%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%======================================= %\section{Cluster algebras from surfaces} \section{Background} \label{sec:cluster_algebras_from_surfaces} %%======================================= %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\subsection{Triangulations of marked surfaces} \subsection{Cluster algebras from surfaces} %\label{subsec:triangulations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%% We provide a brief background on cluster algebras arising from marked surfaces $(S,M)$, following Fomin, Shapiro, and Thurston~\cite{FST08}. %\begin{defn} %[marked surface] Let $S$ be either an annulus or a disk, and %a connected, oriented, Riemann surface with (possibly empty) boundary, and $M$ a non-empty, finite set of marked points in the closure of $S$, such that there is at least one marked point on each boundary component of $S$. The interior marked points in $S$ are called \emph{punctures}. %Then $(S,M)$ is called a \emph{marked surface}, and the interior points of $S$ are called \emph{punctures}. %\end{defn} %For technical reasons, assume that $(S,M)$ is not the following: a sphere with fewer than four punctures; a monogon with zero or one puncture; or a bigon or triangle without punctures. %\begin{defn}[arc] A \emph{generalized ordinary arc} $\gamma$ in $(S,M)$ is a curve in $S$, considered up to isotopy, such that: (1) the endpoints of $\gamma$ are in $M$, (2) the interior of $\gamma$ is disjoint from $M$ and from the boundary of $S$, (3) $\gamma$ does not cut out an unpunctured monogon or bigon. Generalized arcs are allowed to intersect themselves a finite number of times. We consider these up to isotopy of immersed arcs, that is, allowing Reidemeister moves of types II and III but not of type I. In particular, an isotopy cannot remove a contractible kink (see Fig.~\ref{fig:progression_m48}, bottom) from a generalized arc. If an arc intersects itself in its interior, we say that the arc has a \emph{self-crossing}. A \emph{boundary edge} is a curve that connects two marked points and lies entirely on the boundary of $S$ without passing through a third marked point. %Following \cite{DT13,BD14}, we say that A generalized arc $\ga$ is called \emph{peripheral} on a single boundary component $Bd$ of $S$ if: (1) both its endpoints (or its unique endpoint in the case of a loop) are on $Bd$, and (2) $\ga$ is isotopic to a concatenation of two or more boundary edges of a boundary component $Bd$. Our convention is to choose the orientation of $\ga$ so that $Bd$ is to the right of $\ga$ when looking from above. An \emph{ordinary arc} $\gamma$ is a generalized ordinary arc which has no self-crossing. We say that two ordinary arcs $\alpha, \beta$ are \emph{compatible} if there exist representatives $\alpha', \beta'$ in their respective isotopy classes such that $\alpha'$ and $\beta'$ do not intersect in the interior of $S$. An \emph{ideal triangulation} $T$ is a maximal (by inclusion) collection of distinct, pairwise compatible ordinary arcs (see Fig.~\ref{fig:disc_as_AT}, left). %Recall from~\cite[Thm.~7.11]{FST08} that Due to~\cite[Thm.~7.11]{FST08}, we can associate a \emph{signed adjacency matrix} $B_T$, %{\cite[Def. 4.1 and 9.6]{FST08}}, and hence a cluster algebra, to $T$. %Given a cluster algebra $\mathcal{A} = \mathcal{A}(S,M)$ associated to a %triangulation $T$, The ordinary arcs $\tau$ of $(S,M)$ correspond to cluster variables and products of cluster variables, denoted by $x_\tau$ or $x(\tau)$. \subsection{Laurent polynomials associated to generalized arcs and closed loops} In~\cite{MSW11,MSW13}, Schiffler, Williams, and the second author associated to a generalized arc (resp., closed loop) $\gamma$ and an ideal triangulation $T$ a Laurent polynomial $X_\gamma^T$ which is a weighted sum over perfect matchings of a planar \emph{snake graph} (resp., a \emph{band} on a Mobius strip or annulus) $G_{T,\gamma}$. %See [MSW13, Sec. 3] for the full definitions of snake and band graphs. Put simply, such graphs are made out of gluing squares together, with one such square for each arc of T crossed by $\gamma$; see~Fig.~\ref{fig:generalized_arc_snakegraph}, right. A \emph{perfect matching} of a graph $G$ is a subset $P$ of the edges of $G$ such that each vertex of $G$ is incident to exactly one edge of $P$. If $G$ is a snake or band graph, and the edges of a perfect matching $P$ of $G$ are labeled $\tau_{j_1}, \ldots, \tau_{j_r}$, then we define the \emph{weight} $x(P)$ of $P$ to be $x_{\tau_{j_1}} \cdots x_{\tau_{j_r}}$. If $\tau$ is a boundary segment, we let $x_\tau:=1$. \begin{defn} [{\cite[Def. 3.12]{MSW13}}] \label{defn:generalized_arc_laurent_polynomial} Let $T$ be an ideal triangulation, $\mathcal{A}$ the cluster algebra associated to $B_T$, and $\gamma$ be a generalized arc. We define a Laurent polynomial which lies in $\mathcal{A}$. \begin{enumerate} \itemsep-0.1em \item If $\gamma$ cuts out a contractible monogon, then $X_\gamma^T$ is equal to zero. \item If $\gamma$ has a contractible kink, let $\overline{\gamma}$ denote the corresponding generalized arc with this kink removed, and define $X_\gamma^T := (-1)X_{\overline{\gamma}}^T$. \item Otherwise, let $\tau_{i_1}, \tau_{i_2}, \ldots, \tau_{i_d}$ be the sequence of arcs in $T$ which $\gamma$ crosses. Define \[X_\gamma^T := \frac{1}{\tau_{i_1} \, \tau_{i_2} \, \ldots \, \tau_{i_d}} \sum_P x(P),\] where the sum is over all perfect matchings $P$ of $G_{T,\gamma}$. \end{enumerate} \end{defn} \begin{thm}[{\cite[Thm 4.10]{MSW11}}] When $\gamma$ is an ordinary arc with no self-crossings, $X_\gamma^T$ is equal to the Laurent expansion of the cluster variable $x_\gamma$ with respect to $T$. \end{thm} \begin{ex} The snake graph ${G}_{T,\gamma}$ associated to the generalized arc $\gamma$ in~Fig.~\ref{fig:example_generalized_arc} has $11$ perfect matchings. Following Definition \ref{defn:generalized_arc_laurent_polynomial}, we compute \[ X_{\gamma}^T = \frac{x_{0} x_{1} x_{4} + 2 x_{1} x_{3} x_{4} + 2 x_{0}^{2} + 4 x_{0} x_{3} + 2 x_{3}^{2}} {x_{0} x_{1} x_{4}} \] by specializing $x_\tau=1$ for each boundary edge $\tau$. \end{ex} A Laurent polynomial $X_\zeta^T$ is associated to any closed loop $\zeta$ by a similar formula, see~\cite[Def.~3.14]{MSW13}. A closed loop obtained by following a (non-contractible, non-self-crossing, kink-free) loop $k$ times, and thus creating $k-1$ self-crossings, is called a \emph{$k$-bracelet} and is denoted by $Brac_k$, see Fig.~\ref{fig:bracelets}. For the rest of the paper, we will use the notation $x_\gamma$ or $x(\gamma)$ to denote the Laurent polynomial corresponding to $\gamma$, where $\ga$ is a generalized arc or loop.