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%% define your title in the usual way
\title{Snake Graphs and Frieze Patterns from Orbifolds}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Esther Banaian \and Elizabeth Kelley}

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\address{Department of Mathematics, University of Minnesota, Minneapolis, 55455 USA\\}

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\abstract{We give an explicit combinatorial formula for the Laurent expansion of any cluster variable of any generalized cluster algebra from a triangulated orbifold, with respect to any initial seed, and associate frieze patterns to orbifolds.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{orbifold, generalized cluster algebras, snake graphs, frieze}

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\begin{document}

\maketitle
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\section{Introduction}

Our main result is \cref{GenSnake}, a definition of snake graphs from orbifolds that gives combinatorial formulas for the Laurent expansions of cluster variables, with respect to any initial seed, for generalized cluster algebras from orbifolds. In \cref{GenSnakeGenArc,GenBand}, we extend this definition to closed curves and generalized arcs, respectively. This construction also allows us to associate frieze patterns to orbifolds. \cref{OrbFrieze,countingrows} give some properties of these frieze patterns. 
%The paper is organized as follows. In sections 2 and 3, we recall essential background information about generalized algebras and cluster algebras from orbifolds. In Section 4, we state our main result: a definition of generalized snake graphs that gives combinatorial formulas for the Laurent expansions of cluster variables with respect to any initial seed, for Chekhov-Shapiro algebras from orbifolds.
This work is motivated by, but does not use, unpublished work of Gleitz and Musiker  \cite{Gleitz-Musiker}.

\section{Chekhov-Shapiro Algebras}
In 2014, Chekhov and Shapiro defined an extension of ordinary cluster algebras \cite{Fomin-Zelevinsky-I} without the restriction that exchange polynomials be strictly binomial \cite{Chekhov-Shapiro}. Often known as ``generalized cluster algebras'', we refer to these as \emph{Chekhov-Shapiro algebras} or simply \emph{CS algebras}. The motivation for introducing these algebras came from a desire to extend existing work on the Teichm\"{u}ller spaces of Riemann surfaces with holes and orbifold points of order two and three to the more general case of Riemann surfaces with holes and orbifold points of arbitrary order \cite{Chekhov-Mazzocco-Yangians}.

For a fixed semifield $(P,\oplus,\cdot)$, let $\mathcal{F}$ be isomorphic to $\mathbb{QP}[x_1,\dots,x_n].$

\begin{definition}
A \emph{generalized labeled seed} in $\mathcal{F}$ is a quadruple $(\textbf{x}, \textbf{y}, B, \mathbf{Z})$ where $\textbf{x} = (x_1, \dots, x_n)$ is a free generating set for $\mathcal{F}$, $\textbf{y} = (y_1, \dots, y_n)$ is an $n$-tuple with elements in $\mathbb{P}$, $B = (b_{ij})$ is a skew-symmetrizable $n\times n$ integer matrix, and $\mathbf{Z} = (z_1, \dots, z_n)$ is a tuple of exchange polynomials.
%- i.e., there exists a $n\times n$ diagonal matrix $D$ with positive integer entries such that $d_{ii}b_{ij} = -d_{jj}b_{ji}$.
We refer to $\textbf{x}$ as the \emph{cluster} of $(\textbf{x}, \textbf{y}, B)$, $\textbf{y}$ as the \emph{coefficient tuple}, and $B$ as the \emph{exchange matrix}. The elements $x_1, \dots, x_n$ are the \emph{cluster variables} of $(\textbf{x},\textbf{y},B)$ and $y_1, \dots, y_n$ are its \emph{coefficient variables}.
\end{definition}

Mutation in direction $k$ reverses the order of coefficients in the exchange polynomial $z_k$ and does not modify the other exchange relations. When all elements of $\mathbf{Z}$ are binomials, generalized mutation coincides with ordinary mutation. 
%When $z_k$  has reciprocal coefficients, mutation in direction $k$ does not change $z_k$. For this reason, Nakanishi refers to the coefficients as \emph{frozen variables}.

\begin{definition}
Let $\mathbb{T}_n$ be the $n$-regular tree graph, where each vertex has adjacent edges labeled by $\{ 1, \dots, n \}$. A \emph{generalized cluster pattern} is a map $\chi: v \rightarrow \Sigma_v$ that sends each vertex $v$ of $\mathbb{T}_n$ to a labeled generalized seed $\Sigma_v = (\mathbf{x}_v, \mathbf{y}_v, B_v, \mathbf{Z})$, such that $\Sigma_{v'} = \mu_k \Sigma_{v}$ if and only if $\mathbb{T}_n$ contains an edge labeled $k$ between $\Sigma_{v'}$ and $\Sigma_v$. The \emph{generalized cluster algebra} associated with $\chi$ is defined as $\mathcal{A}(\chi) = \mathbb{ZP} \left[ x : x \in \bigcup_{v} \textbf{x}_v \right]$ where $\{ x : x \in \bigcup_{v} \textbf{x}_v  \}$ is the set of all cluster variables appearing in seeds that are (generalized-)mutation-equivalent to $\Sigma_v$.
\end{definition}

%As we might hope to see in any generalization of ordinary cluster algebras, Chekhov-Shapiro algebras also exhibit the Laurent phenomenon.

Chekhov and Shapiro proved that their algebras exhibit the Laurent Phenomenon - i.e., the cluster variables can be expressed in terms of any cluster as a Laurent polynomial with non-negative coefficients. They also prove positivity for a subclass of their algebras from orbifolds; here, we offer an explicit combinatorial proof for the same subclass.

%The question of positivity remains open for general CS algebras, but Chekhov and Shapiro verified positivity for a subclass of CS algebras from orbifolds and offered a conjecture about positivity for a slightly more general subclass.
%In general, the question of positivity for Chekhov-Shapiro algebras remains open. Chekhov and Shapiro proved that 

%giving explicit construction for formulas.. taking away scaffolding 

%\begin{theorem}[\cite{Chekhov-Shapiro}]
%\label{CS-reciprocalDeg2}
%Any CS algebra  whose exchange polynomials are all reciprocal and of degree at most two can be associated with a Riemann surface with at least one puncture and arbitrarily many orbifold points of arbitrary order. Its cluster variables can be expressed in terms of any cluster as a Laurent polynomial with non-negative coefficients in $\mathbb{ZP}$.
%\end{theorem}

%The proof of Theorem \ref{CS-LP} relies on the caterpillar lemma \cite{Fomin-Zelevinsky-LP}, with the majority of the argument focused on showing that the exchange pattern of an arbitrary Chekhov-Shapiro algebra satisfies the conditions of the caterpillar lemma.

%After establishing that the Laurent phenomenon holds for all Chekhov-Shapiro algebras, Chekhov and Shapiro then formulate the analogue of the positivity conjecture and show that it holds for a particular subset of Chekhov-Shapiro algebras.

%In fact, the original paper by Chekhov and Shapiro included proof that a particular subset of Chekhov-Shapiro algebras, which are included in the subset addressed by Nakanishi, not only exhibit the Laurent phenomenon, but satisfy the analogue of the positivity conjecture.

%\begin{conjecture}[\cite{Chekhov-Shapiro}]
%Suppose $\mathcal{A} = (\mathbf{x}, \mathbf{y}, B, \mathbf{Z})$  is a CS algebra with reciprocal exchange polynomials with positive coefficients - 
%i.e., for each $\mathbf{p_i}$, we have ${p_{i,s} = p_{i,d_i - s}}$ and $p_{i,s} \geq 0$. 
%Then $x_1, \dots, x_n$ can be written in terms of any cluster of $\mathcal{A}$ as Laurent polynomials with non-negative coefficients.
%\end{conjecture}

%G: Results above on the chopping block

%G: more important to mention Z's here than p's

%Note that when the normalization condition is met, the algebras addressed in Theorem \ref{CS-reciprocalDeg2} are instances of those considered by Nakanishi.

%Outside of this particular case, the positivity conjecture for Chekhov-Shapiro algebras remains open. %See note on page 19 of CS "We checked by direct inspection that the statement holds for finite type rank 2 cluster algebras." 

\section{Cluster Algebras from Orbifolds}

An orbifold is a generalization of a manifold where the local structure is given by quotients of open subsets of $\mathbb{R}^n$ under finite group actions.
%rather than simply by open subsets of $\mathbb{R}^n$. 
In parallel with the classification of cluster algebras associated with triangulated surfaces \cite{FST-triangulatedSurfaces, FT-II}, Felikson, Shapiro, and Tumarkin established both a notion of triangulating orbifolds and a classification of cluster algebras from orbifolds \cite{FST-orbTriang}. In this section, we briefly review some nomenclature and definitions for triangulations of orbifolds that will be used in later sections. For more details and many examples, we refer the reader to the original paper  \cite{FST-orbTriang}.

\begin{definition}
An \emph{orbifold} $\mathcal{O}$ is a triple $(S,M,Q)$, where $S$ is a bordered surface, $M$ is a finite set of marked points, and $Q$ is a finite set of orbifold points, such that: no point is both a marked point and an orbifold point (i.e., $M \cap Q = \emptyset$); all orbifold points are interior points of $S$; and each boundary component of $S$ contains at least one marked point. For notational convenience, $\partial \mathcal{O}$ is often used to refer to $\partial S$.
\end{definition}

Each orbifold point has an associated order. Unlike in \cite{Felikson-Shapiro-Tumarkin} where all orbifold points are order 2 or $\frac12$, our orbifold points are associated with positive integer orders, $p \geq 2$. 

%Somewhere should mention that our orbifold points come w order a positive integer >= 2, and contrast w papers with orbifold points of orders 2 and 1/2. Later can say orbifold points of order 2 give skew-symm cluster algebras, while order > 2 give CS algebras? 

\begin{definition}
An \emph{arc} $\gamma$ on an orbifold $\mathcal{O} = (S,M,Q)$ is a non-self-intersecting curve in $S$ with endpoints in $M$ that is otherwise disjoint from $M$, $Q$, and $\partial \mathcal{O}$. Curves that are contractible onto $\partial \mathcal{O}$ are not considered arcs. Arcs are considered up to isotopy class. An arc which cuts out an unpunctured monogon with exactly one point in $Q$ is called a \emph{pending arc} (sometimes drawn as an arc between the marked and orbifold points) , while all other arcs are called \emph{ordinary arcs}.
\end{definition}

We will also consider \emph{generalized arcs}, which have the same restrictions except are allowed self-intersections.

%remark about loop vs arc to orb point


\begin{definition}
%Two arcs are considered \emph{compatible} if their isotopy classes contain non-intersecting representatives and, if both are pending arcs, they do not share an orbifold point. A \emph{triangulation} is a maximal collection of pairwise compatible arcs.
Two arcs are considered \emph{compatible} if their isotopy classes contain non-intersecting representatives. A \emph{triangulation} is a maximal collection of pairwise compatible arcs.
\end{definition}

%Could include facts: every triangulation has same # of arcs, and triangulations are all related by flips (although I am not sure if this has been shown for orbifold surfaces, but probably can be inferred from a similar statement about punctured surfaces, using a correspondence pending arcs <-> loop and radius to a puncture)

Although our ultimate goal is to construct snake graphs directly from the triangulated orbifold rather than lifting to some cover, our proof requires considering a particular covering space \cite{Chekhov-Shapiro, L-FV} of an orbifold with a single orbifold point of order $p$:

%will require considering a particular covering space. For an orbifold with a single orbifold point of order $p$, multiple papers give constructions of a $p$-fold covering \cite{Chekhov-Shapiro, L-FV} along the following lines:

\begin{definition}
Let $\Sigma_n$ be an $(n+1)$-gon containing one orbifold point of order $p$ with vertices labeled $v_1, \dots, v_{n+1}$ in counterclockwise order, $\widetilde{\Sigma}_n$ be a regular $p(n+1)$-gon with vertices labeled $v_1, v_2, \dots, v_{pn+p}$ in counterclockwise order, and $T$ be a triangulation of $\Sigma_n$. The triangulation $\widetilde{T}$ is constructed via the following procedure:
\begin{itemize}
\item For arcs in $T$ with endpoints $v_i \neq v_j$, add the $p$ arcs in $\widetilde{T}$ with endpoints $v_{i+k(n+1)}$ and $v_{j+k(n+1)}$, with $k \in [p-1]$. For loops based at $v_i$, add arcs from $v_{i+k(n+1)}$ to $v_{i+(k+1)(n+1)}$.
\item If $p > 3$, $\widetilde{T}$ will still have a central $p$-gon that hasn't been fully triangulated. Denote the vertices of this central polygon as $w_1, \dots, w_p$ and add arcs with endpoints $w_i$ and $w_{i+2}$ (modulo $p$). A complete triangulation can be obtained by iteratively repeating this step until the central polygon has been reduced to a triangle.
%G: if space, mention p = 2 gives centrally symmetric triangulations
\end{itemize}
We refer to $\widetilde{\Sigma}_n$ as either the \emph{$\mathbb{Z}_p$-orbit space of $\widetilde{\Sigma}_n$} or the \emph{p-fold covering}.
\end{definition}

An example of the $3$-fold and $4$-fold coverings for a triangle with a single orbifold point are shown in \cref{fig:pfold}. Note that for all $k \in [p-1]$, restricting $\widetilde{T}$ to the vertex set $\{v_{1+k(n+1)}, \dots, v_{(n+1) + k(n+1)} \}$, restoring the orbifold point, and then identifying $v_{i+k(n+1)}$ and $v_{(n+1)+k(n+1)}$ yields a copy of $T$. 

The case of cluster algebras from surfaces (in our notation, $Q = \emptyset$) is well studied. The correspondence between surfaces and cluster algebras is established by Fomin-Thurston.

%Maybe this could be made shorter?

\begin{figure}
\begin{center}
\includegraphics[scale=1.5]{Diagrams/pfoldcover01}
\includegraphics[scale=1.3]{Diagrams/pfoldcover02}
\includegraphics[scale=1.3]{Diagrams/pfoldcover03}
\caption{An example of a triangulated orbifold with a single orbifold point of order $p$ (left) and the covering spaces for $p=3$ (middle) and $p=4$ (right).}
\label{fig:pfold}
\end{center}
\end{figure}


% Maybe we can just say "there is a way to generalize with multiple orb points, but we only need to look locally near one orb point" 

%It's possible to construct coverings for surfaces with multiple orbifold points using local coverings of each orbifold point. We will use the following construction, due to Chekhov and Shapiro \cite{Chekhov-Shapiro}:

%\begin{definition}
%Let $\mathcal{O} = (S,M,\mathcal{Q})$ be an orbifold with at least one puncture and let $s = |M|$ and $r = |\mathcal{Q}|$. A triangulatable covering for this orbifold can be constructed as follows:
%\begin{itemize}
%    \item Partition $\mathcal{Q}$ into (possibly empty) subsets $\delta_1, \dots, \delta_s$ and assign subset $\delta_i$ to the $i$-th puncture. Within each $\delta_i$, establish an arbitrary cyclic order.
%\end{itemize}
%\end{definition}


\begin{theorem}[Fomin-Thurston \cite{FT-II}]
Given a surface with marked points, $(S,M)$, there exists a unique cluster algebra $\mathcal{A} = \mathcal{A}(S,M)$ with the following properties: (1) the seeds are in bijection with tagged triangulations of $(S,M)$;
(2) the cluster variables are in bijection with tagged arcs in $(S,M)$; and (3) the cluster variable $x_\gamma$ corresponding to arc $\gamma$ is given by the lambda length of $\gamma$, in terms of some initial triangulation. 
\label{SurfaceCA}
\end{theorem}

In this dictionary, mutation of cluster variables in $\mathcal{A}$ is equivalent to flipping arcs in the triangulation $T$. An arc $\tau$ in $T$ acts locally as a diagonal in a quadrilateral. Thus, the result of flipping $\tau$ in $T$ is a new triangulation, $T' = (T - \{\tau\}) \cup \{\tau' \}$ where $\tau'$ is the other diagonal of the quadrilateral surrounding $\gamma$. 
%Part of the theorem above is that the flipping arcs and cluster mutation are compatible operations. 

In \cite{Chekhov-Shapiro}, Chekhov and Shapiro show that flipping pending arcs induces a trinomial exchange relation, which allows them to give an algebraic structure to the dynamics of arcs on orbifolds. We set $\lambda_p$ to be $2\cos(\pi/p)$. 
%More specifically,
%if $\gamma'$ is the result of flipping the pending arc $\gamma$ within a triangulation $T$, then $\lambda(\gamma')\lambda(\gamma) = y^2 + \lambda_p yz + z^2$ where $y$ and $z$ are products of lambda lengths of arcs in $T$ and $\lambda_p:= 2\cos(\pi/p)$ where $p$ is the order of the orbifold point incident to $\gamma$ and $\gamma'$. The work of Chekhov and Shapiro implies the following. 

\begin{lemma}[Chekhov-Shapiro \cite{Chekhov-Shapiro}]
\cref{SurfaceCA} also holds for an orbifold $\mathcal{O}$ where $ \mathcal{A}(\mathcal{O})$ is a CS algebra, with all exchange polynomials being either $z_i(u) = u + 1$ (for ordinary arcs) or ${z_i(u) = u^2 + \lambda_p u + 1}$ (for pending arcs incident to orbifold points of order $p$).
\label{GenFlipLem}
\end{lemma}

%We need to be able to 

%condense this with previous paragraph

\section{Snake Graphs}

Let $T = \{\tau_1,\ldots, \tau_n\}$ be a triangulation of $\mathcal{O}$. Let $\{\tau_{n+1},\ldots,\tau_c\}$ be the boundary arcs of $\mathcal{O}$. For now, we work with orbifolds without any punctures, but we expect our results to hold when we introduce punctures. 

%Choosing an initial triangulation $T$ of $(S,M)$ corresponds to choosing an initial cluster for $\mathcal{A} = \mathcal{A}(S,M)$. Then, the arcs in the initial triangulation give the initial cluster variables. Any other arc on the surface will intersect at least one arc in $T$. We can determine the cluster variable associated to this arc by iteratively applying the \emph{Ptolemy relation}. This relation applies to a quadrilateral with all four vertices lying on a common circle and gives that the the product of the two diagonals is the same as the sum products of opposite side lengths. That is, if $x$ and $x'$ are the side lengths of the two diagonals of a quadrilateral, with side lengths given as $a,b,c,d$ in cyclic order, then $xx' = ac + bd$. 
Given an arbitrary arc $\gamma$ on $\mathcal{O}$ with triangulation $T$, we can use the Ptolemy relation to write $\gamma$ as a rational function in terms of $\{\tau_1,\ldots,\tau_c\}$. %is it clear it is a Laurent polynomial?
However, determining an expansion via this method can be computationally challenging. To simplify this process, we generalize the snake graph construction due to Musiker, Schiffler, and Williams \cite{MSW}, obtaining a combinatorial formula for the expansion in terms of the perfect matchings of a particular weighted graph, called a \emph{snake graph}.

%Musiker, Schiffler, and Williams which gives this expansion in terms of weights of perfect matchings of a certain weighted graph, called a \emph{snake graph}. %plus other stuff 
%By Theorem \ref{SurfaceCA}, we know this is equivalent to giving the expansion of a cluster variable in terms of the initial cluster determined by $T$. 

Fix an arbitrary triangulation $T$ of $\mathcal{O}$ and arc $\gamma$ (either ordinary, pending, or generalized). We  recursively construct the snake graph $G_{T,\gamma}$ by gluing together square \emph{tiles} determined by the intersections of $\gamma$ with arcs of $T$. Further, if $\gamma$ is a closed curve, we can similarly construct a band graph.


\subsection{Tiles}
If $\gamma = \tau_i$ for $1 \leq i \leq n$ (recall the final $c - n$ arcs are boundary arcs), then $G_\gamma$ is a single edge labeled with $\tau_i$. Otherwise, choose some orientation for $\gamma$. Since $T$ is a maximal set of non-intersecting arcs, in this case $\gamma$ will cross at least one arc in $T$.

 Let $\rho_1,\ldots, \rho_m$ be the set of internal arcs of $T$ that $\gamma$ crosses, given a fixed orientation. For each ordinary arc $\rho_i$ that $\gamma$ crosses, we construct a square tile $G_{i}$ by taking the two triangles that $\rho$ borders and gluing them along $\rho$, such that either both have either the same or opposite orientation as in $\mathcal{O}$. We say that the former case has relative orientation $+1$ and the latter case has relative orientation $-1$, denoted as $\textrm{rel}(G_i) = \pm 1.$


\begin{center}
\begin{tikzpicture}[scale = .14cm]
\draw (45:0.3cm) to node[auto]{$a$}(135:0.3cm) to node[auto]{$b$}(180+45:0.3cm) to node[auto]{$c$}(270+45:0.3cm) to node[auto]{$d$} (45:0.3cm);
\draw[gray](45:0.3cm) to node[below]{$\rho_i$} (225:0.3cm);
\draw[xshift = 1 cm] (45:0.3cm) to node[auto]{$b$}(135:0.3cm) to node[auto]{$a$}(180+45:0.3cm) to node[auto]{$d$}(270+45:0.3cm) to node[auto]{$c$} (45:0.3cm);
\draw[xshift = 1 cm, gray](45:0.3cm) to node[below]{$\rho_i$} (225:0.3cm);
\end{tikzpicture}
\end{center}

If $\rho_i$ is a pending arc encircling an orbifold point of order $p$, then it is only incident to one triangle in $\mathcal{O}$. As inspiration for the other triangle, we look to the $p$-fold cover around this orbifold point. For $p > 3$, this surface is not triangulated since we have a central ideal $p$-gon. However, for a regular $p$-gon, with each side length $\rho_i$, the length of the diagonal that bypasses exactly one vertex
( a ``2-diagonal'') is  
$\frac{\sin(2\pi/p)}{\sin(\pi/p)}\rho_i = 2\cos(\pi/p)\rho_i$. Recall this is the same parameter that appears when we mutate pending arcs. We use this 2-diagonal in the $p$-fold cover as an auxiliary arc to define a second triangle that $\rho_i$ borders, as on the left hand side below.

\begin{center}
\begin{tikzpicture}[scale = 0.2cm]
 \draw (0:0.3cm) \foreach \x in {22.5,45,...,359} {
            -- (\x:0.3cm)
        } -- cycle (90:0.3cm);
        \draw[](90:0.3cm) to node[auto]{$\rho_i$} (180:0.3cm);
        \draw[](90:0.3cm) -- (180 - 22.5:0.3cm);
        \draw[](90:0.3cm) -- (180 - 45:0.3cm);
         \draw[](180:0.3cm) to node[auto]{$\rho_i$} (270:0.3cm);
        \draw[](180:0.3cm) -- (270 - 22.5:0.3cm);
        \draw[](180:0.3cm) -- (270 - 45:0.3cm);
           \draw[](270:0.3cm) to node[auto]{$\rho_i$} (0:0.3cm);
        \draw[](270:0.3cm) -- (- 22.5:0.3cm);
        \draw[](270:0.3cm) -- (- 45:0.3cm);
           \draw[](0:0.3cm)to node[auto]{$\rho_i$} (90:0.3cm);
        \draw[](0:0.3cm) -- (90 - 22.5:0.3cm);
        \draw[](0:0.3cm) -- (90 - 45:0.3cm);
        \draw[dashed] (90:0.3cm) to node[auto]{$\rho_i \lambda_p$} (270:0.3cm);
\draw[xshift = 1 cm] (45:0.3cm) to node[auto]{$a$}(135:0.3cm) to node[auto]{$b$}(180+45:0.3cm) to node[auto]{$f \rho_i$}(270+45:0.3cm) to node[auto]{$g \rho_i$} (45:0.3cm);
\draw[xshift = 1 cm, gray](45:0.3cm) to node[below]{$\rho_i$} (225:0.3cm);
\end{tikzpicture}
\end{center}

On the right, $\{f,g\} = \{1, \lambda_p\}$ as sets. We define the orientation of the tile based on whether the orientation of the triangle with edges $a,b, \rho_i$ matches or does not match that on the surface. We determine $f$ and $g$ by the relative orientation of $\gamma$'s path, as below. 

\begin{center}
{\renewcommand{\arraystretch}{2}
\begin{tabular}{|c|c|}
\hline
      \includegraphics[scale=1.5]{Diagrams/SmallPic1.pdf}   &  \includegraphics[scale=1.5]{Diagrams/SmallPic2.pdf} \\ \hline  \includegraphics[scale=1]{Diagrams/Small1.pdf}  &  \includegraphics[scale=1]{Diagrams/Small2.pdf}  \\ \hline
\end{tabular}}
\end{center}



The other labels of the tile labeled 2 will depend on the local configuration around 2. In the configurations above, 2 could be a either ordinary arc or a pending arc. If 2 is a pending arc, then we would need to use such a configuration twice. We use one configuration for both crossings of 1 and one crossing of 2, and the second for one crossing of 1 and both crossings of 2. Below is one example of such gluing - we could do something similar if $\gamma$ matched the orientation in the right hand diagram. 

\includegraphics[scale=1.7]{Diagrams/PuttingPiecesTogether.pdf}

\vspace{-0.3cm}

\includegraphics[scale=1.2]{Diagrams/PuttingSnakesTogether.pdf}

%\oplus = gluing

%Note that, for arcs without self-intersections, crossing a pending arc corresponds to two crossings. This is more apparent if we draw a pending arc as a loop around the orbifold point. 


In the case of ordinary arcs, crossing a pending arc produces two square tiles since it involves two actual crossings. When constructing the snake graph for a generalized arc $\gamma$, however, it's possible to have only one crossing. In this case, the following rules are used to determine $f$ and $g$:

\begin{center}
\begin{tabular}{|c|c|}
\hline
  \includegraphics[scale=1.5]{Diagrams/SmallPic3.pdf}&  \includegraphics[scale=1.5]{Diagrams/SmallPic4.pdf}  \\ \hline  \includegraphics[scale=1]{Diagrams/Small3.pdf}  &  \includegraphics[scale=1]{Diagrams/Small4.pdf}  \\ \hline
\end{tabular}
\end{center}

\subsection{Gluing \texorpdfstring{$G_{T,\gamma}$}{puzzle pieces}}

We then glue these square tiles together, according to the following rule: when crossing two consecutive arcs $\tau_i$ and $\tau_{i+1}$, we glue $G_i$ and $G_{i+1}$ along the shared edge labeled $\tau_{[i]}$, using the appropriate planar embeddings so $\textrm{rel}(T,G_i) \neq \textrm{rel}(T,G_{i+1})$. Note that this rule does not differentiate between ordinary and pending arcs. %If $\gamma$ is a ordinary arc, then tiles corresponding to crossings of pending arcs always appear in pairs. 
%Tiles corresponding to crossings of pending arcs always appear in pairs; this is a natural consequence of the choice to represent pending arcs as loops that cut out a monogon around an orbifold point.

%Because we draw pending arcs as loops around orbifold points, 

%Once we have constructed all of these square graphs, we glue them together. When we cross a pending arc, we use rules as above. If cross two standard arcs consecutively, say $\tau_i$ and $\tau_{i+1}$, then these will border the same triangle in $\mathcal{O}$. Let $\tau_{[i]}$ be the third arc of this triangle. The squares for both $\rho_i$ and $\rho_{i+1}$ will have an edge labeled $\tau_{[i]}$. We glue the squares along $\tau_{[i]}$, choosing opposite orientations for the two squares. Note that, even in our configurations above, the orientation of the tiles switches at each step.




Because the choice of relative orientation for the first tile, $G_1$, isn't fixed, there are two valid planar embeddings of $G_{T,\gamma}$ for any $\gamma$. Our cluster expansion formula produces the same result for either choice of planar embedding, so the choice is unimportant. %We also draw our snake graphs from South-West to North-East

Finally, we can construct generalized band graphs using the same ideas. Band graphs calculate the length of closed curves in a surface.s. Choose a point $p$ on $\gamma$ such that $p$ does not lie on any arc in $T$ or at an intersection of $\gamma$ with itself. For simplicity, we require $p$ to not be in the interior of a pending arc. Then, construct the snake graph for $\gamma$, picking an orientation and starting and ending at $p$. Since the first and last tile correspond to arcs bordering the same triangle, they will have a common edge. We glue the first and tile along this edge. This gives a graph which resembles an annulus or a Mobius strip. Statistics associated to a band graph are the same as for snake graphs

\subsection{Cluster Expansion Formulas}

To state the cluster expansion formula, we need the following definition:

\begin{definition}[\cite{MSW}]
Each snake graph has exactly two perfect matchings which use only boundary edges. One of these matchings uses the south edge of the first tile and the other uses the west edge of the first tile. By convention, we refer to the boundary matching which uses the west edge as the \emph{minimal matching}, $P_{-}$, of P.
\end{definition}

For an arbitrary perfect matching $P$ of $G_{T,\gamma}$, we sometimes consider the symmetric difference $P \ominus P_{-}$. For all $P$, this symmetric difference consists of a set of potentially disjoint cycles.
We now define some statistics associated to $\gamma$ and $G_{T,\gamma}$, following \cite{MSW}:
\begin{itemize}
    \item The \emph{crossing monomial} is $ \textrm{cross}(T,\gamma) = \prod_{\tau_i \in T} x_i^{e(x_i,\gamma)}$, where $e(x_i,\gamma)$ is the minimal number of crossings of $\gamma$ with arcs in the isotopy class of $\tau_i$.
    \item The \emph{weight} $x(P)$ of a perfect matching $P$ of $G_{T,\gamma}$ is the product of all its edge weights - i.e., if the edges of $P$ has labels $\tau_{i_1}, \dots, \tau_{i_k}$, then $x(P) = x_{\tau_{i_1}} \cdots x_{\tau_{i_k}}$.
    \item The \emph{height monomial} $y(P)$ is $y(P) = \prod_{\tau_i \in T} y_i^{m_i}$, where $m_i$ is the multiplicity of tiles labeled $i$ enclosed by cycles of $P \ominus P_{-}$.
\end{itemize}

%The statistics associated to a band graph are the same as for snake graphs.

%In order to describe how the snake graph $G_\gamma$ encodes the cluster variable which $\gamma$ represents, we define several statistics attached to $\gamma$ and $G_\gamma$. The \emph{crossing monomial},  $\text{cross}(T, \gamma)$, for an arc $\gamma$ in a triangulated surface is given by $\Pi_{x_i \in T} x_i^{e(x_i,\gamma)}$ where $e(x_i,\gamma)$ is the minimal number of crossings between arcs in the isotopy classes of $x_i$ and $\gamma$. The weight of a perfect matching $P$, denoted $x(P)$, is the product of the weights of edges in $P$. While originally we labeled these with the arcs in $T$, for the weights we will conflate these arcs with the initial cluster variables they represent. 

%In order to define the height monomial, we first must understand the minimal matching. Given any snake graph, there are exactly two perfect matchings which only use the boundary edges of the graph. One of these will use the south edge of the first square, and the other will use the west edge. As a convention, we will set the latter to be the \emph{minimal matching}, $P_-$. Then, given any perfect matching $P$, the symmetric difference of $P$ with $P_-$ will consist of a set of disjoint cycles. The height monomial $y(P)$ for a matching $P$ is given by $\Pi_{x_i \in T} y_i^{m_i}$ where $m_i$ is the number of tiles enclosed by edges in the symmetric difference between $P$ and $P_-$. 

We are then able to establish the following theorem for Laurent expansions of arcs (both ordinary and pending), a slightly more general version of the theorem from \cite{MSW}.


\begin{theorem}\label{GenSnake}
Let $\mathcal{O} = (S,M,Q)$ be an arbitrary orbifold with triangulation $T$ and $\mathcal{A}$ be the corresponding CS algebra with principal coefficients with respect to $\Sigma_T = (\mathbf{x}_T,\mathbf{y}_T,B_T)$. For an arc $\gamma$ with snake graph $G_{T,\gamma}$, the Laurent expansion of $x_\gamma$ with respect to $\Sigma_T$ is
\[ [x_\gamma]^{\mathcal{A}}_{\Sigma_T} = \frac{1}{\textrm{cross}(T^\circ,\gamma)} \sum_P x(P)y(P) \]
where the summation is indexed by perfect matchings of $G_{T,\gamma}$.
\end{theorem}

\begin{proposition}\label{GenSnakeGenArc}
Let $\gamma$ be a generalized arc in $\mathcal{O}$. 
If $\gamma$ has a contractible kink, let $\gamma'$ be the same curve with the kink removed. Then, $x_\gamma = (-1)x_{\gamma'}$.
Otherwise, $x_\gamma = \frac{1}{\textrm{cross}(T^\circ,\gamma)} \sum_P x(P)y(P) $ where the summation is indexed by perfect matchings of $G_{T,\gamma}$.
\end{proposition}

\begin{proposition}\label{GenBand}
Let $\gamma$ be a a closed curve on $\mathcal{O}$. If $\gamma$ is contractible, then $x_\gamma = -2$. Otherwise, \cref{GenSnake} also holds where $G_{T,\gamma}$ is a generalized band graph.
\end{proposition}


%Flipping a pending arc $\gamma$ produces an arc $\gamma'$ whose snake graph 

%The building blocks of standard snake graphs are single square tiles, which have two perfect matchings and encode the Ptolemy relation, which corresponds to standard cluster mutation. For CS algebras arising from orbifolds, the exchange relations resulting from flipping pending arcs are trinomials.
%We see this sort of relation in our snake graphs also. That is, if $\gamma'$ is the result of flipping a pending arc $\gamma$, then the sum perfect matchings of $G_{\gamma'}$ resembles this exchange relation. 

\noindent \textbf{Example 1.} The table below shows snake graphs for a variety of curves on the triangulated orbifold corresponding to $\mathcal{A} = \left(\mathbf{x},\mathbf{y}, \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, (1+\mu u + u^2, 1 + \lambda u + u^2) \right)$.

\begin{center}
\resizebox{\textwidth}{!}{
\begin{tabular}{|c|c||c|c|}
\hline
%Arc &Snake Graph & Arc & Snake Graph \\ \hline
 \includegraphics[scale=2]{Diagrams/KroneckerEx_orbifold2new.pdf} & \includegraphics[scale=1]{Diagrams/KroneckerEx_orbifold2snake.pdf}  &\includegraphics[scale=2]{Diagrams/KroneckerEx_orbifold_3} & \includegraphics[scale=1]{Diagrams/KroneckerEx_orbifold3snake}  \\ \hline
  \multicolumn{2}{c}{$x_{\gamma_1} = \frac{1}{x_1}(\textcolor{green}{x_a^2} + \mu y_2x_2x_a + y_2^2 x_2^2$)} & \multicolumn{2}{c}{$\begin{array}{l} x_{\gamma_2} = \frac{1}{x_1^2x_2}(x_a^2x_1^2y_1^4y_2^2+\textcolor{green}{\mu\lambda x_a^2 x_1x_2y_1^3y_2}+\lambda x_a x_1x_2^2y_1^2y_2 \\ \qquad \qquad ~~ + \lambda x_a^3x_1 y_1^4 y_2 + \mu^2 x_a^2 x_2^2 y_1^2 + 2\mu x_a x_2^3 y_1 \\ \qquad\qquad~~ + 2 \mu x_a^3 x_2 y_1^3 + x_2^4 + 2x_a^2x_2^2y_1^2 + x_a^4y_1^4) \end{array}$}\\
 \end{tabular}
 }
%can delete these two \end & \begin to return to one table
\resizebox{\textwidth}{!}{
\begin{tabular}{|c|c||c|c|} 
 \hline
\includegraphics[scale=2]{Diagrams/KroneckerEx_nonpending1.pdf} & \includegraphics[scale=1]{Diagrams/Kronecker_nonpending_snake.pdf}&\includegraphics[scale=2]{Diagrams/KroneckerEx_band.pdf}& \includegraphics[scale=1]{Diagrams/KroneckerEx_bandgraph.pdf}\\ \hline
\multicolumn{2}{c}{$\begin{array}{l} x_{\gamma_3} =  \frac{1}{x_1x_2}(x_ax_1^2+ \lambda y_2x_a^2x_1 +  y_2^2 x_a^3 +  \\ \qquad \qquad ~~  \mu  y_1 y_2^2 x_a^2 x_2 + \textcolor{green}{y_1^2y_2^2 x_ax_2^2}) \end{array}$} & 
\multicolumn{2}{c}{$\begin{array}{l} x_{\gamma_4} = \frac{1}{x_1x_2}(x_1^2+ \lambda y_2x_ax_1 +  y_2^2 x_a^2 +  \\ \qquad \qquad ~~  \mu  y_1 y_2^2 x_a x_2 + \textcolor{green}{y_1^2y_2^2x_2^2})\end{array}$}\\ 
\end{tabular}
}
\end{center}

Labels for arcs in the initial triangulation are only shown in the first orbifold diagram, but are consistent throughout. Snake graphs are shown for each (dashed) curve $\gamma_i$, with one perfect matching (and the corresponding term in the Laurent expansion) highlighted. Both $\gamma_1$ and $\gamma_2$ are cluster variables of $\mathcal{A}$, obtained via the respective mutation sequences $\mu_1$ and $\mu_2\mu_1$.

%First figure has minimal perfect matching. 
%Could comment on the fact that terms from our crossing monomial cancel out, and that this always happens. 
%Mutation/flip sequence to get the first two arcs 

The second half of this example illustrates the extension of our results to generalized arcs and closed curves. Since $\gamma_3$ and $\gamma_4$ cross the same arcs in the same orientation, the shapes of the two associated graphs are the same. However, in the band graph associated to $\gamma_4$, we identity $u$ with $u'$ and $v$ with $v'$. In each of these graphs, we have highlighted the maximal matching.
%and highlighted the term it adds to the Laurent expansion. 

Note that in each of these examples, our expression for $x_{\gamma_i}$ is given after canceling a mutual factor from the crossing monomial and the numerator. Although the exact mutual factor depends on the curve being considered, cancellation of this type occurs whenever we cross pending arcs.


%discuss



\section{Frieze Patterns}

Periodic frieze patterns of positive integers have been shown to be in bijection with triangulated surfaces; those with a finite number of rows correspond to triangulated polygons, while those with infinitely many rows correspond to triangulations of annuli. %Cite

Holm and J\o rgensen classified a certain family of finite frieze patterns associated with \emph{polygon dissections} \cite{HJ}. A polygon dissection is a generalization of a triangulation, in the sense that it is a (not necessarily maximal) set of pairwise non-intersecting arcs on a surface. The entries in these frieze patterns are positive integer combinations of $\lambda_p$ for finitely many values of $p$. 
Our snake graphs from orbifolds allow us to construct infinite frieze patterns in the spirit of Holm and J\o rgensen and to recover some of their finite frieze patterns. 

\begin{definition}
A (finite) frieze pattern is a (finite) set of infinite rows, where each row is offset from its neighbors in such a way that the entries of every other row form columns. The first row consists of all zeroes and the second row consists of all ones. The defining property of a frieze pattern is that every diamond  $\begin{array}{ccc} & a & \\ b & & c \\ & d & \end{array}$ satisfies the condition $bc-ad=1$.
\end{definition}

Note that once we specify the first nontrivial row, the entire frieze pattern is determined. We refer to this first nontrivial row as the \emph{quiddity row}. 

We use the following indexing to refer to entries of a frieze pattern. We can also refer to entries in the row of $0$'s as $(i,i)$ and in the row of $1$'s as $(i, i+1)$.
\[
\begin{array}{ccccccccccccccc}
 0&&0&&0&&0&&0&\\
  &1&&1&&1&&1&&1\\
 (1,3)&&(2,4)&&(3,5)&&(4,6)&&(5,7)&\\
  &(1,4)&&(2,5)&&(3,6)&&(4,7)&&(5,8)\\
(0,4)&&(1,5)&&(2,6)&&(3,7)&&(4,8)\\
  &(0,5)&&(1,6)&&(2,7)&&(3,8)&&(4,9)\\
   &&&&\vdots&&&&&\\
 \end{array}
\]

While there are several combinatorial interpretations of entries in frieze patterns of positive integers, we can also interpret each entry as giving the length of an arc on the triangulated surface corresponding to the frieze pattern. In this interpretation, the length of all boundary arcs and arcs in the triangulation is set to 1. In a finite frieze pattern associated to a triangulation of an $n$-gon, entry $(i,j)$ corresponds to the length of the diagonal between vertices $\bar{i}$ and $\bar{j}$ where $\bar{i} \equiv i \pmod{n}$. There is a similar story for infinite frieze patterns, except we must keep track of how many times an arc winds around the annulus. Moreover, the frieze pattern only measures \emph{peripheral arcs} along the outer boundary (although we could also make a frieze pattern with respect to the inner boundary). A peripheral arc is the result of concatenating multiple boundary arcs from the same boundary. In a polygon, all arcs are peripheral. The following Lemma is part of a larger theorem in \cite{GMV}, and holds for more general surfaces than discussed there.

\begin{lemma}
A frieze pattern corresponding to a triangulated surface encodes a map on peripheral arcs on the surface which respects the Ptolemy relation. 
\end{lemma}

Our snake graphs allow us to measure the length of arcs in a triangulated orbifold in terms of the lengths of the arcs in the triangulation. We set all of these lengths to 1; however, we could also recover a frieze pattern of Laurent polynomials by leaving the lengths as variables. 

We construct $\mathcal{F}(\mathcal{O})$, a frieze pattern associated to a triangulated orbifold, $\mathcal{O}$. We distinguish one boundary to work with respect to.
\begin{itemize}
    \item Let $\gamma_1,\ldots,\gamma_m$ be the set of peripheral arcs along one boundary component which are the result of concatenating exactly two boundary arcs. 
    \item Let $X(\gamma_i)$ be the sum of perfect matchings of $G_{\gamma_i}$ with all $x_j = y_j = 1$.
    \item The frieze pattern $\mathcal{F}(\mathcal{O})$ is determined by a quiddity row given by repeating $X(\gamma_1),X(\gamma_2),\ldots, X(\gamma_m)$ infinitely many times.
\end{itemize}

By \cref{GenFlipLem}, we know that an orbifold surface $\mathcal{O}$ encodes a CS algebra $\mathcal{A}$, so some cluster variables and other elements of $\mathcal{A}$ are lambda lengths of arcs on $\mathcal{O}$. By \cref{GenSnake} and \cref{GenSnakeGenArc}, we can express these elements of $\mathcal{A}$ in terms of perfect matchings. This implies the following interpretation of these frieze patterns: 

\begin{proposition}\label{OrbFrieze}
The entries in $\mathcal{F}(\mathcal{O})$ are specializations of elements of $\mathcal{A}(\mathcal{O})$, where we set all initial variables $x_i$ and $y_i$ to 1.
\end{proposition}

Because our edge weights are all positive, we also observe that:
\begin{corollary}
If $p_1,\ldots,p_k$ are the orders of the orbifold points in $\mathcal{O}$, then all entries of $\mathcal{F}(\mathcal{O})$ are positive integer multiples of $1,\lambda_{p_1},\ldots, \lambda_{p_k}$.
\end{corollary}

Our earlier example illustrates that, in the case of an orbifold with two orbifold points, we can have arcs which wind around the two points arbitrarily many times. The same is true for a surface with multiple boundary components, such as an annulus, or a surface with punctures. This leads us to the following.

%our next result

\begin{proposition}\label{countingrows}
 The frieze pattern $\mathcal{F}(\mathcal{O})$ is finite if and only if $S$ is homeomorphic to a disk, $\vert Q \vert = 1$ and $M \cap \partial S = M$. In particular, if $S$ has $n$ marked points on the boundary and the unique orbifold point is order $p$, then $\mathcal{F}(\mathcal{O})$ has $np - 3$ nontrivial rows. 
\end{proposition}

Note that the orbifolds which give finite frieze patterns are exactly those whose cover is an unpunctured polygon. The triangulation of the orbifold lifts to a polygon dissection of the cover. In these cases, the frieze pattern we construct from the orbifold matches the frieze pattern Holm and J\o rgensen associate with the cover. This is the case for the left hand frieze below, which is the frieze pattern associated to the orbifold in \cref{fig:pfold} where the orbifold point has order 4 ($\lambda_4 = \sqrt{2}$). This frieze pattern is 3-periodic and by \cref{countingrows} we know it has 9 nontrivial rows.  

\begin{center}
\begin{tabular}{c||c}
$\arraycolsep=0.5pt\def\arraystretch{0.3}
\begin{array}{ccccccccccc}
     0&&0&&0&&0\\
  &1&&1&&1&&\\
  1 && 2 && 3 + \sqrt{2} &&1\\
  &1&&5 + 2\sqrt{2} &&2 + \sqrt{2}&&\\
  1 + \sqrt{2} && 2 + \sqrt{2} && 3 + 2\sqrt{2} && 1+ \sqrt{2}\\ 
\end{array}$ & 
$\arraycolsep=0.1pt\def\arraystretch{0.1}
\begin{array}{ccccccccccc}
     &0&\\
  1&&1&\\
  & 3 + 2\sqrt{2}&&\\
  16 + 12\sqrt{2} &&  16 + 12\sqrt{2} &\\
  & 93 + 66\sqrt{2} &&\\
\end{array}$
\end{tabular}
\end{center}

The right hand frieze is associated to the orbifold in Example 1, with both orbifold points again order 4. Since this orbifold has two orbifold points, we know that there are infinitely many rows. This frieze pattern is 1-periodic, with the quiddity row determined by the bottom left arc in Example 1, which is the concatenation of arc $a$ with itself. 

Meanwhile, \cite{BFPT} demonstrated a set of invariants for periodic, infinite frieze patterns. Given an infinite frieze pattern $\mathcal{F}$, let $n$ be the minimal period of the quiddity row. Then, using our indexing above and fixing $k \geq 1$, the difference  $(j, j+nk + 1) - (j+1, j+nk - 1)$ is constant for all $j$. These parameters are referred to as \emph{growth coefficients}. In the case of positive integers, where the frieze patterns arise from triangulations of annuli, \cite{GMV} gave a geometric interpretation for these growth coefficients. Specifically, they showed that the $k$-th growth coefficient corresponds to the length of a closed curve which winds around the annulus $k$ times. We conjecture that our generalized band graphs will allow us also to keep track of growth coefficients of the frieze patterns from triangulated orbifolds.

%could do example. maybe something slightly larger than the loop around two orb points. 
\begin{conjecture}
Generalized band graphs determine the growth coefficients of infinite friezes arising from orbifolds.
\end{conjecture}

As evidence to support our conjecture, observe that the first growth coefficient for the infinite frieze above is $3+2\sqrt{2}$, which agrees with $x_{\gamma_4}$ from Example 1 when the Laurent expansion is evaluated at $x_i = 1$ and $\lambda = \mu = \sqrt{2}$.

%as given by the band graph in Example 1. 


%formalize construction, and maybe make lemma that frieze patterns are friezes. 

%\section{Generalized Snake Graphs}

%\subsection{Generalized Snake Graphs for cluster algebras from orbifolds}

%\begin{center}
%\begin{tabular}{|m{2cm}|m{4cm}|m{4cm}|m{5cm}|}
%\hline
%Puzzle Piece & 1 & 2 & 3 \\ \hline
%Diagram & \includegraphics[scale=1.5]{Diagrams/unpunctured_pp01} &   %\includegraphics[scale=1.5]{Diagrams/unpunctured_pp02} &  %\includegraphics[scale=1.5]{Diagrams/unpunctured_pp03}  \\ \hline
%Snake Graph & \includegraphics[scale=1]{Diagrams/unpunctured_sg01}  & %\includegraphics[scale=1]{Diagrams/unpunctured_sg02}  & %\includegraphics[scale=1]{Diagrams/unpunctured_sg03}   \\ \hline
%\end{tabular}
%\end{center}

%\begin{center}
%\begin{tabular}{|c|c|c|}
%\hline
%Puzzle Piece & 4 & 5 \\ \hline
%Diagram & \includegraphics[scale=1.5]{Diagrams/unpunctured_pp04} &   %\includegraphics[scale=1.5]{Diagrams/unpunctured_pp05}  \\ \hline
%Snake Graph & \includegraphics[scale=1]{Diagrams/unpunctured_sg04} & %\includegraphics[scale=1]{Diagrams/unpunctured_sg05-horiz}  \\ \hline
%\end{tabular}
%\end{center}

%\begin{center}
%\begin{tabular}{|c|c|c|}
%\hline
%Puzzle Piece & 6 & 7 \\ \hline
%Diagram & \includegraphics[scale=1.5]{Diagrams/unpunctured_pp06} &   \includegraphics[scale=1.5]{Diagrams/unpunctured_pp07}  \\ \hline
%Snake Graph & \includegraphics[scale=1]{Diagrams/unpunctured_sg06} & \includegraphics[scale=1]{Diagrams/unpunctured_sg07}  \\ \hline
%\end{tabular}
%\end{center}



%\begin{proposition}
%Let $(S,M,\mathcal{O})$ be an arbitrary bordered orbifold with triangulation $T$. For any choice of boundary component with at least one marked point, the above construction can be used to construct a corresponding frieze pattern, $\mathcal{F}$. Let $n$ denote the number of marked points on the chosen boundary component. Then the first $n-1$ rows of $\mathcal{F}$ give specializations of the cluster variables of the Chekhov-Shapiro algebra corresponding to $(S,M,\mathcal{O})$ with respect to the initial cluster determined by $T$. The remaining rows give specializations of elements of the algebra.
%\end{proposition}




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