
\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 Fig.~\ref{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 Figs.~\ref{fig:matching_ex} and~\ref{fig:complement_symmetry}.

\begin{figure}[!hbt]
\centering
\input{fig_skeinrelationsproof}
\caption{Applying skein relations to prove Theorem \ref{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 Theorem~\ref{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 %in $C_{n,m}$ or $D_n$ 
that lifts to the covering by the strip as follows: %(using the notation of Remark \ref{rem:gamIJ}):
$$\gamma_k(i,j) = \begin{cases} \gamma\left(i,j+(k-1)n\right) & \mathrm{~if~} i < j \\ 
\gamma\left(i,j + kn\right) & \mathrm{~if~} i \geq j \end{cases}.$$
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 %(i.e. up to homotopy) 
starting at $j$ and finishing at $i$ and retaining $(k-1)$ self-crossings while following the orientation of the surface. See Fig.~\ref{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 (Theorem \ref{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}

\begin{rem}\label{rem:m1}
In the above theorem,
when $m=1$ (see Fig.~\ref{fig:gamma4})
and $m=k-1$, 
we have
\begin{gather}
%\label{eqn:m_1}
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 (\ref{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 Theorem \ref{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 (resp., $k$ frames) east of $Reg_0$ if $i\neq j$ (resp., if $i=j$). See Fig.~\ref{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 Fig.~\ref{fig:progression_bracelet})
and the curve $\gamma_{k-2m+1}^C$ (see Fig.~\ref{fig:progression_m48}), which correspond to the first and second summands of (\ref{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 Fig.~\ref{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 Sec.~\ref{subsec:infinite_friezes_of_cluster_algebra_elements}.
For each $k \geq 1$, %we have 
\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 Proposition \ref{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}
\scalebox{0.92}{\input{fig_arithmeticIntegerFrieze}}
\caption{Arithmetic progressions in a frieze of type $D$.}
\label{fig:arithmeticIntegerFrieze}
\end{figure}

%%%%%%%%%%%%%%%%%
\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 
%For $k\geq 2$, we have
$c_k = c_1 \left( 1 +  \sum_{i=0}^{k-1}(-1)^{i+\alpha}s_i  \right)$,
\quad
where $\alpha=1$ if $k$ is even 
and $\alpha=0$ otherwise.
\end{inparaenum}

\end{prop}

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

\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 Fig.~\ref{fig:arithmeticIntegerFrieze} highlight two such arithmetic progressions.

\begin{prop}[analog of~{\cite[Thm.~3.11]{Tsc15}}]
%[{Corollary of Theorem \ref{thm:progression_formula}}]
\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
%\displaystyle{
$x(\gamma_k) = x(\gamma_{k-1}) + \left(~ x(\gamma_1) + x(\gamma_1^C) ~\right).$
%}
\end{prop}
