\documentclass[submission]{FPSAC2017}

\articlenumber{76}
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\title[Infinite friezes of cluster algebras from surfaces]{Infinite friezes of cluster algebras from surfaces}

\author[Emily Gunawan, Gregg Musiker and Hannah Vogel]{Emily Gunawan\addressmark{1}\thanks{E. Gunawan and G. Musiker were supported by NSF Grants DMS-1148634 and DMS-1362980. H. Vogel was supported by the Austrian Science Fund (FWF): projects No. P25141-N26 and W1230.}, 
Gregg Musiker\addressmark{2}$^*$ \and Hannah Vogel\addressmark{3}$^*$}

\address{\addressmark{1}
\mbox{Gustavus} Adolphus College, Saint Peter, Minnesota, USA \\
\addressmark{2}School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA\\
\addressmark{3}Department of Mathematics and Scientific Computing, University of Graz, Graz, Austria\\}


\received{\today}

\abstract{Originally studied by Conway and Coxeter, friezes appeared
in various recreational mathematics publications in the 1970s.   More recently, in 2015, Baur, \mbox{Parsons}, and Tschabold 
constructed periodic infinite friezes and related them to matching numbers in the once-punctured disk and annulus.  In this paper, we
study such infinite friezes with an eye towards cluster algebras of type D and affine A, respectively.  By examining infinite friezes 
with Laurent polynomials entries, we discover new symmetries and formulas relating the entries of this frieze to one another. 
}

\keywords{cluster algebra, Conway--Coxeter frieze, 
triangulation, marked surface}


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\begin{document}

\maketitle

%%=======================================
\section{Introduction}
\label{sec:intro}
%%=======================================

A Conway-Coxeter \emph{frieze} $\F=
\{\F_{ij}\}_{i \leq j}$ 
 is an array of rows (arranged and indexed as in \cref{fig:defn_integer_infinite_frieze}) 
such that $\F_{i,i} = 0$ and $\F_{i,i+1} = 1$, and, 
 for every diamond
\[
\begin{array}{ccccccc}
 &c&\\
 a&&b\\
 &d&
\end{array}\]
of entries in the frieze, the equation $ab-cd =1$ is satisfied.

\begin{figure}[!hbt]
\centering
\begin{tikzpicture}[font=\footnotesize] 
\matrix(m) [matrix of math nodes,row sep={1.35em,between origins},column sep={1.55em,between origins},nodes in empty cells]{
&0&&0&&0&&0&&0&&&&&&\\ 
\node{\cdots};&&1&&1&&1&&1&&1&&\node{\cdots};&&\\[-0.25em]
&&&\F_{-1,1}&&\F_{0,2}&&\F_{1,3}&&\F_{2,4}&&\F_{3,5}&&&\\
&&\node{\cdots};&&\F_{-1,2}&&\F_{0,3}&&\F_{1,4}&&\F_{2,5}&&\F_{3,6}&&\node{\cdots};\\
&&&&&\F_{-1,3}&&\F_{0,4}&&\F_{1,5}&&\F_{2,6}&&\F_{3,7}&\\
&&&&&&&&\node[rotate=-6.5,shift={(-0.034cm,-0.08cm)}] {\ddots};&&&&\node[rotate=-6.5,shift={(-0.034cm,-0.08cm)}]  {\ddots};&&\\
};
\end{tikzpicture}
\caption{Arrangement and indices for frieze entries.}
\label{fig:defn_integer_infinite_frieze}
\input{friezepentagonarcyellow}
\input{friezepentagonintyellow}
\caption{Diagonals of a polygon correspond to
entries of a finite frieze. 
}
\label{fig:friezepentagonyellow}
\input{fig_bci_trail}
\caption{The BCI $2$-tuples for $\gamma$ which match vertices $v_1$ \& $v_2$ to their adjacent triangles, and
the corresponding trails  
whose weights sum up to the expansion $x_{\textcolor{red}{\gamma}}=\frac{a}{\textcolor{magenta}{b}} + \frac{1}{\textcolor{magenta}{b}}$.}
\label{fig:bci_trail}
\end{figure}
 
 We say a frieze is \emph{finite} if it is bounded above and below by a row of $1$s.
In the 70s, Conway and Coxeter showed that finite friezes with positive integer entries are in bijection with triangulations of polygons~\cite{Cox71,CC73}.
Given a triangulation $T$ of a polygon, each entry of the second row of the corresponding frieze is the number of triangles adjacent to a vertex.
Broline, Crowe, and Isaacs further studied this in~\cite{BCI74} and 
found that every entry in such a frieze corresponds to a diagonal (see \cref{fig:friezepentagonyellow}).
To any diagonal, they associate a set of vertices $v_{i_1}, \ldots, v_{i_r}$ (those lying to the right) and then match these to a \emph{BCI $r$-tuple} $(t_{1}, \ldots, t_{r})$ of pairwise-distinct triangles in $T$, such that $t_{j}$ is incident to vertex $v_{i_j}$. 
For example, in \cref{fig:bci_trail}, the diagonal from vertex $v_5$ to vertex $v_3$ is associated to the vertices $v_1$ and $v_2$. There are exactly two BCI $2$-tuples corresponding to $v_1$ and $v_2$.
 
\begin{figure}[!hbt]
\centering
\includegraphics[width=\textwidth]{fig4.png}
\caption{First $5$ rows
of the infinite frieze from a pentagon triangulation.}
\label{fig:puncturedpentagon}
\end{figure}
 
%%%%%%%%%%%%%  
More recently,
Caldero and Chapoton in~\cite{CC06} showed that finite frieze patterns appear in the context of Fomin--Zelevinsky cluster algebras~\cite{FZ02} of type $A$. 
Carroll and Price in~\cite{CP03} gave an expansion formula for cluster variables of type $A$ in terms of BCI tuples (see~\cite[Appendix~A]{GMV16} and \cref{fig:bci_trail}). 
Enumerating BCI tuples is equivalent to counting perfect matchings in a bipartite graph whose nodes are the triangles and vertices of (respectively a snake graph associated to) a triangulation; see Sec.~2 (respectively Sec.~4) of~\cite{Pro05}.  

A frieze is said to be \emph{infinite} if it is not bounded below by a row of $1$s.
Infinite friezes of positive integers arising from once-punctured {\disks} were introduced in~\cite{Tsc15} by Tschabold. Given an ideal triangulation $T$ (in the sense of~\cite{FST08}) of a once-punctured {\disk} with $n$ marked boundary vertices labeled $v_1,v_ 2, \ldots, v_n$ counterclockwise around the boundary, 
we can count the number of BCI tuples 
in a similar way, see \cref{fig:puncturedpentagon}.

In~\cite{BPT16}, Baur, Parsons, and Tschabold went further and gave a complete characterization of infinite friezes of positive integers via triangulations of quotients of an infinite strip in the plane. In this classification, periodic friezes arise from triangulations of the annulus or of the once-punctured disk (which can be thought of as a quotient of the infinite strip), see \cref{fig:disc_as_AT}.  
An infinite frieze is said to be of type $D$ or type $\tilde{A}$, if it arises from a once-punctured {\disk} or annulus, respectively. 
Related work on friezes of type D and $\tilde{A}$ 
include~\cite{Sch08Dn, BM09, ARS10,BR10,Smi15,BFPT16,FP16}.

\begin{figure}[!htbp]
\centering
\input{fig_disc_as_AT}
\caption{Triangulation of a once-punctured pentagon drawn as an asymptotic triangulation 
(see~\cite[Lemma 3.6]{BPT16}).}
\label{fig:disc_as_AT}
\end{figure}

This is an extended abstract of~\cite{GMV16},
where
we study infinite friezes whose entries are Laurent polynomials (as opposed to positive integers). 
In \cref{sec:cluster_algebras_from_surfaces}, we give the necessary background.
In \cref{subsec:infinite_friezes_of_cluster_algebra_elements}, we construct
an infinite frieze of Laurent polynomials associated to \emph{generalized peripheral arcs}. 
In \cref{subsec:complementary_arcs_and_progression_formulas}, we introduce \emph{complementary arcs}, which are arcs between the same two vertices in a surface, but of alternate direction. 
We use these complementary arcs to describe \emph{progressions} of arcs in the frieze   (\cref{thm:progression_formula}).
In \cref{subsec:brac_and_growth}, we show that \emph{growth coefficients} (as defined in~\cite{BFPT16}) of the frieze are equal to Laurent polynomials corresponding to certain curves called \emph{bracelets} in the surface. 
We state further combinatorial results in \cref{subsec:diff_comp,subsec:arithmetic_progressions}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%=======================================
%\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}. 
Let $S$ be either an annulus or a disk, 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}. 

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 \cref{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.
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 \cref{fig:disc_as_AT}, left).

Due to~\cite[Thm.~7.11]{FST08},
we can associate a \emph{signed adjacency matrix} $B_T$, 
and hence a cluster algebra, to $T$.
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 (respectively, 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} (respectively, a \emph{band} on a M\"obius strip or annulus) $G_{T,\gamma}$. 
Put simply, such graphs are made out of gluing squares together, with one such square for each arc of T crossed by $\gamma$; see \cref{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 \cref{fig:example_generalized_arc} has $11$ perfect matchings.
Following \cref{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 \cref{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.




\begin{figure}[!hbt]
\centering
\input{fig_bci_triangulation}
\hfill
\input{fig_snake_laurent_example}
\caption{Left: An ideal triangulation $T$ and a generalized arc $\gamma$ of a once-punctured {\disk}. Center: drawn on a strip. Right: Snake graph $G_{T,\gamma}$.}
\label{fig:example_generalized_arc}
\label{fig:generalized_arc_snakegraph}
\scalebox{0.7}{\input{fig_bracelets}}
\caption{Bracelets $Brac_1$, $Brac_2$, and $Brac_3$.}\label{fig:bracelets}
\end{figure}

\section{Results}
\subsection{Infinite friezes of cluster algebra elements}
\label{subsec:infinite_friezes_of_cluster_algebra_elements}

\begin{thm}
Let $T$ be an ideal triangulation of a once-punctured {\disk} or an annulus.
Let $Bd$ be a boundary component with $n$ marked points, where $n\geq 2$.
Then the Laurent polynomials corresponding to generalized peripheral arcs on $Bd$ form an infinite frieze pattern.
\label{thm:Frieze_Laurent}
\end{thm}

We prove this theorem by applying skein relations, as illustrated in \cref{fig:skeinrelationsproof}. 
Further, we lift arcs from a once-punctured disk (or annulus) to a covering space given by the infinite strip.  
Given a triangulation of the infinite strip with marked points on a boundary $\partial$,
the peripheral arc $\gamma(i,j)$ from $i$ to $j$ on $\partial$
corresponds to the $(i,j)$-th entry in the infinite frieze pattern arising from this triangulation. 
See \cref{fig:matching_ex,fig:complement_symmetry}.

\begin{figure}[!hbt]
\centering
\input{fig_skeinrelationsproof}
\caption{Applying skein relations to prove \cref{thm:Frieze_Laurent}}
\label{fig:skeinrelationsproof}
\end{figure}

\begin{figure}[!hbt]
\centering
\input{fig_matching_ex}
\caption{Triangulation of a strip and an arc $\gamma(i,j)$ from $i$ to $j$.}
\label{fig:matching_ex}
\end{figure}

\begin{figure}[!hbt]
\centering
\input{fig_complement_symmetry}
\caption{The first six rows of an infinite frieze of elements of the cluster algebra corresponding to peripheral arcs of a punctured disk.}
\label{fig:complement_symmetry}
\end{figure}

\subsection{Complementary arcs and progression formulas}
\label{subsec:complementary_arcs_and_progression_formulas}
In this section, we present formulas governing relations among the Laurent polynomial entries of the frieze of \cref{thm:Frieze_Laurent}. 
These generalize the relations given in~\cite[Thm. 2.5]{BFPT16}.

For $1 \leq i, j \leq n$ and $k=1,2,\dots$, we let $\gamma_k(i,j)$ denote the generalized peripheral arc 
that lifts to the covering by the strip as follows: 
$$\gamma_k(i,j) = \begin{cases} \gamma\left(i,j+(k-1)n\right) & \mathrm{~if~} i < j \\ 
\gamma\left(i,j + kn\right) & \mathrm{~if~} i \geq j \end{cases}.$$
That is, $\gamma_k(i,j)$ is the generalized peripheral arc that starts at marked point $i$ and finishes at the marked point $j$ (possibly with $i=j$) with $(k-1)$ self-crossings such that the boundary Bd is to the right of the curve as we trace it.  

\begin{defn}[Complementary arc] Using the above shorthand notation, we define the arc complementary to $\gamma_k = \gamma_k(i,j)$ as 
$$\gamma_k(i,j)^C =  \begin{cases} \gamma\left(j, i+kn \right) & \mathrm{~if~} i < j \\ 
\gamma\left(j, i + (k-1)n\right) & \mathrm{~if~} i \geq j \end{cases}.$$
\end{defn}

\begin{rem}
When $i\not = j$, the complementary arc $\gamma_k^C$ to $\gamma_k=\gamma_k(i,j)$ is the generalized arc 
starting at $j$ and finishing at $i$ and retaining $(k-1)$ self-crossings while following the orientation of the surface. See \cref{fig:complementary_arcs}. In this case, $(\gamma_k^C)^C = \gamma_k$. 
On the other hand, when $i=j$, complementation is non-involutive and simply decreases the number of self-intersections by one. 
\end{rem}

\def\ScaleForProgressionFormulaFigs{0.41}
\begin{figure}[!htbp]
\centering
\input{fig_complementaryarcs}
\caption{Examples of involutive complementary arcs $\gamma_1$, $\gamma_1^C$ and $\gamma_3$, $\gamma_3^C$.}\label{fig:complementary_arcs}
\input{fig_gamma2}
\hfill
\input{fig_gamma4}
\caption{Case $m=1$ for the progression formula (\cref{thm:progression_formula}).
Left: $x(\gamma_2) = x(\gamma_1) x(Brac_{1}) + x(\gamma_{1}^C)$.
Right: $x(\gamma_4) = x(\gamma_1) x(Brac_{3}) + x(\gamma_{3}^C)$.}
\label{fig:gamma4}
\end{figure}

\begin{thm}[Progression formulas]
\label{thm:progression_formula}
Let $\ga_1$ be a peripheral arc or a boundary edge of $(S,M)$ starting and finishing at points $i$ and $j$. 
For $k=2,3,\dots$ and $1\leq m \leq k-1$, we have
\begin{equation}
\label{eq:thm:progression_formula}
x(\gamma_k) = x(\gamma_m) \, x(Brac_{k-m}) + x(\gamma_{k-2m+1}^C).
\end{equation}
For $r \geq 0$, $\gamma_{-r}^C$ is defined to be the curve $\gamma_{r+1}$ with a kink, so that $x(\gamma_{-r}^C) = - x(\gamma_{r+1})$.
\end{thm}

\vspace{-1ex}
\begin{rem}\label{rem:m1}
In the above theorem,
when $m=1$ (see \cref{fig:gamma4})
and $m=k-1$, 
we have
\begin{gather}
x(\gamma_k) = x(\gamma_1) \, x(Brac_{k-1}) + x(\gamma_{k-1}^C) \text{  and  }
x(\gamma_k) = x(\gamma_{k-1}) \, x(Brac_{1}) - x(\gamma_{k-2}).
\label{eqn:m_k_min_1}
\end{gather}
Compare \eqref{eqn:m_k_min_1}, right, with~\cite[Thm. 2.5]{BFPT16}.  
\end{rem}


\begin{figure}[!htbp]
\centering
\input{fig_progression_unresolved}
\caption{Lift of $\gamma_k$ for $k=10$, $m=4$ drawn on the strip.}
\label{fig:progression_unresolved}

\input{fig_progression_bracelet}
\caption{Lifts of $\gamma_{m}$ and $Brac_{k-m}$ for $k=10$, $m=4$ drawn on the strip.}
\label{fig:progression_bracelet}

\input{fig_progression}
\input{fig_progression_kink}
\caption{Lift of $\gamma_{k-2m+1}^C$ for $k=10$ drawn on the strip. Top: $m=4$, Bottom: $m=8$.}
\label{fig:progression_m48}
\end{figure}

We give a sketch of our proof of \cref{thm:progression_formula}.
Let $\gamma_k:=\gamma_k(i,j)$.
We draw $\gamma_k$ so that it first closely follows the other boundary (or the puncture) and then spirals out.
In the covering via the infinite horizontal strip, we draw the lower boundary $Bd$ so that $i$ is drawn to the left of $j$ in each frame.
Each representative of $\gamma_k$ is drawn 
starting from a vertex labeled $i$ at a frame $Reg_0$.
We go north, passing through all of the $(k-1)$ crossings.  
We then turn southeast and finish at a vertex labeled $j$, which is located in the frame $k-1$ frames (respectively, $k$ frames) east of $Reg_0$ if $i\neq j$ (respectively, if $i=j$). See \cref{fig:progression_unresolved}. 

We order the crossings of $\gamma_k$ so that the first crossing is the one closest to $Bd$ and the $(k-1)$-th crossing is the one furthest away from $Bd$.
Resolving each representative of the $m$-th crossing in each frame,
we get $\gamma_{m}$ and $Brac_{k-m}$ (see \cref{fig:progression_bracelet})
and the curve $\gamma_{k-2m+1}^C$ (see \cref{fig:progression_m48}), which correspond to the first and second summands of \eqref{eq:thm:progression_formula}, respectively.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Bracelets and growth coefficients}
\label{subsec:brac_and_growth}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

According to~\cite[Thm. 2.2]{BFPT16}, for an n-periodic infinite frieze of positive integers, the difference between the entries in rows $(nk+1)$ \& $(nk-1)$ and the same column is a constant (see \cref{fig:growth_frieze}).
These differences are also constant in our infinite friezes of Laurent polynomials, and we give geometric interpretations to these differences. 

\def\mylevel{k}
\def\myremainder{j}

\begin{prop}
\label{prop:see_BFPT16_thm2_2}
Let $\mathcal{F}=\{\mathcal{F}_{i,j}\}$ 
be an $n$-periodic frieze as described in \cref{subsec:infinite_friezes_of_cluster_algebra_elements}.
For each $k \geq 1$, 
\begin{gather*}
 x(Brac_{k})  = \mathcal{F}_{i,i+1+kn} - \mathcal{F}_{i+1,i+kn}
 \text{ for all $i\in\mathbb{Z}$.}
\end{gather*}
\end{prop}

Following \cite[Def. 2.3]{BFPT16}, 
for $k \geq 0$, we define the \emph{$k$th growth coefficient} for $\mathcal{F}$
to be $s_0:=2$, and $s_k:= \mathcal{F}_{i,i+1+kn} - \mathcal{F}_{i+1,i+kn}$, otherwise. 
We say that \emph{level $\mylevel$} of a frieze consists of the entries of the frieze indexed by $(i,i+(\mylevel-1)n+\myremainder)$ where $\myremainder=1,\dots,n$.
Note that $s_k$ measures the difference between entries in the first row of the $(k+1)$st level and the penultimate row of the $k$th level. 
Also, per \cref{prop:see_BFPT16_thm2_2}, $s_k=x(\Brac_k)$ whenever $k\geq 1$, so we can use the two terms interchangeably.

Given a triangulation of an annulus, we get two different friezes 
corresponding to the outer and inner boundaries. 
We see that their growth coefficients $s_k$ coincide since 
 $\Brac_k$ is defined independently of the choice of the boundary of an annulus.
This agrees with \cite[Thm. 3.4]{BFPT16}. 


\begin{figure}[!hbt]
\centering
\scalebox{0.8}{\input{fig_growth_frieze}}
\caption{Growth coefficients in a frieze of type $\tilde{A}$.}
\label{fig:growth_frieze}
\end{figure}

\begin{figure}[!hbt]
\hspace{3cm}\scalebox{0.9}{\input{fig_arithmeticIntegerFrieze}}
\caption{Arithmetic progressions in a frieze of type $D$.}
\label{fig:arithmeticIntegerFrieze}
\end{figure}

\vspace{-2ex}
%%%%%%%%%%%%%%%%%
\subsection{Differences from complement symmetry}
\label{subsec:diff_comp}
%%%%%%%%%%%%%%%%%

We consider the difference between frieze entries associated to complementary arcs. 
For the once-punctured {\disk}, this difference is constant across all levels and is determined only by the endpoints of each arc. 

\begin{prop}\label{prop:complementary_diff}
Let $\mathcal F$ be a frieze coming from a triangulation of a once-punctured {\disk} or annulus.
Let $\gamma_1=\gamma$ be an ordinary arc from $i$ to $j$ (possibly $i=j$) or a boundary edge from $i$ to $i+1$. 
Define $c_{k,\gamma} :=  x\left(\gamma_k\right) - x\left(\gamma^C_k \right)$, and
write $c_k:=c_{k,\gamma}$.
Then, for $k \geq 2$, we have the relations

\smallskip
\begin{inparaenum}[$(1)$]
\item
$c_k = (s_{k-1}-s_{k-2}) c_1 + c_{k-2}$, \quad where we define $c_0 = c_1$;

\item 
$c_k = c_1 \left( 1 +  \sum_{i=0}^{k-1}(-1)^{i+\alpha}s_i  \right)$,
\quad
where $\alpha=1$ if $k$ is even 
and $\alpha=0$ otherwise.
\end{inparaenum}

\end{prop}

Note that, if $i=j$, then $c_1=x(\gamma_1)$.

\vspace{-1ex}
\subsection{Arithmetic progressions}
\label{subsec:arithmetic_progressions}

Tschabold showed that each diagonal of a frieze (of positive integers) arising from a once-punctured {\disk} is made up of a collection of arithmetic progressions~{\cite[Prop. 3.11]{Tsc15}}. 
The 
dotted
and 
dashed
circles in \cref{fig:arithmeticIntegerFrieze} highlight two such arithmetic progressions.

\begin{prop}[Analog of~{\cite[Thm.~3.11]{Tsc15}}]
\label{prop:arithmetic_progression}
Suppose $(S,M)$ is a once-punctured \disk. 
Let $\gamma_1=\gamma$ be an ordinary arc from $i$ to $j$ (possibly $i=j$) or a boundary edge from $i$ to $i+1$. 
Then, for $k \geq 2$, we have
$x(\gamma_k) = x(\gamma_{k-1}) + \left(~ x(\gamma_1) + x(\gamma_1^C) ~\right).$
\end{prop}

\vspace{-2ex}
\acknowledgements{
We thank Karin Baur for helpful comments, Manuela Tschabold for allowing us to use her TikZ figures, and
  University of Minnesota for hosting Hannah Vogel.
during her stay in the Winter of 2016.
This work benefited from computations using {\sc SageMath}~\cite{sage} and code written by Ana Garc\'{i}a Elsener and Jorge Nicol\'{a}s L\'{o}pez.
}

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