes the index of the
source and $j$ the index of the taret of the map.

To compute $\chi(Gr_{\underline{e}}(M))$, we will construct all possible
subrepresentations with dimension vector $\underline{e}$. When a
representation $N=(N_{i})$ has a dimension vector with
$\underline{e}\leq\underline{d}$, the condition that each $N_{i}$ is a
subspace of $M_{i}$ is satisfied. Therefore, to check that $N$ is a
subrepresentation of $M$, it suffices to check that

$\varphi_{i,j}(N_{i})\subseteq N_{j}\text{ if }i\rightarrow j\text{ , and
}\varphi_{j,i}(N_{j})\subseteq N_{i}\text{ if }j\rightarrow i$ (1)

for all $i$ and $j$ adjacent in $Q$.

Recall from Definition 4.1 that $C=\\{i\in Q_{0}:(d_{i},e_{i})=(2,1)\\}$. Then
for any $i\notin C$, there is only one possible subspace of $M_{i}$ of
dimension $e_{i}$. For a connected component $S$ of $C$, if $i,j$ are vertices
in $S$ and $i\rightarrow j$, then condition (1) above together with the fact
that $\underline{e}\leq\underline{d}$ imply that $N_{i}\cong N_{j}$.
Therefore, when a subspace of dimension 1 is chosen for one vertex of the
component $S$, all vertices in that component must be assigned the same
subspace.

If $p\leq i,j\leq n$ with $i\rightarrow j\in Q_{1}$ are such that at least one
of $i,j$ are not in $C$ and greater than or equal to $r$ and less than or
equal to $m-2$, then it is straightforward to check that
$\varphi_{i,j}(N_{i})\subseteq N_{j}$. For example, if $r\leq i\leq m-2$ and
$i\notin C$, then by definition of $C$, since $d_{i}=2$, we have that
$e_{i}=0$ or $e_{i}=2$. If $e_{i}=0$, then $N_{i}=0$ so
$\varphi_{i,j}(N_{i})=0$. If $e_{i}=2$, since $i\rightarrow j$ is acceptable,
by Lemma 4.5, we have that $e_{j}=d_{j}$, so $N_{j}=M_{j}$, which contains
$\varphi_{i,j}(N_{i})$. Therefore, it only remains to show that the property
(1) holds for $(i,j)=(r-1,r)$ if $r\in C$, and that the property holds for
$(i,j)=(m-2,m-1),(m-2,m)$ if $m-2\in C$.

We will consider which one-dimensional subspaces can be assigned to each
component. To do this, we construct three distinct one-dimensional subspaces
of the 2-dimensional subspace $M_{m-2}$. For one-dimensional $M_{j}$ and
2-dimensional $M_{k}$ with $j\rightarrow k\in Q_{1}$, let
$V_{\\{j,k\\}}=Im(\varphi_{j,k})$. In the case that $k\rightarrow j\in Q_{1}$,
let $V_{\\{j,k\\}}=Ker(\varphi_{k,j})$. Then $V_{\\{r-1,r\\}}$,
$V_{\\{m,m-2\\}}$, and $V_{\\{m-1,m-2\\}}$ are three distinct one-dimensional
subspaces of $M_{m-2}$.

We call a pair of vertices $(i,j)$ a critical pair when the arrow between them
in $Q_{1}$ is a critical arrow. Note that when $(i,j)=(r-1,r)$ is a critical
pair, property (1) is satisfied for $N_{r-1}$ and $N_{r}$ if and only if
$N_{r}=V_{\\{r-1,r\\}}$. When $(i,j)=(m-2,m)$ is a critical pair, property (1)
is satisfied if and only if $N_{m-2}=V_{\\{m,m-2\\}}$. Analogously, when
$(i,j)=(m-2,m-1)$, (1) is satisfied if and only if
$N_{m-2}=V_{\\{m-1,m-2\\}}$.

Let $S$ again be a connected component of $C$. Recall that once a subspace of
dimension 1 is chosen for one vertex of $S$, all vertices must of $S$ must be
assigned the same subspace. We consider cases for the number of critical
arrows in $S$. If $\nu(S)=0$, then any one-dimensional subspace of $\Bbbk^{2}$
may be assigned to all of the vertices of $S$. This corresponds to
$\mathbb{P}^{1}$, so $\chi(\mathbb{P}^{1})=1$. Next suppose that $\nu(S)=1$.
If $(m-2,m)$ is a critical pair and $m-2\in S$, then the chosen subspace for
$S$ must be $V_{m}$. Similarly, if $(m-2,m-1)$ is a critical pair and $m-2,\in
S$, then the chosen subspace assigned to the vertices of $S$ must be
$V_{m-1}$. Since there is no choice of subspace, the Euler characteristic is
1. Finally, if $\nu(S)\geq 2$, there is no one-dimensional subspace which can
be assigned to the vertices of $S$ which will satisfy condition (1). This
implies that the Euler characteristic is 0. ∎

With this, the only thing that remains to show to prove Theorem 4.1 is to
verify that the number of cycles on a mixed dimer configuration $D$ associated
to a submodule-indexing vector $\underline{e}$ is given by the number of
connected components $S$ of $C$ such that with $\nu(S)=0$.

###### Proof.

First note that we cannot have a cycle on a tile that is a dimer
configuration, i.e. we need $d_{i}=2$ in order for $i$ to potentially be
enclosed by a cycle. Also, note that there are no cycles in $M_{-}$ by the
convention of the black and white coloring in the definition of $M_{-}$, so in
order for tile $i$ to be enclosed by a cycle, we must have that $e_{i}\geq 1$.
This tells us that the set of all tiles that can potentially be enclosed by a
cycle are contained within $S=\\{i\in Q~{}:~{}(d_{i},e_{i})=(2,1)\\}$.

Fix $D$ associated to $(\underline{d},\underline{e})$. Let $C$ be a connected
component of $S$. Suppose that $\nu(C)\neq 0$, that is, there is some $c\in C$
such that $c$ is the vertex of a critical arrow. Namely, this $c$ must be
connected to a tile outside of $S$ by definition of the $d,e$ coordinates of
the other vertex of the critical arrow. Let $c^{\prime}$ be the other vertex
of the critical arrow involving $c$. Note that if $c\to c^{\prime}$ about the
edge $\alpha$, then $(d_{c^{\prime}},e_{c^{\prime}})=(1,0)$ and we have that
$n_{c,c^{\prime}}=\max(d_{c}-d_{c^{\prime}},0)+(e_{c^{\prime}}-e_{c})=1-1=0$.
If $c^{\prime}\to c$ about the edge $\alpha$, then
$(d_{c^{\prime}},e_{c^{\prime}})=(1,1)$ and we have that
$n_{c^{\prime},c}=\max(d_{c}^{\prime}-d_{c},0)+(e_{c}-e_{c^{\prime}})=0$.
Therefore, by Lemma 4.1, the edge $\alpha$ cannot have an edge from $D$ which
tells us that no cycle in $D$ can enclose tile $c$. This tells us that if a
cycle encloses any tile or collection of tiles in $C$, then $\nu(C)=0$.

Now, suppose that $C$ is a connected component of $S$ with $\nu(C)=0$, i.e. no
critical arrows involve any vertex $c$ from $C$. Suppose that $C$ is comprised
of the tiles $i,\dots,m$ where $i$ is the minimal tile in $C$ and $i^{\prime}$
is the maximal tile in $C$ with respect to the indexing in $Q$. We aim to show
that there exists a cycle in $D$ enclosing all of $C$. Note that for $i\leq
j\leq i^{\prime}-1$, we have that $(d_{j},e_{j})=(2,1)$, so there are no edges
in $D$ are on the edge straddling tiles $j$ and $j+1$ by Lemma 4.1 since
$n_{j,j+1}=0$. This implies that if we can show that all the boundary edges of
$C$ are in $D$, that there is a cycle enclosing all of $C$ in $D$.

Note that as $e_{j}=1$ for all $i\leq j\leq i^{\prime}$, we have that each of
the tiles in $C$ have been flipped exactly once from $M_{-}$ using our
bijection of adding 1’s to the $\underline{e}$-vector coinciding with flipping
the corresponding tile from Theorem 4.4. Note that if a boundary edge $\alpha$
on tile $t$ is oriented black to white clockwise with respect to $G_{2}$, then
by definition of $M_{-}$, $\alpha$ must have two edges distinguished in
$M_{-}$. After one flip at tile $t$, $w(\alpha)$ decreases by 1. This gives
that this edge now appears distinguished as a single edge in $D$. On the other
hand, if $\alpha$ is oriented white to black clockwise with respect to
$G_{2}$, then by definition of $M_{-}$, $\alpha$ has no edges distinguished in
$M_{-}$. After performing a flip on tile $t$, we have that $w(\alpha)$
increases by 1. This gives that $\alpha$ now has exactly one edge
distinguished in $D$. Hence, all the boundary edges on $C$ have exactly one
edge on them in $D$.

Now, we need to show that the cycle closes up on the interior edges of $G$,
i.e. if $i>0$ and/or $i^{\prime}<m-1$, we need to verify that the edges
straddling tiles $i,i-1$ and $i^{\prime},i^{\prime}+1$ have one edge
distinguished in $D$. Note that as $i,i^{\prime}\in C$ with $\nu(C)=0$, no
arrow involving $i$ or $i^{\prime}$ is critical. By the structure of
$\underline{d}\in\Phi_{+}$, we have that $d_{i-1}\geq 1$ and $d_{m+1}\geq 1$.
Seeking a contradiction, suppose that $n_{i,i-1}=0$. Then,
$\max(d_{i}-d_{i-1},0)=e_{i}-e_{i-1}$, i.e. $\max(2-d_{i-1},0)=1-e_{i-1}$.
Since $e_{i-1},d_{i-1}\in\\{0,1,2\\}$, this only occurs if either $d_{i-1}=1$
giving that $e_{i-1}=0$ or if $d_{i-1}=2$ giving that $e_{i-1}=1$. In the
first case, this gives that the arrow $i\to i-1$ is critical which contradicts
that $\nu(C)=0$. In the second case, $i-1\in C$ which contradicts the
minimality of $i$. Hence, $n_{i,i-1}>0$ and by Lemma 4.4, this means there
must be an edge straddling tiles $i,i-1$ in $D$. Note that this same argument
holds for showing that there must be an edge in $D$ straddling tiles
$i^{\prime},i^{\prime}+1$. Therefore, we see that one cycle is formed around
all of $C$ when $\nu(C)=0$. Hence, the number of cycles in $D$ is exactly the
number of connected components $C$ of $S$ with $\nu(C)=0$. ∎

Therefore, we have shown that the $F$-polynomial is indeed given by the mixed
dimer configuration generating function given in Theorem 4.1.

###  4.1 $g$-Vectors

As detailed in [FZ07], see Definition 2.9 above, a cluster variable can be
reconstructed from an $F$-polynomial, but only if it is also accompanied by
the extra data of a $g$-vector. In our prequel to this paper [MW20], we were
able to define a weighting scheme on the edges of the base graph and then show
that the $g$-vector can be realized using weight of the minimal mixed dimer
configuration in the case of an acyclic quiver of type $D_{n}$. We conjecture
that a similar weighting scheme exists in the non-acyclic case, which would
lead to a similar interpretation for $g$-vectors, in order to have a full
dimer theoretic interpretation of the Laurent expansions for these cluster
variables.

######  Conjecture 4.1.

The $g$-vector associated to $Q$ and $\underline{d}$-vector, denoted
$\underline{g}=\underline{g}(Q,\underline{d})$, is given by

$\underline{g}=\deg\left(\frac{\text{wt}(M_{-})}{\underline{x}^{\underline{d}}}\right)$

where $\deg(x_{0}^{a_{0}}x_{1}^{a_{1}}\cdots
x_{n-1}^{a_{n-1}}):=(a_{0},a_{1},\dots,a_{n-1})$ and $\text{wt}(M_{-})$ is
some weighting scheme given on the base graph.

Based on preliminary computations, it appears that this weighting scheme must
be different than the one in [MW20] to account for singular arrows. We also
conjecture that the edge weights may have to be allowed to be more complicated
terms than just a single $x_{i}$, which differs from the acyclic case. The
proof of our result for $g$-vectors in the acyclic case also relies on a
classification of $g$-vectors found in [Tra09], that we would have to extend
to the cyclic case.

##  Appendix A Classification of $\underline{d}$-vectors

In this appendix, we classify all crossing vectors, $\underline{d}$-vectors,
in all non-acyclic type $D_{n}$ cluster algebras. To accomplish this, we rely
on the surface model for type $D_{n}$ cluster algebras - a once-punctured disk
with $n$ marked points. We split our work into four types following Vatne’s
classification of type $D_{n}$ quivers [Vat10]. In each type, we rely on the
computing the crossing vector associated to an arc $\gamma$ in a triangulation
of the once-punctured disk with $n$ marked points by keeping track of the arcs
that $\gamma$ crosses in the given triangulation [FST06].

If $\gamma$ crosses any arc in a triangulation twice, we streamline the Vatne
categorization to make our catalog more concise. To this end, we describe the
precise situations where this can occur. Any arc of the triangulation that is
crossed twice by $\gamma$ must be a peripheral arc i.e. not incident to the
puncture. Moreover, if we follow the path of $\gamma$, out of all the
peripheral arcs that are crossed twice, there is a unique peripheral arc that
is closest to the puncture, up to isotopy. Without loss of generality, we let
$c$ denote this arc. This arc splits the triangulation into two parts. On one
side of $c$, the sub-triangulation is that of a unpunctured polygon i.e. has
corresponding sub-quiver $Q^{\prime}$ is of type $A_{m}$ with vertex $c$ is
one of its endpoints. On the other side of $c$, the sub-triangulation is that
of a punctured bi-gon, with arcs $a$ and $b$ incident to the puncture such
that the only peripheral arc that $\gamma$ crosses is $c$ (following Vatne’s
labeling of vertices in Figure 4 that we repeat here for convenience). Arc
$\gamma$ must cross arcs $a$ and $b$ along its path, and hence the
corresponding sub-quiver consists of a fork of $a$ and $b$ attached to $c$,
with any of the four orientations on the edges $a-c$ and $b-c$ depending on
the four possible triangulations of the bigon.

Figure 18:

With these observations above, the types II and III surfaces degenerate to the
type I surface. Moreover, arc $\gamma$ must cross at most three other arcs,
each of which corresponds to a vertex of the quiver. Either the vertices
$a,c,d$ and not $b$ or the vertices $b,c,d$ and not $a$, which implies the
type $D_{n}$ parts of both of these surfaces degenerate to the fork in the
type $D_{n}$ part of the type I surface.

Furthermore, we can streamline these types of surfaces even more so. Observe
that if an arc $\gamma$ is supported on some subset of $a,b,c$, but not $d$ or
some subset of $a,b,d$, but not $c$, we can treat the arcs $d,c$ respectively
as boundary arcs and the arc can be considered a type I arc.

The last reduction we make reduces case work for the type A parts of the
surfaces. Namely, we rely on sectioning off the triangulations of our once-
punctured disk into its type $A_{m}$ parts. In our catalog, we focus on the
general structure of the $\underline{d}$-vectors and leave space for all
potential type $A_{m}$ crossing vectors that correspond to triangulations of
polygons.

With those reductions, we begin by cataloging the type I surface. There are
nuances between the orientation of the arrows between $a,b,c$ with what the
surface model looks like; however, there are six families of arcs that are
produced in any type I case.

######  Lemma A.1.

Consider any orientation of the quiver corresponding to the type I surface.
Suppose that first three coordinates of the $\underline{d}$-vector are given
by $a,b,c$ respectively. The fourth coordinate is the first coordinate in the
type $A_{m}$ part of the surface. Then, there are six families:

$(0,0,0,\text{type }A)~{}~{}(0,0,1,\text{type }A)~{}~{}(1,1,1,\text{type }A)$
$(0,1,1,\text{type }A)~{}~{}(1,0,1,\text{type }A)~{}~{}(1,1,2,\text{type }A)$

In order to see the proof of this lemma, we systemically go through the
possibilities for arcs when the arrows between $a$ and $c$ and $b$ and $c$ are
pointed in the same direction and when they are pointing in opposite
directions.

####  A.0.1 Type IA

Let type IA be the orientation of the type I quiver where $a\to c$ and $c\to
b$. We begin by categorizing the families of arcs that do not cross any arc in
a triangulation twice. Namely, we analyze the possibilities for the support of
$a,b,c$.

Family I. Any arc that does not cross the type $D_{n}$ part of the surface
i.e. the arcs $a,b,c$ will be of the form: $\underline{d}=(0,0,0,\text{type
}A_{m})$.

Family II. Arcs that emanate from the marked point $y$ and wrap clockwise
around the puncture: $\underline{d}=(0,0,1,\text{type }A)$.

Family III. Arcs that emanate from the marked point $x$ and wrap
counterclockwise around the puncture: $\underline{d}=(1,1,1,\text{type }A)$.

Family IV. Arcs going into the puncture, tagged $\underline{d}_{\Bowtie}$ and
untagged $\underline{d}$ are given by:
$\underline{d}_{\Bowtie}=(0,1,1,\text{type }A)$ and
$\underline{d}=(1,0,1,\text{type }A)$.

These are all the families of arcs that cross existing arcs in the
triangulation at most once, pictured in Figure 19.

Figure 19: Examples of arcs in families for surface IA: I, II, III and IV from
left to right drawn in red.

We now consider when arcs can cross multiple times i.e. where 2’s can appear
in the $\underline{d}$-vectors. Up to isotopy, the only type $D_{n}$ arc that
can be crossed twice is $c$; namely, $a$ and $b$ cannot be crossed twice.
Observe that arcs whose endpoints only involve marked points in the “type
$D_{n}$” part i.e. endpoints of arcs $a,b,c$ cannot cross any arc twice, so at
least one of the endpoints must be involved the “type $A_{m}$” part i.e. a
marked point in the triangulation of the $n$-gon. In particular, any such arc
must wrap counterclockwise around the puncture. This gives the last family of
arcs in Type 1A: $\underline{d}=(1,1,2,\overrightarrow{\text{type
}A_{\alpha}}+\overrightarrow{\text{type }A_{\beta}})$, where we split up our
arc into two pieces: $\alpha$, an arc in the $n$-gon involving the crossings
that appear before the first crossing of $c$ and $\beta$, an arc in the
$n$-gon involving the crossings that appear after the second crossing of $c$.
See Figure 20.

Figure 20: Example of arc in that cross twice in surface type IA drawn in red,
where $\alpha,\beta$ components are drawn dashed in gray.

######  Remark A.1.

We note that for crossing vectors $\underline{d}$ of this format, a quiver
theoretic formula for F-polynomials was explored by unpublished work of the
second author and Lauren Williams where the F-polynomial was conjecturally
written as a difference. The contributions being subtracted therein appear to
correspond to the node polychromatic mixed dimer configurations that appear in
the larger poset $\overline{P}$ defined in Definition 3.8 but not poset $P$
from Definition 3.9.

####  A.0.2 Type IB

Let type IB be the orientation of the type I quiver where $a\to c$ and $b\to
c$. We begin by categorizing the families of arcs that do not cross any arc in
a triangulation twice. Namely, we analyze the possibilities for the support of
$a,b,c$.

Family I. Any arc that does not cross the type $D_{n}$ part of the surface
i.e. the arcs $a,b,c$ will be of the form: $\underline{d}=(0,0,0,\text{type
}A)$.

Family II. Arcs that emanate from the marked point $y$ and wrap clockwise
around the puncture: $\underline{d}=(0,1,1,\text{type }A)$.

Family III. Arcs that emanate from the marked point $x$ and wrap
counterclockwise around the puncture: $\underline{d}=(1,0,1,\text{type }A)$.

Family IV. Arcs going into the puncture, tagged $\underline{d}_{\Bowtie}$ and
untagged $\underline{d}$ that involved at least one endpoint in the “type
$A_{m}$” part of the surface: $\underline{d}_{\Bowtie}=(0,1,1,\text{type }A)$
and $\underline{d}=(1,0,1,\text{type }A)$.

We also have two additional arcs that are the tagged versions of the arcs
$a,b$ which give two additional $\underline{d}$-vectors:
$\underline{d}_{a\Bowtie}=(0,1,0,\dots,0)$ and
$\underline{d}_{b\Bowtie}=(1,0,\dots,0)$.

These are all the families of arcs that cross existing arcs in the
triangulation at most once, pictured in Figure 21.

Figure 21: Examples of arcs in families for surface IB: I, II, III and IV from
left to right drawn in red.

We now consider when arcs can cross multiple times i.e. where 2’s can appear
in the $\underline{d}$-vectors. As in Type IA, we see again that the only
family $\underline{d}$-vector in this case is:
$\underline{d}=(1,1,2,\overrightarrow{\text{type
}A_{\alpha}}+\overrightarrow{\text{type }A_{\beta}})$, where $\alpha,\beta$
are defined as in Type 1A. See Figure 22.

Figure 22: Example of arc in that cross twice in surface type IA drawn in red,
where $\alpha,\beta$ components are drawn dashed in gray.

####  A.0.3 Type II

The surface model in type II has two potential type $A_{m}$ parts
corresponding to the quivers $Q^{\prime},Q^{\prime\prime}$ i.e. two
triangulations of polygons $P^{\prime},P^{\prime\prime}$. We begin by
cataloging the families of arcs that do not cross any arc in a triangulation
twice. We index the $\underline{d}$-vectors as follows: the first four
coordinates in order are $a,b,c,d$, the next set of coordinates are indexed by
arcs in the polygon $P^{\prime}$ and the last set of coordinates are indexed
by arcs in the polygon $P^{\prime\prime}$. Recall any arc that crosses an arc
in the triangulation more than once or any arc that is supported on some
subset of $a,b,c$, but not $d$ or some subset of $a,b,d$, but not $c$, are
previously cataloged in the type I surface. So, it suffices to catalog any
arcs in the type II surfaces that are supported on both $c$ and $d$. It turns
out that there are only two families of arcs that are supported on both $c$
and $d$ shown in Figure 23.

Family I. Arcs that start in one polygon and end in the other, wrapping
clockwise around the puncture. These arcs cannot start or end at the marked
points $x$ or $y$:
$\underline{d}=(1,1,1,1,\text{type }A,\text{type }A)$.

Family II. Arcs that start in one polygon and end in the other, wrapping
counterclockwise around the puncture. These arcs cannot start or end at the
marked points $x$ or $y$: $\underline{d}=(0,0,1,1,\text{type }A,\text{type
}A)$.

Figure 23: Examples of arcs in families I and II from left to right in the
type II surface drawn in red.

####  A.0.4 Type III

Similarly to type II, the surface model in type III has two potential type
$A_{m}$ parts corresponding to the quivers $Q^{\prime},Q^{\prime\prime}$ i.e.
two triangulations of polygons $P^{\prime},P^{\prime\prime}$. We begin by
cataloging the families of arcs that do not cross any arc in a triangulation
twice. We index the $\underline{d}$-vectors as follows: the first four
coordinates in order are $a,b,c,d$, the next set of coordinates are indexed by
arcs in the polygon $P^{\prime}$ and the last set of coordinates are indexed
by arcs in the polygon $P^{\prime\prime}$.