# Periodic Infinite Frieze Patterns of Type $\Lambda_{p_{1},\ldots,p_{n}}$ and
Dissections on Annuli

Esther Banaian and Jiuqi Chen

###### Abstract

Finite frieze patterns with entries in
$\mathbb{Z}[\lambda_{p_{1}},\ldots,\lambda_{p_{s}}]$ where
$\\{p_{1},\ldots,p_{s}\\}\subseteq\mathbb{Z}_{\geq 3}$ and
$\lambda_{p}=2\cos(\pi/p)$ were shown to have a connection to dissected
polygons by Holm and Jørgensen. We extend their work by studying the
connection between infinite frieze patterns with such entries and dissections
of annuli and once-punctured discs. We give an algorithm to determine whether
a frieze pattern with entries in
$\mathbb{Z}[\lambda_{p_{1}},\ldots,\lambda_{p_{s}}]$, finite or infinite,
comes from a dissected surface. We introduce quotient dissections as a
realization for some frieze patterns unrealizable by an ordinary dissection.
We also introduce two combinatorial interpretations for entries of frieze
patterns from dissected surfaces. These interpretations are a generalization
of matchings introduced by Broline, Crowe, and Isaacs for finite frieze
patterns over $\mathbb{Z}$.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Finite Frieze Patterns of Positive Integers
    2. 2.2 Infinite Frieze Patterns of Positive Integers
    3. 2.3 Finite Frieze Patterns from Dissected Polygons
  3. 3 Frieze Patterns of Type $\Lambda_{p_{1},\ldots,p_{s}}$
  4. 4 Dissections of Once-Punctured Discs and Annuli
    1. 4.1 Infinite Strips
  5. 5 Frieze Patterns from dissected Once-Punctured Discs and Annuli
    1. 5.1 Realizability of Skeletal Frieze Patterns
    2. 5.2 Quotient Dissections
    3. 5.3 Cutting
    4. 5.4 Realization Algorithm
  6. 6 Matchings
    1. 6.1 Background
    2. 6.2 Local Weighting
    3. 6.3 Traditional Weighting
  7. 7 Growth Coefficients
    1. 7.1 Comparing Inner and Outer Frieze Patterns
  8. 8 Connection To T-Paths
  9. 9 Positivity
  10. 10 Acknowledgements

##  1 Introduction

Frieze patterns, defined in the finite case by Coxeter [10] and in the
infinite case by Tschabold [20], have a rich connection with the geometry of
triangulated and dissected surfaces. For a beautiful survey on frieze patterns
and their relation to both geometry and algebra see [16]. The goal of this
paper is to delve further into the geometric connections, principally in the
case of dissected surfaces.

We define finite and infinite frieze patterns simultaneously.

######  Definition 1.

A (finite) frieze pattern $\mathcal{F}$ over a ring $R$ is a (finite) array
$(m_{i,j})_{i,j\in\mathbb{Z},j\geq i}$ of shifted infinite rows such that

  1. 1.

for all $i,j$, $m_{i,j}\in R$;

  2. 2.

$m_{i,i}=0$ and $m_{i,i+1}=1$ for all $i\in\mathbb{Z}$; and

  3. 3.

every diamond in $\mathcal{F}$ of the form $\begin{array}[]{ccc}&m_{i,j}&\\\
m_{i-1,j}&&m_{i,j+1}\\\ &m_{i+1,j+1}&\end{array}$ satisfies the unimodular
rule: $m_{i-1,j}m_{i,j+1}-m_{i,j}m_{i+1,j+1}=1$.

We illustrate the indexing of a frieze pattern. Notice that all entries of the
form $m_{i,j}$ for fixed $i$ lie along a SE diagonal while all entries with
fixed $j$ lie along a SW diagonal.

$\begin{array}[]{ccccccccccccccccccccccccc}\ldots&0&&0&&0&&0&&0&&\ldots\\\
\ldots&&1&&1&&1&&1&&&\ldots\\\
\ldots&m_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{-1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{1}}&&m_{0,2}&&m_{1,3}&&m_{2,4}&&m_{3,\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{5}}&&\ldots\\\
\ldots&&m_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{-1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{2}}&&m_{0,3}&&m_{1,4}&&m_{2,\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{5}}&&&\ldots\\\
\ldots&m_{-2,2}&&m_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{-1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{3}}&&m_{0,4}&&m_{1,\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{5}}&&m_{2,6}&&\ldots\\\
\ldots&&m_{-2,3}&&m_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{-1},\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{4}}&&m_{0,\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{5}}&&m_{1,6}&&&\ldots\\\
&&&.^{.^{.}}&&m_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{-1},\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{5}}&&\ddots&&&&&&\end{array}$

We define several terms related to frieze patterns.

######  Definition 2.

  1. 1.

The first nontrivial row is called the _quiddity row_. We will simply refer to
this as the first row.

  2. 2.

The number of nontrivial rows of a finite frieze pattern is the _width_.

  3. 3.

If the quiddity row $\ldots,a_{0},a_{1},\ldots,a_{n},a_{n+1},\ldots$ of a
frieze pattern is $n$-periodic, so that $a_{i}=a_{i+n}$ for all
$i\in\mathbb{Z}$, we call $(a_{1},\ldots,a_{n})$ a _quiddity cycle_.

By the unimodular rule, the quiddity sequence of a frieze pattern determines
the entire frieze pattern. Hence a quiddity cycle also determines a frieze
pattern.

Soon after Coxeter defined frieze patterns in [11], Conway and Coxeter showed
that finite frieze patterns of positive integers are in bijection with
triangulated polygons [10]; see Section 2.1. When Caldero and Chapoton
demonstrated a connection between cluster algebras and frieze patterns [6],
there was renewed interest in frieze patterns. One resulting generalization
was from Baur, Parsons, and Tschabold, who showed infinite frieze patterns of
positive integers are in bijection with triangulated annuli and once-punctured
discs [1]. See Section 2.2 for more details.

Holm and Jørgensen generalized Conway and Coxeter’s result in a different
direction and investigated finite frieze patterns from dissections of polygons
[14]. In place of positive integers, they work over the ring of algebraic
integers of the field $\mathbb{Q}(\lambda_{p_{1}},\ldots,\lambda_{p_{s}})$
where $p_{1},\ldots,p_{s}$ is a list of sizes of subgons in the dissection and
$\lambda_{p}=2\cos(\pi/p)$. More specifically, their frieze patterns have
quiddity cycles where each entry is of the form
$\sum_{i=1}^{s}c_{i}\lambda_{p_{i}}$ for $c_{i}\in\mathbb{Z}_{\geq 0}$. We say
that such frieze patterns are of _Type $\Lambda_{p_{1},\ldots,p_{s}}$_. We
give more details about this work in Section 2.3.

We combine these directions of generalization by investigating infinite frieze
patterns of Type $\Lambda_{p_{1},\ldots,p_{s}}$. Some of these frieze patterns
correspond to dissections of annuli and once-punctured discs. In Section 3 we
discuss some preliminary results about frieze patterns of Type
$\Lambda_{p_{1},\ldots,p_{s}}$ while in Section 4 we discuss the details of
dissections of annuli and once-punctured discs.

In Section 5, we build towards determining which infinite frieze patterns of
Type $\Lambda_{p_{1},\ldots,p_{s}}$ arise from dissected annuli and once-
punctured discs. As a first step, we give an explicit description of some
frieze patterns which do not arise in this way (i.e., are “unrealizable”) in
Proposition 1. When we restrict to the subclass of _skeletal_ frieze patterns
of Type $\Lambda_{p_{1},\ldots,p_{s}}$, we give a direct characterization of
which frieze patterns which are realizable by a dissection of an annulus or
once-punctured disc (Proposition 2). See Definition 9 where skeletal
dissections are described.

There is a class of skeletal frieze patterns which are neither unrealizable by
Proposition 1 nor realizable by Proposition 2. We demonstrate these frieze
patterns in the table below. We consider some 2-periodic frieze patterns with
quiddity cycle $(a,b)$. All resulting frieze patterns are skeletal. We write
✓when the frieze pattern generated by quiddity cycle $(a,b)$ is realizable by
Proposition 2, we write $\times$ when the resulting frieze pattern is
unrealizable by Proposition 1, and otherwise we write ?.

$a$ \$b$ | 3 | $2+\sqrt{2}$ | $1+2\sqrt{2}$ | $3\sqrt{2}$
---|---|---|---|---
3 | ✓ | ✓ | ? | $\times$
$2+\sqrt{2}$ | ✓ | ✓ | ✓ | ?
$1+2\sqrt{2}$ | ? | ✓ | ✓ | ✓
$3\sqrt{2}$ | $\times$ | ? | ✓ | ✓

The presence of these frieze patterns between the realizable and unrealizable
frieze patterns motivated the definition of _quotient dissections_ , which
provide a realization for all skeletal frieze patterns which are neither
unrealizable by Proposition 1 nor realizable by Proposition 2. See Section 5.2
for details on quotient dissections. Then, Section 5.4 provides an algorithm
to determine whether an arbitrary (not necessarily infinite) frieze pattern of
Type $\Lambda_{p_{1},\ldots,p_{s}}$ is realizable by a dissection or a
quotient dissection.

In Section 6, we provide two combinatorial interpretations of entries of
frieze patterns from dissections of surfaces. These interpretations are sums
over sets of assignments of subgons to vertices with two different rules for
weighting the matchings. Theorem 8 shows that the _local weighting_ provides a
combinatorial interpretation for the entries of a frieze pattern from a
dissection or quotient dissection. We show that summing over the two
weightings (local and traditional) always gives the same result in Theorem 9.

Section 7 discusses a third type of weighting, _annulus weighting_ , which
gives the _growth coefficients_ of infinite frieze patterns from dissections.
See Theorem 12. Growth coefficients are a sequence of numbers from a periodic
infinite frieze pattern which measure the growth rate of its entries and were
first defined in [2].

In Section 8, we compare our weighted matchings to weak T-paths recently
defined in [8]. We conclude in Section 9 with a discussion about which
infinite frieze patterns of Type $\Lambda_{p_{1},\ldots,p_{s}}$ have all
positive entries. We cannot guarantee that all frieze patterns from quotient
dissections have positive entries, but we conjecture this is true in
Conjecture 1.

##  2 Background

###  2.1 Finite Frieze Patterns of Positive Integers

The correspondence between triangulated polygons and finite frieze patterns
over $\mathbb{Z}_{\geq 0}$ is a model for the way we will connect other
surfaces to frieze patterns. Given a triangulated $n$-gon, label the vertices
$v_{1},\ldots,v_{n}$ in counterclockwise order. Suppose that $v_{i}$ is
incident to $a_{i}$ triangles. Then, we associate the frieze pattern with
quiddity cycle $(a_{1},\ldots,a_{n})$ to the triangulated polygon. See Figure
1 for an example. Conway and Coxeter prove that this map is a bijection.

######  Theorem 1 ([10]).

Finite frieze patterns of positive integers with width $n$ are in bijection
with triangulated $(n+3)$-gons.

51313221 $\begin{array}[]{cccccccccccccccccccc}&&0&&0&&0&&0&&0&&0&&0&&0&&0&\\\
&&&1&&1&&1&&1&&1&&1&&1&&1&&1\\\ &&1&&3&&1&&3&&2&&2&&1&&5&&1&\\\
&&&2&&2&&2&&5&&3&&1&&4&&4&&2\\\ &&7&&1&&3&&3&&7&&1&&3&&3&&7&\\\
&&&3&&1&&4&&4&&2&&2&&2&&5&&3\\\ &&1&&2&&1&&5&&1&&3&&1&&3&&1&\\\
&&&1&&1&&1&&1&&1&&1&&1&&1&&1\\\ &&0&&0&&0&&0&&0&&0&&0&&0&&0&\\\ \end{array}$

Figure 1: A triangulated octagon and the corresponding frieze pattern over
$\mathbb{Z}_{\geq 0}$ of width 5

###  2.2 Infinite Frieze Patterns of Positive Integers

Tschabold produced infinite frieze patterns by using the same algorithm as
Conway and Coxeter but starting with a triangulated once-punctured disc [20].
Soon after, Tschabold along with Baur and Parsons extended this work by also
considering triangulated annuli [1]. With these surfaces, we count adjacent
triangles by drawing a small circle around each vertex, then counting the
number of triangles this circle passes through. When the triangulation
includes a self-folded triangle, the small circle will see the two sides of
the triangle as two separate triangles. When forming a quiddity cycle from a
triangulated annulus, one picks one boundary to work with respect to. As a
convention, we will work with respect to the outer boundary. However, in
Section 7.1 we will discuss how the frieze patterns from the outer boundary
and inner boundary are related. See Figure 2 for an example of a frieze
pattern from a triangulated annulus.

It turns out that all periodic infinite frieze patterns of positive integers
arise from a triangulation of either a once-punctured disc or annulus.

######  Theorem 2 ([1]).

Periodic infinite frieze patterns of positive integers are in bijection with
triangulated once-punctured discs and annuli.

441 $\begin{array}[]{cccccccccccccccccccc}0&&0&&0&&0&&0&&0&&0&&0\\\
&1&&1&&1&&1&&1&&1&&1&\\\ 1&&4&&4&&1&&4&&4&&1&&4\\\ &3&&15&&3&&3&&15&&3&&3&\\\
8&&11&&11&&8&&11&&11&&8&&11\\\ &29&&8&&29&&29&&8&&29&&29&\\\
105&&21&&21&&105&&21&&21&&105&&21\\\ &&&&&&&\vdots&&&&&\\\ \end{array}$

Figure 2: A triangulated annulus and the corresponding infinite frieze pattern
over $\mathbb{Z}_{\geq 0}$

###  2.3 Finite Frieze Patterns from Dissected Polygons

Holm and Jørgensen defined a class of frieze patterns arising from dissected
polygons [14].

######  Definition 3.

A _dissection_ $\mathcal{D}$ of a surface with marked points $(S,M)$ is a set
of non-crossing arcs which divide $S$ into polygons. We refer to these
polygons as _subgons_. A _$p$ -angulation_ is a dissection which breaks
$(S,M)$ into $p$-gons.

Note that a $n$-gon can only be $p$-angulated if $n+2$ is divisible by $p$.

Given a polygon $P$ with dissection $\mathcal{D}$, let $v_{1},\ldots,v_{n}$ be
the vertices of P, labeled in counterclockwise order. Let $A_{i}$ be the set
of subgons in the dissection incident to $v_{i}$. For any $p\geq 3$, let
$\lambda_{p}=2\cos(\pi/p)$. Then, we set $m_{i-1,i+1}=\sum_{\mathcal{P}\in
A_{i}}\lambda_{|\mathcal{P}|}$, where $|\mathcal{P}|$ is the number of sides
of $\mathcal{P}$. For example, in Figure 3, if the top left vertex of the
hexagon is $v_{1}$, $A_{1}=\\{3,3,4\\}$.

######  Remark 1.

The number $\lambda_{p}$ is the ratio of the length of a 1-diagonal (a
diagonal which skips 1 vertex) and the length of a side of a regular $p$-gon.
The first few values of $\lambda_{p}$ are familiar numbers.

$p$ | $\lambda_{p}$
---|---
3 | 1
4 | $\sqrt{2}$
5 | $\frac{1+\sqrt{5}}{2}=\phi$
6 | $\sqrt{3}$

In particular, since $\lambda_{3}=1$, the Holm and Jørgensen algorithm reduces
to that of Conway and Coxeter when the dissection is a triangulation.

Lemma 9 gives expressions for all diagonals in a regular polygon.

######  Theorem 3 ([14]).

Using the above algorithm, there is an injection from the set of dissections
of an $(n+3)$-gon to width $n$ frieze patterns.

$1+\sqrt{2}$$1$$2+\sqrt{2}$$1$$1+\sqrt{2}$$\sqrt{2}$
$\begin{array}[]{cccccccccccccccccccc}0&&0&&0&&0&&0&&0&&0\\\
&1&&1&&1&&1&&1&&1&\\\ 1&&2+\sqrt{2}&&1&&1+\sqrt{2}&&\sqrt{2}&&1+\sqrt{2}&&1\\\
&\sqrt{2}&&1+\sqrt{2}&&1+\sqrt{2}&&\sqrt{2}&&1+\sqrt{2}&&1+\sqrt{2}&\\\
\sqrt{2}&&1+\sqrt{2}&&1&&2+\sqrt{2}&&1&&1+\sqrt{2}&&\sqrt{2}\\\
&1&&1&&1&&1&&1&&1&\\\ 0&&0&&0&&0&&0&&0&&0\\\ \end{array}$

Figure 3: A dissected hexagon and the corresponding frieze pattern of width 3

Holm and Jørgensen do not provide a description of which frieze patterns are
in bijection with all dissected polygons. However, when narrowing the focus to
$p$-angulations, they have a stronger result.

######  Definition 4.

Let $p\in\mathbb{Z}\geq 3$. A frieze pattern is of _Type $\Lambda_{p}$_ if the
quiddity row consists of (positive) integral multiples of
$\lambda_{p}=2\cos(\frac{\pi}{p})$.

See Figure 4 for an example of a frieze pattern of Type $\Lambda_{4}$ and the
corresponding 4-angulation.

######  Theorem 4 ([14]).

There is a bijection between $p$-angulated $(n+3)$-gons and frieze patterns of
Type $\Lambda_{p}$.

$2\sqrt{2}$$\sqrt{2}$$\sqrt{2}$$2\sqrt{2}$$2\sqrt{2}$$\sqrt{2}$$\sqrt{2}$$2\sqrt{2}$
$\begin{array}[]{cccccccccccccccccc}0&&0&&0&&0&&0&&0&&0&&0&&0\\\
&1&&1&&1&&1&&1&&1&&1&&1&\\\
2\sqrt{2}&&\sqrt{2}&&\sqrt{2}&&2\sqrt{2}&&2\sqrt{2}&&\sqrt{2}&&\sqrt{2}&&2\sqrt{2}&&2\sqrt{2}\\\
&3&&1&&3&&7&&3&&1&&3&&7&\\\
5\sqrt{2}&&\sqrt{2}&&\sqrt{2}&&5\sqrt{2}&&5\sqrt{2}&&\sqrt{2}&&\sqrt{2}&&5\sqrt{2}&&5\sqrt{2}\\\
&3&&1&&3&&7&&3&&1&&3&&7&\\\
2\sqrt{2}&&\sqrt{2}&&\sqrt{2}&&2\sqrt{2}&&2\sqrt{2}&&\sqrt{2}&&\sqrt{2}&&2\sqrt{2}&&2\sqrt{2}\\\
&1&&1&&1&&1&&1&&1&&1&&1&\\\ 0&&0&&0&&0&&0&&0&&0&&0&&0\\\ \end{array}$

Figure 4: A 4-angulated octagon and the corresponding frieze pattern of Type
$\Lambda_{4}$

##  3 Frieze Patterns of Type $\Lambda_{p_{1},\ldots,p_{s}}$

We further study frieze patterns whose quiddity sequence consists of sums of
$\lambda_{p}$ for possibly multiple values of $p$. Inspired by Holm and
Jørgensen’s notation, we introduce a term for shorthand.

######  Definition 5.

Let $p_{1},\ldots,p_{s}\in\mathbb{Z}_{\geq 3}$. An infinite frieze pattern is
of _Type $\Lambda_{p_{1},\ldots,p_{s}}$_ if each entry in its quiddity row is
the sum of positive integral multiples of
$\lambda_{p_{1}},\ldots,\lambda_{p_{s}}$.

For example, the frieze pattern in Figure 3 is Type $\Lambda_{3,4}$. Given a
frieze pattern $\mathcal{F}$ of Type $\Lambda_{p_{1},\ldots,p_{s}}$, we can
write each entry in the quiddity row in the form of

$m_{i-1,i+1}=\sum_{p\in A_{i}}\lambda_{p}$

where $A_{i}$ is a multiset. For example, given the quiddity cycle
$(3,2+\sqrt{2},\sqrt{2}+\sqrt{3})$, we have that
$A_{3k}=\\{3,3,3\\},A_{3k+1}=\\{3,3,4\\},$ and $A_{3k+2}=\\{4,6\\}$ for all
$k\in\mathbb{Z}$.

We record some useful results about frieze patterns. First, we study
1-periodic frieze patterns. These results are most conveniently stated in
terms of Chebyshev polynomials.

######  Definition 6.

Let $U_{k}(x)$ denote the _normalized Chebyshev polynomials of the second
kind_ , determined by initial conditions $U_{-1}(x)=0,U_{0}(x)=1$ and the
recurrence,

$U_{k}(x)=U_{1}(x)U_{k-1}(x)-U_{k-2}(x)$

For example, $U_{1}(x)=x$ and $U_{2}(x)=x^{2}-1$. These normalized Chebyshev
polynomials can be obtained by evaluating the ordinary Chebyshev polynomials
of the second kind at $\frac{x}{2}$.

######  Lemma 1.

Let $\mathcal{F}$ be the 1-periodic frieze pattern determined by the quiddity
cycle $(x)$, where $x$ is an indeterminate. Then, $\mathcal{F}$ is an infinite
frieze pattern, with the $k$-th row given by $(U_{k}(x))$.

This lemma is quickly proven given the following well-known result, which
shows that rows of frieze patterns satisfy a recurrence similar to the
Chebyshev recurrence.

######  Lemma 2.

Let $\mathcal{F}=\\{m_{i,j}\\}$ be a frieze pattern. Then, for all $m_{i,j}$
with $|j-i|\geq 2$,

$m_{i,j}=m_{i,i+2}m_{i+1,j}-m_{i+2,j}$

and dually,

$m_{i,j}=m_{j-2,j}m_{i,j-1}-m_{i,j-2}.$

Thus, we can study one-periodic frieze patterns by studying evaluations of
Chebyshev polynomials. Lemma 3 easily follows from knowledge of roots of
Chebyshev polynomials.

######  Lemma 3.

Evaluations of Chebyshev polynomials at $\lambda_{p}$ are periodic, in the
sense that $U_{k+p}(\lambda_{p})=-U_{k}(\lambda_{p})$. In particular,
$U_{p-1}(\lambda_{p})=-U_{-1}(\lambda_{p})=0$.

From Lemma 3, we see for every $n\geq 0$, there is exactly one 1-periodic
frieze pattern of width $n$, and it is given by quiddity cycle
$(\lambda_{n+3})$. Moreover, this provides us with a necessary criterion for
infinite frieze patterns of Type $\Lambda_{p_{1},\ldots,p_{s}}$. We will see a
similar statement about unrealizable frieze patterns in Proposition 1.

######  Lemma 4.

If a sequence of numbers contains more than $p-2$ consecutive entries
$\lambda_{p}$, then it cannot be a quiddity sequence of an infinite frieze
pattern of positive entries.

###### Proof.

Suppose we have $m_{0,2}=m_{1,3}=\cdots=m_{p-2,p}=\lambda_{p}$. . Then, by
Lemma 2 and Lemma 3, $m_{0,p}=0$. Thus, either this is finite frieze pattern
of width $p-3$, from an empty dissection of a $p$-gon, or it is an infinite
frieze pattern with non-positive entries. ∎

##  4 Dissections of Once-Punctured Discs and Annuli

Our main goal is to study frieze patterns from dissections of once-punctured
discs and annuli. Accordingly, we must describe these surfaces and their
dissections more carefully.

We let $A_{n,m}$ denote an annulus with $n$ vertices on the outer boundary and
$m$ vertices on the inner boundary. We require $n,m\geq 1$. We let $S_{n}$
denote a once-punctured disc with $n\geq 1$ vertices on the boundary. For
convenience, we will refer to the boundary of $S_{n}$ as the outer boundary.
As a convention, we will let $v_{i}$ denote a vertex on the outer boundary of
either surface; if working with $A_{n,m}$, $w_{j}$ will denote a vertex on the
inner boundary.

There are two types of arcs in a dissection of $A_{n,m}$ or $S_{n}$.

######  Definition 7.

Let $\mathcal{D}$ be a dissection of the annulus $A_{n,m}$. An arc connecting
an outer vertex and an inner vertex is called _bridging arc_. If $\mathcal{D}$
is instead a dissection of $S_{n}$, an arc from the boundary to the puncture
will be called _bridging_. For either surface, an arc connecting two vertices
on the same boundary is called a _peripheral arc_.

Arcs in $S_{n}$ from the boundary to the puncture are instead called “central”
in [1].

As a convention, we will assume there are never peripheral arcs which are
incident to the inner boundary. Since our frieze patterns are determined by
the number of subgons incident to each vertex, we will consider dissections up
to Dehn twist.

We also describe several special types of subgons in a dissection. In general,
a dissection can have many subgons which are not one of these types.

######  Definition 8.

  * •

A subgon of size $p$ in a dissection with $p-1$ edges on the same boundary is
called an _ear_.

  * •

A subgon in a dissection which has at least one edge on the outer boundary and
at least one vertex on the inner boundary is called an _outer subgon_.

We will see in Section 5 that working with skeletal dissections is much
simpler than working with general dissections. Baur, Çanakçi, Jacobsen,
Kulkarni, and Todorov introduced the notion of a skeletal triangulation in
[3].

######  Definition 9.

A dissection of $A_{n,m}$ or $S_{n}$ is said to be _skeletal_ if it contains
only bridging arcs.

Importantly, a skeletal dissection will not contain any ears.

###  4.1 Infinite Strips

Our matching formulas in Section 6 for frieze patterns from dissections of
$A_{n,m}$ or $S_{n}$ will require looking at the universal cover of these
surfaces.

We describe first the universal cover of a dissected annulus. Let
$\mathcal{D}$ be a dissection of $A_{n,m}$. Every dissection of $A_{n,m}$ must
contain at least one bridging arc since the dissection divides the surface
into polygons. Pick a bridging arc and call it $e$. Call the vertex of $e$ on
the outer boundary $v_{1}$ and the vertex on the inner boundary $w_{1}$. Label
the remainder of the vertices in counterclockwise order: $v_{2},\ldots,v_{n}$
on the outer boundary and $w_{2},\ldots,w_{m}$ on the inner boundary. Cut the
annulus along $e$, producing a dissected $(n+m+2)$-gon, with two copies of the
edge $e$ and two copies of the vertices $v_{1},w_{1}$. Call this dissected
polygon $F$.

In this $(n+m+2)$-gon, one set of vertices $v_{1},w_{1}$ are neighbors to
$v_{2}$ and $w_{2}$ and the other are neighbors to $v_{n}$ and $w_{m}$. Call
the vertices $v_{1},w_{1}$ which are next to $v_{2},w_{2}$ $v_{1}^{0}$ and
$w_{1}^{0}$ respectively. For all $2\leq i\leq n$ and $2\leq j\leq m$ relabel
$v_{i}$ and $w_{j}$ as $v_{i}^{0}$ and $w_{j}^{0}$. Finally, relabel the
vertices $v_{1},w_{1}$ which are neighbors to $v_{n},w_{m}$ as $v_{1}^{1}$ and
$w_{1}^{1}$.

$v_{1}$$v_{2}$$v_{3}$$w_{1}$$w_{2}$$w_{3}$$e$ | $e$$e$$v_{1}^{0}$$v_{2}^{0}$$v_{3}^{0}$$v_{1}^{1}$$w_{1}^{0}$$w_{2}^{0}$$w_{3}^{0}$$w_{1}^{1}$$\rightarrow$
---|---

We draw $F$ as a rectangle, with the boundary edges labeled $e$ vertical and
all other edges horizontal so that all vertices $v_{i}^{0}$ and $v_{1}^{1}$
are on the bottom and all vertices $w_{j}^{0}$ and $w_{1}^{1}$ are on the top.
We create a periodic dissection of the infinite strip by translating $F$
infinitely many times horizontally to both the right and the left. We glue
these copies of $F$ along the edge $e$. In the copy of $F$ which is $k$ shifts
to the right, we label the vertices on the bottom
$v_{1}^{k},\ldots,v_{n}^{k},v_{1}^{k+1}$ and similarly on the top. We label
the vertices in the copy of $F$ $k$ shifts to the left with $-k$.

There is a covering map $\rho$ from this dissected infinite strip to
$\mathcal{D}$; for all $k\in\mathbb{Z}$, $\rho(v_{i}^{k})=v_{i}$ and
$\rho(w_{j}^{k})=w_{j}$. This will result in a map on edges: if $g_{k}$ is an
edge from $v_{i}^{k}$ to $w_{j}^{k^{\prime}}$ (note $k^{\prime}$ could be $k$
or $k+1$), then $\rho(g_{k})=g$. We also extend this map to the subgons in the
dissection.

$\alpha$$\beta$$\gamma$$\delta$ | $v_{3}^{-1}$$v_{1}^{0}$$v_{2}^{0}$$v_{3}^{0}$$v_{1}^{1}$$w_{1}^{0}$$w_{2}^{0}$$w_{3}^{0}$$w_{1}^{1}$$w_{2}^{-1}$$w_{2}^{-1}$$\alpha_{0}$$\beta_{0}$$\delta_{0}$$\delta_{1}$$\gamma_{0}$$\gamma_{-1}$$\cdots$$\cdots$$\rightarrow$
---|---

Next, suppose that $\mathcal{D}$ is a dissection of $S_{n}$. Label the
vertices $v_{1},\ldots,v_{n}$ in counterclockwise order. Draw an infinite
strip with vertices $v_{i}^{k}$ for all $1\leq i\leq n$ and all
$k\in\mathbb{Z}$ on the lower boundary, in order
$\ldots,v_{1}^{i},\ldots,v_{n}^{i},v_{1}^{i+1},\ldots$ and no vertices on the
upper boundary.

Following Lemma 3.6 in [1], from $\mathcal{D}$ we will construct a dissection
of the infinite strip using _asymptotic arcs_. On the infinite strip, an
asymptotic arc has one endpoint at a marked point then goes infinitely far to
the right or left. Our asymptotic arcs will travel to the right, and we
consider that they meet at $+\infty$.

For each $i$ such that there is an arc from $v_{i}$ to the puncture in
$\mathcal{D}$ and for all $k\in\mathbb{Z}$, draw an asymptotic arc with one
endpoint at $v_{i}^{k}$ which travels infinitely far to the right. Note that
we can draw all of these asymptotic arcs so that they do not cross.

In our dissection of $S_{n}$, suppose $v_{i}$ and $v_{j}$ are such that each
are connected to the puncture by an edge in $\mathcal{D}$ and, sweeping
counterclockwise from $v_{i}$ to $v_{j}$, there are no other arcs to the
puncture. Then, these two arcs to the puncture cut out a dissected polygon
from $S_{n}$. Let $k^{\prime}=k$ if $j>i$ and $k^{\prime}=k+1$ if $j\leq i$.
In the infinite strip, for all $k\in\mathbb{Z}$ we have a polygon of the same
size between $v_{i}^{k}$ and $v_{j}^{k^{\prime}}$, using $+\infty$ as one of
the endpoints. We dissect each of these polygons in the infinite strip so that
they have the same dissection as the polygon in $S_{n}$. That is, if there is
a peripheral arc between $v_{a}$ and $v_{b}$, where $v_{a}$ and $v_{b}$ are
weakly between $v_{i}$ and $v_{j}$ traveling counterclockwise, we connect
$v_{a}^{k_{1}}$ and $v_{b}^{k_{2}}$ between $v_{i}^{k}$ and
$v_{j}^{k^{\prime}}$ where $k_{1}$ and $k_{2}$ are $k$ or $k+1$. If there are
at least two bridging arcs in $\mathcal{D}$, we do this between each
consecutive pair of bridging arcs. If there is exactly one bridging arc, we do
this process once using $v_{i}=v_{j}$.

We define a map $\rho$ from this asymptotic dissection of the infinite strip
to $\mathcal{D}$ such that $\rho(v_{i}^{k})=v_{i}$. If, for some $i$, each
$v_{i}^{k}$ is incident to an asymptotic arc, $e_{k}$, then $\rho(e_{k})=e$
where $e$ is an arc from $v_{i}$ to the puncture. Let $g$ be an arc between
$v_{a}^{k}$ and $v_{b}^{k^{\prime}}$, where either $a<b$ and $k^{\prime}=k$ or
$a\geq b$ and $k^{\prime}=k+1$. In particular, $v_{a}^{k}$ appears to the left
of $v_{b}^{k^{\prime}}$ with our convention of drawing the infinite strip.
Then, $\rho(g)$ is the arc between $v_{a}$ and $v_{b}$ which follows the
boundary in the counterclockwise direction from $v_{a}$ to $v_{b}$. The image
of $\rho$ on vertices and edges will extend to a map on subgons. A subgon in
the infinite strip with a vertex at $+\infty$ will be mapped to a subgon in
$S_{n}$ with a vertex at the puncture.

$v_{1}$$v_{2}$$v_{3}$$\alpha$$\beta$ | $v_{3}^{-1}$$v_{1}^{0}$$v_{2}^{0}$$v_{3}^{0}$$v_{1}^{1}$$\alpha_{0}$$\beta_{-1}$$\beta_{0}$$\alpha_{1}$$\cdots$$\cdots$$\rightarrow$
---|---

##  5 Frieze Patterns from dissected Once-Punctured Discs and Annuli

For the remainder of this paper, we will study the relationship between
periodic infinite frieze patterns and dissections of annuli and once-punctured
discs. From here on, all infinite frieze patterns will be assumed to be
periodic. One could study non-periodic infinite frieze patterns of Type
$\Lambda_{p_{1},\ldots,p_{s}}$ by studying non-periodic dissections of the
infinite strip.

We use Holm and Jørgensen’s algorithm to form a quiddity cycle, and thus a
frieze pattern, from a dissection of one of these surfaces. To be explicit,
given a dissection $\mathcal{D}$ of $S_{n}$ or $A_{n,m}$ with vertices
$v_{1},\ldots,v_{n}$ on the outer boundary, let Poly($v_{i}$) be the set of
subgons incident to $v_{i}$. As in the triangulation case, we will determine
this set by drawing a small half circle around each vertex and looking at the
subgons this half circle intersects. This can result in counting a subgon
twice, as in the instance of a self-folded triangle, or more generally, a
self-folded subgon.

For each subgon $\mathcal{P}$ in $\mathcal{D}$, we will use $|\mathcal{P}|$
denote the number of edges of this subgon. We set
$A_{i}=\\{|\mathcal{P}|:\mathcal{P}\in\text{Poly($v_{i}$)}\\}$, as was defined
for the polygon case in Section 2.3. Then, we form a quiddity cycle
$(m_{0,2},\ldots,m_{n-1,n+1})$ by setting $m_{i-1,i+1}=\sum_{p\in
A_{i}}\lambda_{p}$.

######  Example 1.

For example, in the dissection of $A_{3,3}$ below, $A_{1}=\\{3,3,4\\}$,
$A_{2}=\\{3\\}$, and $A_{3}=\\{3,3,4,4\\}$.

$v_{1}$$v_{2}$$v_{3}$

We give the first few rows of the frieze pattern from this dissection. This is
the frieze pattern with quiddity cycle $(2+\sqrt{2},1,2+2\sqrt{2})$.

$\begin{array}[]{cccccccccccccccccccc}0&&0&&0&&0&&0&&\\\ &1&&1&&1&&1&\\\
1&&2+\sqrt{2}&&2+2\sqrt{2}&&1&&2+\sqrt{2}&&\\\
&1+\sqrt{2}&&7+6\sqrt{2}&&1+2\sqrt{2}&&1+\sqrt{2}&\\\
4+3\sqrt{2}&&5+4\sqrt{2}&&5+5\sqrt{2}&&4+3\sqrt{2}&&5+4\sqrt{2}&&\\\
&21+7\sqrt{2}&&4+3\sqrt{2}&&13+9\sqrt{2}&&21+7\sqrt{2}&\\\ &&&&\vdots&&&&&\\\
\end{array}$

Our goal in this section is determining when an infinite frieze pattern of
Type $\Lambda_{p_{1},\ldots,p_{s}}$ comes from a dissection of an annulus or
once-punctured disc.

######  Definition 10.

An infinite frieze pattern of period $n$ is called realizable if there exists
a dissection $\mathcal{D}$ on an annulus $A_{n,m}$ or a once-punctured disc
$S_{n}$ such that each entry $m_{i-1,i+1}$ in the quiddity sequence is given
by

$m_{i-1,i+1}=\sum_{p\in A_{i}}\lambda_{p}$

where $A_{i}$ is the multiset of sizes of subgons incident to $v_{i}$. If a
frieze pattern is not realizable, we call it _unrealizable_.

Similarly, we say a quiddity cycle $(m_{0,2},\ldots,m_{n-1,n+1})$ is
_realizable_ if the frieze pattern it generates is realizable, and otherwise
we call it _unrealizable_.

######  Remark 2.

We will also call any frieze pattern with a negative entry or a zero outside
the boundary row(s) of zeroes unrealizable. See Section 9 for more discussion
on when the entries of frieze pattern may not be positive.

We begin by determining some frieze patterns which are unrealizable.

######  Proposition 1 (Realizability test).

Let $\\{m_{i-1,i+1}\\}$ be such that $m_{i-1,i+1}=\sum_{p\in
A_{i}}\lambda_{p}$ for all $i$. Then, the frieze pattern generated by
$\\{m_{i-1,i+1}\\}$ is unrealizable if at least one of the following is true.

  1. 1.

There exists $i$ such that $A_{i}\cap A_{i+1}=\emptyset$.

  2. 2.

For some $p\in\mathbb{Z}_{\geq 3}$, there exists more than $p-2$ consecutive
indices $i$ such that $A_{i}=\\{p\\}$, and there also exists some $j$ such
that $A_{j}\neq\\{p\\}$.

###### Proof.

In a dissection, each pair of adjacent vertices are incident to a common
subgon. Therefore, a quiddity sequence from a dissection must have $A_{i}\cap
A_{i+1}\neq\emptyset$ for all $i$. Similarly, it is impossible for the
quiddity sequence from a realizable frieze pattern to have more than $p-2$
consecutive entries $\lambda_{p}$ unless the sequence comes from an empty
dissection of a $p$-gon. ∎

We say that a frieze pattern which does not satisfy either condition of
Proposition 1 “passes the realizability test”. Recall that by Lemma 4, if a
quiddity sequence satisfies 2 above, the frieze pattern it generates contains
negative entries.

###  5.1 Realizability of Skeletal Frieze Patterns

In order to describe which frieze patterns are realizable by dissections of
$A_{n,m}$ or $S_{n}$, it will be useful to first restrict to a certain class
of frieze patterns.

######  Definition 11.

If $\mathcal{F}$ is an infinite frieze pattern with quiddity sequence
$\\{m_{i-1,i+1}\\}$ such that the length any string of consecutive entries
$\lambda_{p}$ in the quiddity sequence is smaller than $p-2$, we call
$\mathcal{F}$ a _skeletal frieze pattern_.

Recall by Definition 9 that a dissection is skeletal if it contains no
peripheral arcs. The following statement follows from the fact that if a
dissection has any peripheral arcs, it will have ears.

######  Lemma 5.

A skeletal dissection will always produce a skeletal frieze pattern.

We can directly characterize realizability of skeletal frieze patterns.

######  Proposition 2 (Skeletal Realizability Criteria).

Let $\mathcal{F}=\\{m_{i,j}\\}$ be a skeletal frieze pattern of Type
$\Lambda_{p_{1},\ldots,p_{s}}$ with quiddity cycle
$(m_{0,2},\ldots,m_{n-1,n+1})$. Let $A_{i}$ be the multiset of integers, all
at least 3, such that $m_{i-1,i+1}=\sum_{p\in A_{i}}\lambda_{p}$. Then,
$\mathcal{F}$ is realizable by a dissection of $A_{n,m}$ or $S_{n}$ if and
only if

  1. i)

the sequence $\\{m_{i-1,i+1}\\}$ does not satisfy either condition of
Proposition 1, and

  2. ii)

we can choose $p_{i,i+1}\in A_{i}\cap A_{i+1}$ such that, if $|A_{i}|>1$ and
$p_{i-1,i}=p_{i,i+1}$, then $\\{p_{i-1,i},p_{i,i+1}\\}\subseteq A_{i}$ as
multisets.

###### Proof.

We will first prove that if $\mathcal{F}$ is a skeletal frieze pattern
realizable by dissection $\mathcal{D}$, then the quiddity cycle
$(m_{0,2},\ldots,m_{n-1,n+1})$ of $\mathcal{F}$ satisfies i and ii.

Since $\mathcal{F}$ is skeletal, the quiddity sequence will not satisfy part 2
of Proposition 1. In any dissection, each pair of consecutive vertices
$v_{i},v_{i+1}$ will be vertices of a common subgon, $\mathcal{P}_{i,i+1}$.
Hence, $|\mathcal{P}_{i,i+1}|\in A_{i}\cap A_{i+1}$, and the quiddity cycle
does not satisfy 1 in Proposition 1. Moreover, if
$|\mathcal{P}_{i-1,i}|=|\mathcal{P}_{i,i+1}|$ and $|A_{i}|>2$, then
$|\mathcal{P}_{i-1,i}|$ will appear with multiplicity at least 2 in $A_{i}$
since $v_{i}$ is adjacent to at least two distinct polygons of size
$|\mathcal{P}_{i-1,i}|$. This shows item ii.

For the other direction, given a skeletal frieze pattern $\mathcal{F}$
generated by a quiddity cycle $(m_{0,2},\ldots,m_{n-1,n+1})$ which satisfies
the conditions in the statement of the Proposition, we will construct a
dissection which is the realization of $\mathcal{F}$. We start by determining
the shape of the outer subgons.

Choose a sequence $p_{i,i+1}$, for $1\leq i\leq n$ which satisfies condition
ii. For some $i$, there may be multiple choices of $p_{i,i+1}$; choose one
arbitrarily. Let $\\{i_{1},\ldots,i_{\ell}\\}\subseteq\\{1,\ldots,n\\}$ be the
set of indicies such that $|A_{i_{j}}|>1$, with the ordering
$i_{1}<i_{2}<\cdots<i_{\ell-1}<i_{\ell}$. Since for any indices
$i_{j}<s<i_{j+1}$, $|A_{s}|=1$, we have that
$p_{i_{j},i_{j}+1}=p_{i_{j}+1,i_{j}+2}=\cdots=p_{i_{j+1}-1,i_{j+1}}$. We
define this number to be $p_{i_{j},i_{j+1}}$. Since we work modulo $n$, we
similarly define $p_{i_{\ell},i_{1}}.$

First, if all $i_{j}$ are such that $|A_{i_{j}}|=2$ and for all consecutive
pairs $(i_{j},i_{j+1})$, modulo $n$, $p_{i_{j},i_{j+1}}-2=i_{j+1}-i_{j}$, then
this frieze pattern will correspond to a dissection of $S_{n}$. We label the
vertices of $S_{n}$ $v_{1},\ldots,v_{n}$ in counterclockwise order, and we
draw an arc from each $v_{i_{j}}$ to the puncture. This dissection of $S_{n}$
will correspond to the original frieze pattern $\mathcal{F}$.

Suppose then that there exists $i_{j}$ such that $|A_{i_{j}}|>2$ or a pair
$(i_{j},i_{j+1})$ such that $p_{i_{j},i_{j+1}}-2>i_{j+1}-i_{j}$. Then,
$\mathcal{F}$ will correspond to a dissection of $A_{n,m}$, where $m$ will be
determined in the following construction. We label the vertices on the outer
boundary $v_{1},\ldots,v_{n}$ in counterclockwise order. For each $i_{j}$, we
draw two vertices on the inner boundary, $w_{i_{j}}^{-}$ and $w_{i_{j}}^{+}$
where, in counterclockwise order $w_{i_{j}}^{+}$ follows $w_{i_{j}}^{-}$ and
$w_{i_{j+1}}^{\pm}$ both follow $w_{i_{j}}^{\pm}$. If $|A_{i_{j}}|=2$, we
identify $w_{i_{j}}^{-}$ and $w_{i_{j}}^{+}$. If
$p_{i_{j},i_{j+1}}-2=i_{j+1}-i_{j}$, then we identify $w_{i_{j}}^{+}$ and
$w_{i_{j+1}}^{-}$. If $p_{i_{j},i_{j+1}}-2>i_{j+1}-i_{j}$, we include
$p_{i_{j},i_{j+1}}-3-(i_{j+1}-i_{j})$ vertices on the inner boundary between
$w_{i_{j}}^{+}$ and $w_{i_{j+1}}^{-}$. Once we have made these identifications
and drawn these vertices, we draw arcs from each $v_{i_{j}}$ to
$w_{i_{j}}^{-}$ and $w_{i_{j}}^{+}$. If $|A_{i_{j}}|=2$, then since
$w_{i_{j}}^{-}$ and $w_{i_{j}}^{+}$ are identified we only draw one arc from
$v_{i_{j}}$.

At this point, we have established all outer subgons; we add the remaining
subgons of the dissection. We do not need to add anything to vertices
$v_{i_{j}}$ such that $|A_{i_{j}}|\leq 3$. Let $v_{i_{j}}$ be a vertex such
that $|A_{i_{j}}|>3$. We draw $|A_{i_{j}}|-3$ additional arcs from $v_{i_{j}}$
to distinct vertices on the inner boundary, between $w_{i_{j}}^{-}$ and
$w_{i_{j}}^{+}$. We add additional vertices between the vertices on the inner
boundary connected to $v_{i_{j}}$ so that $A_{i_{j}}$ records the sizes of
polygons incident to $v_{i_{j}}$. ∎

We used the following fact in the proof of Proposition 2. We record it for use
in the Realizability Algorithm.

######  Lemma 6.

A skeletal frieze pattern $\mathcal{F}$ with quiddity cycle
$(m_{0,2},\ldots,m_{n-1,n+1})$ which satisfies Proposition 2 is realizable by
a dissection of $S_{n}$ if and only if all $A_{i}$ are such that
$|A_{i}|\in\\{1,2\\}$, and for each $p\in\mathbb{Z}$, any string of entries,
$m_{i-1,i+1}=\cdots=m_{i+\ell-1,i+\ell+1}=\lambda_{p}$ in the quiddity cycle
has length $p-3$.

As an example of Proposition 2, consider the quiddity cycle
$(2+\sqrt{2},\sqrt{2}+\sqrt{3},\sqrt{3},1+\sqrt{3})$. This quiddity cycle
satisfies all conditions of the Proposition. Here,
$A_{1}=\\{3,3,4\\},A_{2}=\\{4,6\\},A_{3}=\\{6\\},$ and $A_{4}=\\{3,4\\}$. Our
subsequence of indices where $A_{i}$ is larger than a singleton is $1,2,4$. We
only have one choice for the sequence of $p_{i,i+1}$:
$p_{1,2}=4,p_{2,3}=p_{3,4}=6,p_{4,1}=3$. Thus, we set $p_{2,4}=6$. Since we
have $|A_{1}|>2$, $p_{1,2}-2>2-1$, and $p_{2,4}-2>4-2$, this frieze pattern
will come from a dissected annulus. We place vertices
$v_{1},v_{2},v_{3},v_{4}$ in counterclockwise order on the outer boundary.

We start with adding 6 vertices, $w_{1,2}^{\pm},w_{2,4}^{\pm},w_{4,1}^{\pm}$
to the inner boundary, with counterclockwise order given by
$w_{1,2}^{-},w_{1,2}^{+},w_{2,4}^{-},\ldots$. Since $p_{4,1}-2=5-4$, we
identify $w_{1,2}^{-}$ and $w_{4,1}^{+}$. Since $|A_{2}|=2$, we identify
$w_{2,4}^{-}$ and $w_{2,4}^{+}$; similarly, we identify $w_{4,1}^{-}$ and
$w_{4,1}^{+}$. Finally, we add $p_{2,4}-3-(4-2)=1$ vertices on the inner
boundary between $w_{2,4}^{+}$ and $w_{4,1}^{-}$. Since $p_{1,2}-3-(2-1)=0$,
we do not add any vertice between $w_{1,2}^{+}$ and $w_{2,4}^{-}$. Now, we
connect each $v_{i_{j}}$ to $w_{i_{j}}^{\pm}$, with these identifications in
place.

At this point, we have established all outer subgons of the dissection. From
here, we check whether we need additional arcs from any $v_{i_{j}}$ so that
each $A_{i}$ correctly records all incident subgons at $v_{i}$. However, in
this case we do not need to add any additional arcs or vertices on the inner
boundary. We draw this dissection below.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$w_{1,2}^{+}$$w_{2,4}^{-}=w_{2,4}^{+}$$w_{1,2}^{-}=w_{4,1}^{+}=w_{4,1}^{-}$$1$$1$$\sqrt{2}$$\sqrt{3}$

Given a frieze pattern which satisfies the conditions of Proposition 2, the
corresponding dissection is not unique. First, since the quiddity sequence is
$n$-periodic for some $n$, one could take a longer quiddity cycle
$(m_{0,2},\ldots,m_{kn-1,kn+1})$ for any $k\in\mathbb{Z}_{>0}$. We call the
resulting dissection the _kth power_ of the dissection from
$(m_{0,2},\ldots,m_{n-1,n+1})$, as is described in Definition 21.

There is a more substantial way that these dissections can be non-unique,
which only appears in some dissections with mixed sizes of subgons. Given a
quiddity cycle $(m_{0,2},\ldots,m_{n-1,n+1})$ there might be several distinct
sequences $p_{1,2},p_{2,3},...,p_{n,n+1}$ that satisfy the conditions of
Proposition 2. For instance, see the two dissections with the same quiddity
cycle in Example 2.

######  Example 2.

We show two distinct dissections of annuli, each of which corresponds to the
quiddity cycle $(1+2\sqrt{2},2+2\sqrt{2})$.

$1+2\sqrt{2}$$2+2\sqrt{2}$ | $1+2\sqrt{2}$$2+2\sqrt{2}$
---|---

###  5.2 Quotient Dissections

There are skeletal frieze patterns that pass the realizability test but do not
satisfy the requirements in Proposition 2. One example is the frieze pattern
generated by the quiddity cycle $(1+\phi,1+\sqrt{2},1+\phi,\sqrt{2}+\phi)$;
recall $\lambda_{3}=1,\lambda_{4}=\sqrt{2},$ and $\lambda_{5}=\phi$. We see
that $A_{1}=\\{3,5\\},A_{2}=\\{3,4\\}$, and $A_{3}=\\{3,4\\}$. Thus,
$A_{1}\cap A_{2}=A_{2}\cap A_{3}=\\{3\\}$, but 3 only appears with
multiplicity 1 in $A_{2}$. We introduce another type of geometric construction
for such frieze patterns.

######  Definition 12.

Given $\mathcal{D}$, a dissection of $A_{n,m}$, we can form a _quotient
dissection_ , $\overline{\mathcal{D}}$, by iteratively identifying pairs of
subgons of the same size which share a vertex on the outer boundary. For our
purposes, we forbid identifying two subgons which share an edge or identifying
subgons which have all vertices on outer boundary. We may end up identifying
many subgons of the same size as one depending on which pairs we identify.

Given a quotient dissection of $A_{n,m}$, we construct a quiddity cycle in the
same way as for a ordinary dissection. That is, for all vertices $v_{i}$, let
$A_{i}$ be the (multi)set of sizes of subgons incident to $v_{i}$, using the
identifications. Then, we form a quiddity sequence
$(m_{0,2},\ldots,m_{n-1,n+1})$ by setting $m_{i-1,i+1}=\sum_{p\in
A_{i}}\lambda_{p}$.

######  Example 3.

Below, we identify the 4-gons labeled $a_{1}$ and $a_{2}$. This changes the
quiddity cycle from $(2+2\sqrt{2},3\sqrt{2})$ to $(2+\sqrt{2},2\sqrt{2}$)

$2+2\sqrt{2}$$3\sqrt{2}$$a_{1}$$a_{2}$ | $\rightarrow$ | $2+\sqrt{2}$$2\sqrt{2}$$a$$a$
---|---|---

######  Remark 3.

  1. 1.

We do not allow identifying subgons which share an edge to avoid constructing
quotient dissections which realize quiddity cycles that do not pass the
realizability test. In particular, we could realize a quiddity cycle with more
than $p-2$ consecutive entries $\lambda_{p}$ by gluing many $p$-gons together
along shared edges. See Figure 5.

  2. 2.

We also do not allow identifying subgons with all vertices on one boundary to
avoid constructing realizations of unrealizable frieze patterns. As an
example, see Figure 6 where we identify two ears to produce the quiddity cycle
$(2\sqrt{2},\sqrt{2},1,1+2\sqrt{2})$. This quiddity cycle is unrealizable,
which is verified by the Realizability Algorithm in Section 5.4.

  3. 3.

Due to these restrictions, we will never form quotient dissections starting
with a dissection of $S_{n}$. In a once-punctured disc, given two subgons
which share a vertex in $S_{n}$ either these subgons share an edge or at least
one subgon has all vertices on the outer boundary.

$2\sqrt{2}$$\sqrt{2}$$\sqrt{2}$$2\sqrt{2}$$\sqrt{2}$ Figure 5: If we identify
three consecutive pairs of 4-gons, which pairwise share edges, the resulting
quiddity cycle does not pass the realizability test. We forbid quotient
dissections which glue two subgons that share an edge.
$1+2\sqrt{2}$$1+2\sqrt{2}$$1+\sqrt{2}$$1$$1$ Figure 6: A dissection where we
identify two ears. The resulting quiddity sequence is a $3$-gluing to the
unrealizable sequence $(2\sqrt{2},\sqrt{2},1,1+2\sqrt{2})$. We forbid such
dissections.

######  Definition 13.

We say that a frieze pattern $\mathcal{F}=\\{m_{i,j}\\}$ has a _quotient
dissection realization_ if there exists some quotient dissection
$\overline{\mathcal{D}}$ on $A_{n,m}$ such that, if $A_{i}$ is the (multi)set
of sizes of subgons incident to $v_{i}$, for all $m_{i-1,i+1}$ we have

$m_{i-1,i+1}=\sum_{p\in A_{i}}\lambda_{p}.$

For example, in Example 3 we see that the frieze pattern generated by
$(2+\sqrt{2},2\sqrt{2})$ has a quotient dissection realization. There are
skeletal frieze patterns which satisfy Proposition 2 and also can be realized
by a quotient dissection. We will focus on frieze patterns which have a
quotient dissection realization and are not realizable by an ordinary
dissection.

We now show that quotient dissections are the missing piece in the sense that
the realizability test (Proposition 1) is sufficient to determine if a
skeletal frieze pattern is realizable by a dissection or quotient dissection.

######  Theorem 5.

Every skeletal frieze pattern which passes the realizability test (Proposition
1) is realizable by a dissection of a once-punctured disc or annulus, or a
quotient dissection of an annulus.

###### Proof.

Let $\mathcal{F}=\\{m_{i,j}\\}$ be a frieze pattern which does not satisfy
either condition of Proposition 1. If moreover, $\mathcal{F}$ satisfies
Proposition 2, then $\mathcal{F}$ is realized by a dissection of $S_{n}$ or
$A_{n,m}$.

Now, suppose that $\mathcal{F}$ does not satisfy item ii in Proposition 2.
Then, it is not possible to pick a sequence $p_{1,2},p_{2,3},\ldots,p_{1,n}$
such that $p_{i-1,i}\in A_{i}$, $p_{i,i+1}\in A_{i}$, and if
$p_{i-1,i}=p_{i,i+1}$, then this number appears with at least multiplicity two
in $A_{i}$.

However, since $\mathcal{F}$ passes the realizability test, we know that for
all $i$, $A_{i}\cap A_{i+1}\neq\emptyset$. Therefore, it is possible to pick a
sequence $p_{i,i+1}\in A_{i}\cap A_{i+1}$. Let $j_{1},\ldots,j_{\ell}$ be the
subsequence of $[n]$ where $p_{j_{i}}:=p_{j_{i}-1,j_{i}}=p_{j_{i},j_{i}+1}$,
but this number appears only with multiplicity 1 in $A_{j_{i}}$. We know that
for all choices of $p_{i,i+1}$ this set of indices must be nonempty;
otherwise, this frieze pattern would be realizable by a dissection of
$A_{n,m}$ or $S_{n}$ already.

Let $(m_{0,2},\ldots,m_{n-1,n+1})$ be the quiddity cycle for $\mathcal{F}$. We
introduce a modified quiddity cycle,
$(\widehat{m}_{0,2},\ldots,\widehat{m}_{n-1,n+1})$. If there exists either at
least two indices $i$ or no indices $i$ such that $|A_{i}|>1$ and
$i\notin\\{j_{1},\ldots,j_{\ell}\\}$, define $\widehat{m}_{i-1,i+1}$ for
$1\leq i\leq n$ as

$\widehat{m}_{i-1,i+1}=\begin{cases}m_{i-1,i+1}&i\notin\\{j_{1},\ldots,j_{\ell}\\}\\\
m_{i-1,i+1}+\lambda_{p_{j_{k}}}&i=j_{k}\\\ \end{cases}$

If there exists exactly one index $i$ such that $|A_{i}|>1$ and $i\neq j_{k}$,
then it must be that $p_{j_{1}}=p_{j_{2}}=\cdots=p_{j_{\ell}}$. In this case,
we set $\widehat{m}_{i-1,i+1}=m_{i-1,i+1}+\lambda_{p_{1}}$ for all $1\leq
i\leq n$

By construction, $\widehat{F}$, the frieze pattern generated by the quiddity
cycle $(\widehat{m}_{0,2},\ldots,\widehat{m}_{n-1,n+1})$ is realizable. We
claim that the corresponding dissection, $\widehat{D}$, is on an annulus
$A_{n,m}$ for some $m>0$, and not on $S_{n}$.

First, note that it would be impossible for $\widehat{F}$ to be non-skeletal.
A frieze pattern is non-skeletal if its quiddity cycle has at least one string
of $p-2$ entries $\lambda_{p}$. We produce $\widehat{F}$ by beginning with a
skeletal frieze pattern $\mathcal{F}$ which passes the realizability test,
then we add values $\lambda_{p}$ to several entries of the quiddity cycle;
thus, we will not produce longer strings of entries $\lambda_{p}$.

Now, note that the quiddity cycle from a skeletal dissection of $S_{n}$ will
be of the form

$(\ldots,\lambda_{p_{1}},\ldots,\lambda_{p_{1}},\lambda_{p_{1}}+\lambda_{p_{2}},\lambda_{p_{2}},\ldots,\lambda_{p_{2}},\lambda_{p_{2}}+\lambda_{p_{3}},\ldots)$
(1)

with a string of $p_{i}-3$ entries $\lambda_{p_{i}}$ between each
$\lambda_{p_{i}}+\lambda_{p_{i+1}}$. The entries $\lambda_{p_{i}}$ could not
have been the result of adding $\lambda_{p_{i}}$ to the entry in the original
quiddity cycle, or else the original entry would be zero. Similarly, if one of
the entries $\lambda_{p_{i}}+\lambda_{p_{i+1}}$ resulted from adding either
$\lambda_{p_{i}}$ or $\lambda_{p_{i+1}}$ to the original quiddity sequence,
then the original frieze pattern would have failed the realizability test;
that is, it would have satisifed one of the conditions of Proposition 1. If
$\lambda_{p_{i}}=\lambda_{p_{i+1}}$, then the original quiddity cycle would
have had a string of over $p-2$ entries $\lambda_{p}$. If
$\lambda_{p_{i}}\neq\lambda_{p_{i=1}}$, then the original quiddity cycle would
have had adjacent entries $m_{j-1,j+1},m_{j,j+2}$ such that $A_{j}\cap
A_{j+1}=\emptyset$.

Thus, $\widehat{\mathcal{D}}$ is a dissection of $A_{n,m}$, for some $m\geq
1$. At each vertex $j_{k}$, $v_{j_{k}}$ is incident to two distinct outer
$p_{j_{k}}$-subgons, one of which is shared with $v_{j_{k}}-1$ and the other
with $v_{j_{k}}+1$. At each $j_{k}$, we identify these two subgons, forming a
quotient dissection $\overline{\mathcal{D}}$. By construction, for all $i$,
$m_{i-1,i+1}$ is the weighted sum of subgons incident to $v_{i}$ in
$\overline{\mathcal{D}}$. ∎

We end this section by describing the universal cover of quotient dissections.
As in the case of ordinary dissections, we will use the universal cover in our
matching formulas in Section 6.

######  Definition 14.

Let $\overline{\mathcal{D}}$ be a quotient dissection of $A_{n,m}$, and let
$\mathcal{D}$ be the dissection of $A_{n,m}$ obtained by considering all
identified pairs of subgons as distinct. Let $I$ be the dissected infinite
strip, which is the universal cover of $\mathcal{D}$.

Let $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ be two subgons in $\mathcal{D}$
which are both incident to $v_{i}$ and are identified in
$\overline{\mathcal{D}}$. In $I$, for all $k\in\mathbb{Z}$ there are subgons
$\mathcal{P}_{1}^{k}$ and $\mathcal{P}_{2}^{k}$ incident to $v_{i}^{k}$ such
that $\rho(\mathcal{P}_{j}^{k})=\mathcal{P}_{j}$ for $j=1,2$. We identify all
pairs $\mathcal{P}_{1}^{k}$ and $\mathcal{P}_{2}^{k}$, and repeat this process
for all other pairs of subgons identified in $\overline{\mathcal{D}}$. Call
this dissection of the infinite strip with identifications $\overline{I}$. We
call $\overline{I}$ the universal cover of $\overline{\mathcal{D}}$.

By construction, if $\mathcal{P}_{i}$ and $\mathcal{P}_{j}$ are two subgons of
$\overline{I}$ which are identified, then $\rho(\mathcal{P}_{i})$ and
$\rho(\mathcal{P}_{j})$ are identified in $\overline{\mathcal{D}}$.

###  5.3 Cutting

The conditions of Proposition 2 are not sufficient for non-skeletal frieze
patterns. For example, the quiddity cycle $(1+\sqrt{2},1,2,1+\sqrt{2})$
satisfies all conditions, but we claim it is not realizable. Since this frieze
pattern is not skeletal, we can _cut_ it in order to make it simpler. Baur,
Parsons, and Tschabold introduced the notion of cutting a frieze pattern when
proving one of their main results [1]. They were working with frieze patterns
of positive integers and triangulations of surfaces; we extend the notion of
cutting to general dissections.

######  Definition 15.

Let $\mathcal{F}$ be a frieze pattern with quiddity cycle
$(m_{0,2},\ldots,m_{n-1,n+1})$ where, for some $p\in\mathbb{Z}_{\geq 3}$,
there exists $1\leq i\leq n-(p-3)$ such that
$m_{i-1,i+1}=m_{i,i+2}=\cdots=m_{i+(p-3)-1,i+(p-3)+1}=\lambda_{p}$. We can
_cut_ at the interval $[i,i+p-3]$ and produce a new quiddity cycle
$(m^{\prime}_{0,2},\ldots,m^{\prime}_{n-(p-2)-1,n-(p-2)+1})$. If $n>p-1,$ then
we assume without loss of generality $1<i<n-(p-3)$, and define

$m^{\prime}_{j-1,j+1}=\begin{cases}m_{j-1,j+1}&j<i-1\\\
m_{i-1,i+1}-\lambda_{p}&j=i-1\\\ m_{i+(p-2)-1,i+(p-2)+1}-\lambda_{p}&j=i\\\
m_{j+(p-2)-1,j+(p-2)+1}&j>i\\\ \end{cases}$

If $n=p-1$, assume without loss of generality that $i=2$. Then, the new
quiddity cycle is 1-periodic, with $m_{0,2}^{\prime}=m_{0,2}-2\lambda_{p}$.

If $n=p-2$, then this is the 1-periodic frieze pattern with quiddity cycle
$(\lambda_{p})$, which corresponds to an empty dissection of a $p$-gon. Such a
frieze pattern is finite by results of Holm and Jørgensen [14]. We do not
define the cut of such a frieze pattern.

As an example, we look again at our example of the quiddity sequence
$(1+\sqrt{2},1,2,1+\sqrt{2})$. We can cut at the interval $[2,2]$, producing
$(\sqrt{2},1,1+\sqrt{2})$. The resulting quiddity cycle does not pass the
realizability test. We will see in Lemma 7 that this implies the original
quiddity cycle is also not realizable.

######  Remark 4.

  1. 1.

It may be necessary to cyclically shift a quiddity cycle to cut it using
Definition 15.

  2. 2.

If a quiddity cycle does not pass the realizability test, Proposition 1, then
the result of cutting could give negative entries in the adjusted cycle. For
example, if our quiddity cycle is $(1,\sqrt{2},\sqrt{2},\sqrt{3})$, then
cutting at interval $[2,3]$ results in $(1-\sqrt{2},\sqrt{3}-\sqrt{2})$.

We define $p$-_gluing_ , an inverse operation to cutting an interval of length
$p-2$. Our definition follows that of Holm and Jørgensen in [14]. Unlike in
the case of cutting, any frieze pattern admits a $p$-gluing for any
$p\in\mathbb{Z}_{\geq 3}$ and at any interval.

######  Definition 16.

Let $\mathcal{F}$ be a frieze pattern with quiddity cycle
$(m_{0,2},\ldots,m_{n-1,n+1})$, and let $p\in\mathbb{Z}_{\geq 3}$. For some
$1\leq i\leq n$, we $p$-_glue_ between $i$ and $i+1$, producing a new quiddity
cycle $(m^{\prime}_{0,2},\ldots,m^{\prime}_{n-1+(p-2),n+1+(p+2)})$ defined by

$m_{j-1,j+1}^{\prime}=\begin{cases}m_{j-1,j+1}&j<i\\\
m_{i-1,i+1}+\lambda_{p}&j=i\\\ \lambda_{p}&i<j<i+(p-1)\\\
m_{i,i+2}+\lambda_{p}&j=i+(p-1)\\\ m_{j-1-(p-1),j+1-(p-1)}&j>i+(p-1)\\\
\end{cases}$

For example, we can 6-glue between $1$ and $2$ on the quiddity cycle
$(2\sqrt{2},\sqrt{2})$, producing
$(2\sqrt{2}+\sqrt{3},\sqrt{3},\sqrt{3},\sqrt{3},\sqrt{3},\sqrt{2}+\sqrt{3})$.
For any $p\in\mathbb{Z}_{\geq 3}$, $p$-gluing between $i$ and $i+1$ and
cutting at the interval $[i,i+p-3]$ are inverse operations.

######  Lemma 7.

Suppose a frieze pattern $\mathcal{F}$ with quiddity cycle
$(m_{0,2},\ldots,m_{n-1,n+1})$ is realizable by a dissection of
$A_{n,m},S_{n}$, or an $n$-gon, $P_{n}$.

  1. 1.

If we can cut this quiddity cycle at $[i,i+p-3]$, then resulting frieze
pattern is realizable by a dissection of $A_{n-(p-2),m},S_{n-(p-2)}$, or
$P_{n-(p-2)}$.

  2. 2.

For any $p\in\mathbb{Z}_{\geq 3}$ and any $1\leq i\leq n$, the frieze pattern
resulting from $p$-gluing between $i$ and $i+1$ is realizable by a dissection
of $A_{n+(p-2),m},S_{n+(p-2)}$, or $P_{n+(p-2)}$.

###### Proof.

(1) Since we can cut $(m_{0,2},\ldots,m_{n-1,n+1})$ at $[i,i+p-3]$, it must be
that $m_{i-1,i+1}=\cdots=m_{i+(p-3)-1,i+(p-3)+1}=\lambda_{p}$; since
$\mathcal{F}$ is realizable, $m_{i-2,i}=\sum_{q\in A_{i-1}}\lambda_{q}$ and
$m_{i+p-3,i+p-1}=\sum_{q\in A_{i+p-2}}\lambda_{q}$ where $p\in A_{i-1}$, $p\in
A_{i+p-2}$, $|A_{i-1}|>1$, and $|A_{i+p-2}|>1$ . In the realization of
$\mathcal{F}$, vertices $v_{i},\ldots,v_{i+p-3}$ are all adjacent to an ear.
This ear’s unique non-boundary edge is between $v_{i-1}$ and $v_{i+p-2}$.
Performing the algebraic operation of cutting $(m_{0,2},\ldots,m_{n-1,n+1})$
at $[i,i+p-3]$ is equivalent to removing this ear from the dissection. If the
realization of $\mathcal{F}$ is a dissection of an $n$-gon, then removing this
ear produces an $(n-(p-2))$-gon and the resulting frieze pattern is width
$n-(p-2)-3=n-p-5$. If the realization of $\mathcal{F}$ is a dissection of
$S_{n}$ or $A_{n,m}$, then removing this ear produces $S_{n-(p-2)}$ or
$A_{n-(p-2),m}$ respectively.

Note that, if $m_{i-1,i+1}=\lambda_{p}$ for all $i$, then this is a finite
frieze pattern of width $p-3$ which corresponds to an empty dissection of a
$p$-gon. We do not define the cut of such a frieze pattern in Definition 15.

(2) A $p$-gluing between $i$ and $i+1$ corresponds to attaching an ear of size
$p$ to the realization of $\mathcal{F}$ such that the edge between vertices
$v_{i}$ and $v_{i+1}$ is the unique non-boundary edge of the ear. ∎

$\sqrt{2}$$\sqrt{2}$$2\sqrt{2}+\phi$$2\sqrt{2}+\phi$$\phi$ | $\xrightarrow{cut}$ | $\sqrt{2}+\phi$$\sqrt{2}+\phi$$\phi$
---|---|---
Figure 7: The algebraic operation of cutting a frieze pattern corresponds to
removing an ear. Gluing a frieze pattern corresponds to adding an ear.

When a frieze pattern $\mathcal{F}$ is only realizable by a quotient
dissection, cutting and gluing have the same geometric meaning.

######  Lemma 8.

Suppose a frieze pattern $\mathcal{F}$ with quiddity cycle
$(m_{0,2},\ldots,m_{n-1,n+1})$ is realizable by a quotient dissection
$\overline{\mathcal{D}}$ of $A_{n,m}$.

  1. 1.

If we can cut this quiddity cycle at $[i,i+p-3]$, then the resulting frieze
pattern is realizable by a quotient dissection of $A_{n-(p-2),m}$.

  2. 2.

For any $p\in\mathbb{Z}_{\geq 3}$ and any $1\leq i\leq n$, the frieze pattern
resulting from $p$-gluing between $i$ and $i+1$ is realizable by a quotient
dissection of $A_{n+(p-2),m}$.

###### Proof.

Recall we do not allow identifying subgons with all vertices on the same
boundary. Thus, if we can cut $\mathcal{F}$ at the interval $[i,i+p-3]$, then
in the quotient dissection, $\overline{\mathcal{D}}$, vertices
$v_{i},\ldots,v_{i+p-3}$ are incident to an ear which has not been identified
to any other subgon. Therefore, we can remove this ear and this produces a
valid quotient dissection of $A_{n-(p-2),m}$. Similarly, we can always add a
ear of size $p$ to a quotient dissection, which would produce a quotient
dissection of $A_{n+(p-2),m}$. ∎

###  5.4 Realization Algorithm

Since we have direct realizability criteria for skeletal frieze patterns in
Proposition 2, we can use cutting to characterize the realizability of
arbitrary frieze patterns of Type $\Lambda_{p_{1},\ldots,p_{s}}$.

Let $\mathcal{F}_{0}$ be any frieze pattern of Type
$\Lambda_{p_{1},\ldots,p_{s}}$ with a period $n$ quiddity cycle.If
$\mathcal{F}_{0}$ is finite, of width $n-3$, we must consider the quiddity row
of $\mathcal{F}_{0}$ as $n$-periodic. If $\mathcal{F}_{0}$ is infinite, we can
consider any valid period of its quiddity row. We run the following algorithm
to determine whether $\mathcal{F}_{0}$ is realizable; moreover, if
$\mathcal{F}_{0}$ is realizable, we will be able to determine which surface
can be dissected to realize $\mathcal{F}_{0}$ or whether $\mathcal{F}_{0}$ is
only realizable by a quotient dissection.

Let $i\geq 0$.

  1. 1.

If $\mathcal{F}_{i}$ satisfies at least one condition of Proposition 1 or has
any entries in the quiddity sequence which are less than or equal to 0, it is
unrealizable.

  2. 2.

If $\mathcal{F}_{i}$ is generated by the quiddity cycle $(\lambda_{p})$, then
$\mathcal{F}_{i}$ is realizable by an empty dissection of a $p$-gon.
$\mathcal{F}_{0}$ is realizable by a dissection of an $n$-gon.

  3. 3.

If $\mathcal{F}_{i}$ passes the realizability test, is not generated by the
quiddity cycle $(\lambda_{p})$, and is not skeletal, let $\mathcal{F}_{i+1}$
be the result of cutting $\mathcal{F}_{i}$ at one interval. Return to step 1.

  4. 4.

If $\mathcal{F}_{i}$ passes the realizability test and is skeletal, but does
not satisfy Proposition 2, then $\mathcal{F}_{i}$ is realizable by a quotient
dissection. $\mathcal{F}_{0}$ is realizable by a quotient dissection of
$A_{n,m}$.

  5. 5.

If $\mathcal{F}_{i}$ satisfies Proposition 2, implying that it passes the
realizability test and is skeletal, then $\mathcal{F}_{i}$ is realizable by a
dissection of an annulus or once-punctured disc. $\mathcal{F}_{0}$ is
realizable by a dissection of $A_{n,m}$ (for some $m$) or $S_{n}$.

    1. (a)

Lemma 6 provides one way to determine whether $\mathcal{F}_{i}$ is realizable
by $S_{n}$ or $A_{n,m}$

    2. (b)

Proposition 3 gives another method of checking: $\mathcal{F}_{i}$ is
realizable by $S_{n}$ if only if the _principal growth coefficient_ (see
Theorem 10) of $\mathcal{F}_{i}$ is 2.

When we end in cases 2,4, or 5, Lemma 7 and Lemma 8 give the guarantee that
$\mathcal{F}_{0}$ is also realizable by the same type of dissection as
$\mathcal{F}_{i}$. In each case, these realizations can be constructed by
geometrically gluing subgons onto the skeletal realization of
$\mathcal{F}_{i}$ in the reverse order of the cuts performed to get from
$\mathcal{F}_{0}$ to $\mathcal{F}_{i}$.

######  Example 4.

  1. 1.

Consider $\mathcal{F}_{0}$ with quiddity cycle
$(1+\sqrt{2}+\sqrt{3},\sqrt{3},\sqrt{3},\sqrt{3},1+\sqrt{3},1,1+\sqrt{2}+\sqrt{3},1+\sqrt{2})$.
This quiddity cycle passes the realizability test, but is not skeletal. We cut
at $[6,6]$ to produce $\mathcal{F}_{1}$ with quiddity cycle
$(1+\sqrt{2}+\sqrt{3},\sqrt{3},\sqrt{3},\sqrt{3},\sqrt{3},\sqrt{2}+\sqrt{3},1+\sqrt{2})$.
This new quiddity cycle still passes the realizability test and still is not
skeletal. We cut at $[2,5]$ to produce $\mathcal{F}_{2}$ with quiddity cycle
$(1+\sqrt{2},\sqrt{2},1+\sqrt{2})$. By Lemma 6, we see that $\mathcal{F}_{2}$
is realizable by a skeletal dissection of $S_{3}$. Thus, $\mathcal{F}_{0}$ is
realizable by a dissection of $S_{8}$.

  2. 2.

Consider $\mathcal{F}_{0}$ with quiddity cycle $(1+\sqrt{2},1,2,1+\sqrt{2})$.
We cut at $[2,2]$ to produce $(\sqrt{2},1,1+\sqrt{2})$. Since this quiddity
cycle does not pass the realizability test, $\mathcal{F}_{0}$ is not
realizable.

##  6 Matchings

###  6.1 Background

We now introduce combinatorial interpretations of the entries of frieze
patterns of Type $\Lambda_{p_{1},\ldots,p_{n}}$ from dissected surfaces. Our
interpretations will hold in both infinite and finite cases. Each entry in a
frieze pattern will correspond to to a sum of weighted _matchings_ on the
surface. We will in fact introduce two distinct weightings for matchings;
these two weightings shed light on different features of these frieze
patterns.

######  Definition 17.

Let $S$ be a dissected surface, with vertices $v_{1},\ldots,v_{n}$ on one
boundary, labeled in counterclockwise order. A _matching_ or _path_ $w$
between $v_{i}$ and $v_{j}$ is a sequence of subgons
$w=\mathcal{P}_{i+1},\ldots,\mathcal{P}_{j-1},$ (working modulo $n$) such that
$\mathcal{P}_{k}$ is incident to $v_{k}$ for all $i+1\leq k\leq j-1$.

We will use $\mathbf{P}_{i,j}$ to denote the set of all paths from $v_{i}$ to
$v_{j}$. We will also set the convention that
$\mathbf{P}_{i,i+1}=\\{\emptyset\\}$ while $\mathbf{P}_{i,i}=\emptyset$. These
conventions will allow us to extend our combinatorial interpretation to the
upper boundary rows of 0’s and 1’s in a frieze pattern.

Traditionally, a matching $w=\mathcal{P}_{i+1},\ldots,\mathcal{P}_{j-1}$ in a
triangulated surface is only admissible if all $\mathcal{P}_{k}$ are pairwise
distinct. For purposes of this section, we denote the subset of matchings with
this property $\mathbf{P}_{i,j}^{d}$, with $d$ for “distinct”. Later, we will
rephrase this property by weighting matchings which use the same triangle
twice with weight 0 (see Definition 19).

Broline, Crowe and Isaacs were the first to connect matchings to frieze
patterns [5].

######  Theorem 6 ([5]).

Let $\mathcal{F}=\\{m_{i,j}\\}$ be a finite frieze pattern of positive
integers of width $n-3$, and let $P$ be the triangulated $n$-gon which is the
realization of $\mathcal{F}$. Label the vertices of $P$ in counterclockwise
order $v_{1},\ldots,v_{n}$, treating these labels modulo $n$. Then,

$m_{i,j}=\sum_{w\in\mathbf{P}_{i,j}^{d}}1=|\mathbf{P}_{i,j}^{d}|$

Baur, Parsons, and Tschabold extended this correspondence to infinite frieze
patterns of positive integers in [1]. In these cases, one must work in the
universal cover of the triangulated surface which realizes the frieze pattern.
Even though this surface is infinite, given any two vertices $v_{i}^{k}$ and
$v_{j}^{\ell}$ on the infinite strip with a dissection, we can work in a
finite piece (a polygon) which contains $v_{i}^{k}$ and $v_{j}^{\ell}$.

In the statement of the theorem, we use the following notation. Let
$i\in\mathbb{Z}$, $n\in\mathbb{Z}_{>0}$. Then, set

$i_{n}=\begin{cases}i\pmod{n}&n\nmid i\\\ n&n\mid i\end{cases}.$

Recall our notation for infinite strips established in Section 4.1.

######  Theorem 7 ([1]).

Let $\mathcal{F}=\\{m_{i,j}\\}$ be an infinite, $n$-periodic frieze pattern of
positive integers. Let $S$ be the triangulated surface which is the
realization of $\mathcal{F}$. Let $I$ be the triangulated infinite strip which
is the universal cover of $S$ . Let $\\{v_{i}^{k}\\}$ be the infinite family
of vertices of $I$ which map to the vertices on the outer boundary of $S$
under the covering map. Then,

$m_{i,j}=\sum_{w\in\mathbf{P}_{i,j}^{d}}1=|\mathbf{P}_{i,j}^{d}|$

where $\mathbf{P}_{i,j}$ is the set of matchings between
$v_{i_{n}}^{\lfloor\frac{i}{n}\rfloor}$ and
$v_{j_{n}}^{\lfloor\frac{j}{n}\rfloor}$.

Our weighting scheme will use evaluations of normalized Chebyshev polynomials
at $\lambda_{p}$. See Section 3 for the definition of these Chebyshev
polynomials and some basic results.

In a polygon, we say a _$k$ -diagonal_ is one which skips $k$ vertices
counterclockwise. A side of a polygon is a 0-diagonal. The following result
comes from work in [15] and adds motivation for the presence of the terms
$U_{k}(\lambda_{p})$ in our weightings.

######  Lemma 9.

In a regular $p$-gon, the ratio of the length of a $k$-diagonal and a side is
$U_{k}(\lambda_{p})$. In particular,

  1. 1.

since we could equivalently look at vertices skipped in clockwise order,
$U_{k}(\lambda_{p})=U_{p-2-k}(\lambda_{p})$, and

  2. 2.

since $k$-diagonals in a polygon will be at least as long as the sides, for
$0\leq k\leq p-2,U_{k}(\lambda_{p})\geq 1$.

We calculate some values of $U_{k}(\lambda_{p})$. Recall that by Lemma 3,
$U_{k}(\lambda_{p})=-U_{k+p}(\lambda_{p})$.

k \p | 3 | 4 | 5 | 6
---|---|---|---|---
0 | 1 | 1 | 1 | 1
1 | 1 | $\sqrt{2}$ | $\phi$ | $\sqrt{3}$
2 | 0 | 1 | $\phi$ | 2
3 | -1 | 0 | 1 | $\sqrt{3}$
4 | -1 | -1 | 0 | 1
5 | 0 | $-\sqrt{2}$ | -1 | 0

###  6.2 Local Weighting

We now define our first method to weight paths in dissected polygons. We will
use $w[a:b]=\mathcal{P}_{a},\ldots,\mathcal{P}_{b}$ to denote a submatching of
$w=\mathcal{P}_{i+1},\ldots,\mathcal{P}_{j-1}$, where $i+1\leq a\leq b\leq
j-1$. If $a>b$, then $w[a:b]=\emptyset$.

######  Definition 18 (Local Weighting).

Given a matching $w=\mathcal{P}_{i+1},\ldots,\mathcal{P}_{j-1}$ in a dissected
polygon, we define the local weight $wt_{L}(w)$ recursively. Suppose $k$ is
the largest positive integer such that the first $k$ subgons in $w$ are the
same, i.e.
$\mathcal{P}_{i+1}=\mathcal{P}_{i+2}=\cdots=\mathcal{P}_{i+k}\neq\mathcal{P}_{i+k+1}$,
for some $1\leq k\leq j-i-1$. Then, we set

$wt_{L}(w)=U_{k}(\lambda_{|\mathcal{P}_{i+1}|})wt_{L}(w[i+k+1:j-1]).$

We set $wt_{L}(\emptyset)=1$.

This weighting is “local” in the sense that it only depends on consecutive
occurrences of the same subgon. See Example 6 and Figure 8 for examples of
calculating $wt_{L}$.

Lemma 10 gives an equivalent definition of the local weighting by recursively
checking if two consecutive subgons in a matching are equal.

######  Lemma 10.

Given a matching $w=\mathcal{P}_{i+1},\ldots,\mathcal{P}_{j-1}$ in a dissected
polygon, we define $wt_{L^{\prime}}$ recursively. If
$\mathcal{P}_{i+1}=\mathcal{P}_{i+2},$ set

$wt_{L^{\prime}}(w)=\lambda_{|\mathcal{P}_{i+1}|}wt_{L^{\prime}}(w[i+2:j-1])-wt_{L^{\prime}}(w[i+3:j-1]),$

and otherwise we set

$wt_{L^{\prime}}(w)=\lambda_{|\mathcal{P}_{i+1}|}wt_{L^{\prime}}(w[i+2:j-1]).$

We set $wt_{L^{\prime}}(\emptyset)=1$. Then, for any $w$,
$wt_{L}(w)=wt_{L^{\prime}}(w)$.

###### Proof.

Let $w\in\mathbf{P}_{i,j}$. We will induct on $j-i$.

If $j=i$, there are no matchings in $\mathbf{P}_{i,j}$. If $j=i+1$, then
$\mathbf{P}_{i,j}=\\{\emptyset\\}$, and
$wt_{L}(\emptyset)=wt_{L^{\prime}}(\emptyset)=1$. If $j=i+2$, then each
matching $w\in\mathbf{P}_{i,j}$ is of the form $w=\mathcal{P}_{i+1}$. In this
case as well,
$wt_{L}(\mathcal{P}_{i+1})=wt_{L^{\prime}}(\mathcal{P}_{i+1})=\lambda_{|\mathcal{P}_{i+1}|}$.

Now, suppose we have established the claim for all $w\in\mathbf{P}_{i,j}$ with
$j\leq i+\ell-1$, $\ell\geq 3$ and all possible values of $k$, the number of
consecutive subgons at the beginning of a matching which are the same. Let
$j=i+\ell$ and let $w\in\mathbf{P}_{i,j}$. Suppose the first $k\geq 1$ subgons
of $w$ are the same. If $k=1$, then by induction we immediately have

$wt_{L^{\prime}}(w)=\lambda_{|\mathcal{P}_{i+1}|}wt_{L^{\prime}}(w[i+2,j-1])=wt_{L}(w)$

Now let $k>1$. Then,

$wt_{L^{\prime}}(w)=\lambda_{|\mathcal{P}_{i+1}|}wt_{L^{\prime}}(w[i+2:j-1])-wt_{L^{\prime}}(w[i+3:j-1])$

By induction, we can rewrite the righthand side in terms of $wt_{L}$,

$\displaystyle
wt_{L^{\prime}}(w)=\lambda_{|\mathcal{P}_{i+1}|}wt_{L^{\prime}}(w[i+2:j-1])-wt_{L^{\prime}}(w[i+3:j-1])$
$\displaystyle=\lambda_{|\mathcal{P}_{i+1}|}U_{k-1}(\lambda_{|\mathcal{P}_{i+1}|})wt_{L}(w[i+k+1:j-1])-U_{k-2}(\lambda_{|\mathcal{P}_{i+1}|})wt_{L}(w[i+k+1:j-1])$
$\displaystyle=U_{k}(\lambda_{|\mathcal{P}_{i+1}|})wt_{L}(w[i+k+1:j-1]).$

∎

The formulation of the local weight in Lemma 10 resembles the recurrences for
frieze pattern entries in Lemma 2. This fact is crucial for verifying that
matchings with the local weighting give a combinatorial interpretation of
frieze pattern entries.

######  Theorem 8.

  1. 1.

Let $\mathcal{F}=\\{m_{i,j}\\}$ be a finite frieze pattern of Type
$\Lambda_{p_{1},\ldots,p_{s}}$ and of width $n-3$ which is realizable by a
dissected polygon, $P$. Let $v_{1},\ldots,v_{n}$ be the vertices of $P$. Then
for all $i,j$ with $0\leq j-i\leq n$

$m_{i,j}=\sum_{w\in\mathbf{P}_{i,j}}wt_{L}(w)$

  2. 2.

Let $\mathcal{F}=\\{m_{i,j}\\}$ be an infinite, $n$-periodic frieze pattern of
Type $\Lambda_{p_{1},\ldots,p_{s}}$ which is realizable by a dissection of
$S_{n}$ or $A_{n,m}$ or a quotient dissection. Let $I$ be the dissected
infinite strip which is the universal cover. Let $\\{v_{i}^{k}\\}$ be the
infinite family of vertices of $I$ which map to the vertices on the outer
boundary of $S$ under the covering map. Then,

$m_{i,j}=\sum_{w\in\mathbf{P}_{i,j}}wt_{L}(w)$

where $\mathbf{P}_{i,j}$ is the set of matchings between
$v_{i_{n}}^{\lfloor\frac{i}{n}\rfloor}$ and
$v_{j_{n}}^{\lfloor\frac{j}{n}\rfloor}$.

###### Proof.

We will consider finite and infinite frieze patterns simultaneously. We prove
this theorem by induction. We have that

$\sum_{w\in\mathbf{P}_{i,i}}wt_{L}(w)=0=m_{i,i}$

since $\mathbf{P}_{i,i}=\emptyset$. Similarly,
$\mathbf{P}_{i,i+1}=\\{\emptyset\\}$, giving

$\sum_{w\in\mathbf{P}_{i,i+1}}wt_{L}(w)=wt_{L}(\emptyset)=1=m_{i,i+1}.$

By definition, we also know that every entry in the quiddity row satisfies the
formula

$m_{i-1,i+1}=\sum_{w\in\mathbf{P}_{i-1,i+1}}wt_{L}(w)=\sum_{\mathcal{P}\in\text{Poly($v_{i}$)}}\lambda_{|\mathcal{P}|}.$

Suppose we have verified the claim for all $m_{i,j}$ with $j-i<k$. If our
frieze pattern is finite of width $k-4$, then for all $i$, $m_{i,i+(k-1)}=0$.
Hence, we have checked all rows of the frieze pattern and we are done.

Now, suppose our frieze pattern is infinite or finite with width larger than
$k-4$, and consider $m_{i,j}$ with $j=i+k$. By Lemma 2, we have that
$m_{i,j}=m_{i,i+2}m_{i+1,j}-m_{i+2,j}$. Thus, it suffices to prove

$\sum_{w\in\mathbf{P}_{i,j}}wt_{L}(w)=\bigg{(}\sum_{w\in\mathbf{P}_{i,i+2}}wt_{L}(w)\bigg{)}\bigg{(}\sum_{w\in\mathbf{P}_{i+1,j}}wt_{L}(w)\bigg{)}-\sum_{w\in\mathbf{P}_{i+2,j}}wt_{L}(w)$

where, by induction, we already know that the right hand side is equal to
$m_{i,i+2}m_{i+1,j}-m_{i+2,j}$.

Let $\hat{\mathcal{P}}$ be the unique subgon which is incident to both
$v_{i+1}$ and $v_{i+2}$. There is a bijection between the set
$\mathbf{P}_{i+2,j}$ and the subset of $\mathbf{P}_{i,j}$ consisting of
matchings $w=\mathcal{P}_{i+1},\ldots,\mathcal{P}_{j-1}$ with
$\mathcal{P}_{i+1}=\mathcal{P}_{i+2}=\hat{\mathcal{P}}$. This bijection is
given by sending $h\in\mathbf{P}_{i+2,j}$ to
$w_{h}:=\hat{\mathcal{P}},\hat{\mathcal{P}},h\in\mathbf{P}_{i,j}$. Let
$\mathbf{P}_{i,j}^{\prime}\subseteq\mathbf{P}_{i,j}$ be the complement of the
set of matchings of the form $w_{h}$. It is possible that
$\mathbf{P}_{i,j}^{\prime}$ is empty.

We break up our sum based on this partition of $\mathbf{P}_{i,j}$

$\sum_{w\in\mathbf{P}_{i,j}}wt_{L}(w)=\sum_{h\in\mathbf{P}_{i+2,j}}wt_{L}(w_{h})+\sum_{w\in\mathbf{P}_{i,j}^{\prime}}wt_{L}(w).$

Lemma 10 transforms the two terms on the right hand side as

$\sum_{h\in\mathbf{P}_{i+2,j}}wt_{L}(w_{h})=\sum_{h\in\mathbf{P}_{i+2,j}}(\lambda_{|\hat{\mathcal{P}}|}wt_{L}(\hat{\mathcal{P}},h)-wt_{L}(h))$
(2)

and

$\sum_{w\in\mathbf{P}_{i,j}^{\prime}}wt_{L}(w)=\bigg{(}\sum_{\mathcal{P}\in\mathbf{P}_{i,i+2}}wt_{L}(\mathcal{P})\bigg{)}\bigg{(}\sum_{w^{\prime}\in\mathbf{P}_{i+1,j}}wt_{L}(w^{\prime})\bigg{)}-\sum_{h\in\mathbf{P}_{i+2,j}}\lambda_{|\hat{\mathcal{P}}|}wt_{L}(\hat{\mathcal{P}},h).$
(3)

The subtracted term in Equation 3 is removing the contribution of any
matchings of the form $w_{h}$ for $h\in\mathbf{P}_{i+2,j}$.

From equations 2 and 3, we conclude that

$\sum_{w\in\mathbf{P}_{i,j}}wt_{L}(w)=\bigg{(}\sum_{\mathcal{P}\in\mathbf{P}_{i,i+2}}wt_{L}(\mathcal{P})\bigg{)}\bigg{(}\sum_{w^{\prime}\in\mathbf{P}_{i+1,j}}wt_{L}(w^{\prime})\bigg{)}-\sum_{h\in\mathbf{P}_{i+2,j}}wt_{L}(h).$

∎

In the proof of Theorem 8, if we were considering a finite frieze pattern, we
stopped the induction once we reached the lower boundary row of 0’s. However,
if we use the recurrence in Lemma 2 instead of the diamond relation, we can
extend a finite frieze pattern to an infinite frieze pattern with some
negative rows, as in Example 5. Such infinite frieze patterns are related to
_$SL_{2}$ -tilings_, as discussed in [7]. This allows us to consider extending
the inductive argument in Theorem 8 in the case of a frieze pattern from a
polygon.

######  Corollary 1.

Let $\mathcal{F}=\\{m_{i,j}\\}$ be an _infinite_ frieze pattern of Type
$\Lambda_{p_{1},\ldots,p_{s}}$ from a dissection of an $n$-gon with vertices
$v_{1},\ldots,v_{n}$. Then,

$m_{i,j}=\sum_{w\in\mathbf{P}_{i,j}}wt_{L}(w)$

where if $|j-i|\geq n$, then $\mathbf{P}_{i,j}$ is the set of matchings
between $v_{i_{n}}$ and $v_{j_{n}}$ which visits vertices
$v_{(i+1)_{n}},\ldots,v_{(j-1)_{n}}$.

######  Example 5.

See below for an example of an extension of a finite frieze pattern to an
infinite frieze pattern. On the left we give the dissected polygon which
realizes this frieze pattern.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$\beta$$\gamma$$\alpha$
$\begin{array}[]{cccccccccccccccccccc}&&0&&0&&0&&0&&0&&0&&0&&0&&0&\\\
&&&1&&1&&1&&1&&1&&1&&1&&1&&1\\\ &&1&&3&&1&&2&&2&&1&&3&&1&&2&\\\
&&&2&&2&&1&&3&&1&&2&&2&&1&&3\\\ &&1&&1&&1&&1&&1&&1&&1&&1&&1&\\\
&&&0&&0&&0&&0&&0&&0&&0&&0&&0\\\ &&-1&&-1&&-1&&-1&&-1&&-1&&-1&&-1&&-1&\\\
&&&-2&&-2&&-1&&-3&&-1&&-2&&-2&&-1&&-3\\\
&&-1&&-3&&-1&&-2&&-2&&-1&&-3&&-1&&-2&\\\
&&&-1&&-1&&-1&&-1&&-1&&-1&&-1&&-1&&-1\\\ &&0&&0&&0&&0&&0&&0&&0&&0&&0&\\\
&&&&&&&&&&&\vdots&&&&&&&&\\\ \end{array}$

As an example of Corollary 1, note that all entries $m_{i,i+6}=-1$. We look at
the case $i=2$; that is, $m_{2,8}=\sum_{w\in\mathbf{P}_{2,8}}wt_{L}(w)$. The
set $\mathbf{P}_{2,8}$ consists of matchings of the form
$\mathcal{P}_{3},\mathcal{P}_{4},\mathcal{P}_{5},\mathcal{P}_{6},\mathcal{P}_{7}$,
where $\mathcal{P}_{i}$ is incident to $v_{i_{n}}$. Amongst the 12 matchings
in $\mathbf{P}_{2,8}$, $wt_{L}(\alpha\beta\gamma\gamma\gamma)=-1$ and all
other matchings have weight 0. Recall that $\lambda_{3}=1$, $U_{2}(1)=0$ and
$U_{3}(1)=-1$.

For an example with more nonzero matchings, we compute
$\sum_{w\in\mathbf{P}_{4,10}}wt_{L}(w)=-1$.

$w$ | $wt_{L}(w)$ | $w$ | $wt_{L}(w)$
---|---|---|---
$\beta\gamma\gamma\alpha\alpha$ | 0 | $\gamma\gamma\gamma\alpha\alpha$ | 0
$\beta\gamma\beta\alpha\alpha$ | 0 | $\gamma\gamma\beta\alpha\alpha$ | 0
$\beta\gamma\gamma\alpha\beta$ | 0 | $\gamma\gamma\gamma\alpha\beta$ | $-1$
$\beta\gamma\alpha\alpha\alpha$ | $-1$ | $\gamma\gamma\alpha\alpha\alpha$ | 0
$\beta\gamma\beta\alpha\beta$ | 1 | $\gamma\gamma\beta\alpha\beta$ | 0
$\beta\gamma\alpha\alpha\beta$ | 0 | $\gamma\gamma\alpha\alpha\beta$ | 0

###  6.3 Traditional Weighting

We now have one combinatorial interpretation for entries of a realizable
frieze pattern of Type $\Lambda_{p_{1},\ldots,p_{s}}$. However, there are some
matchings $w$ such that $wt_{L}(w)<0$, even though the frieze pattern entries
appear to be all positive. We introduce another weighting, $wt_{T}$, which
satisfies $wt_{T}(w)\geq 0$ for all matchings $w$.

######  Definition 19 (Traditional Weighting).

Consider a dissected polygon, with vertices $v_{1},\ldots,v_{n}$. Let
$w=(\mathcal{P}_{i+1},\ldots,\mathcal{P}_{j-1})$ be a matching between
vertices $v_{i}$ and $v_{j}$. For all $\mathcal{P}$ in the dissection, let
$k_{\mathcal{P}}$ be the number of times that $\mathcal{P}$ is used in $w$. If
for any $\mathcal{P}$, $k_{\mathcal{P}}>|\mathcal{P}|-2$, then $wt_{T}(w)=0$.
Otherwise, we set

$wt_{T}(w)=\prod_{\mathcal{P}\in
w}U_{k_{\mathcal{P}}}(\lambda_{|\mathcal{P}|})$

We define this weighting on a dissected polygon; when working with a
dissection of $A_{n,m}$ or $S_{n}$, we will calculate the weight of a matching
from the universal cover.

This weighting scheme is “traditional” in the sense that, in the case of
triangulations, it forbids using a triangle more than one time in a matching,
as was the case in [5] and [1]. In the case of dissections, Bessenrodt also
only permitted using a $p$-gon up to $p-2$ times in a matching in [4]. This
weighting differs from Bessenrodt’s though, as she used formal variables and
did not use Chebyshev polynomials to track using a subgon multiple times.

######  Remark 5.

We do not define traditional weighting on a quotient dissection. We do not yet
know a consistent way to weight matchings in a quotient dissection, other than
with $wt_{L}$, which would give an interpretation of the entries of the
corresponding frieze pattern. In particular, we cannot verify that all frieze
patterns from quotient dissections have positive entries; see Conjecture 1.

See Example 6 and Figure 8 for sample calculations of $wt_{T}$.

######  Example 6.

In Figure 8, we exhibit the three weightings, $wt_{L},wt_{T},wt_{A}$ on
matchings in $\mathbf{P}_{0,4}$ from the following dissection. See Definitions
18, 19, and 20 respectively.

$v_{1}$$v_{2}$$v_{3}$$\alpha$$\beta$$\gamma$$\delta$

We draw the universal cover of this dissection in Figure 8. As an example of
our calculations, compare how we calculate $wt_{L}(\beta\alpha\beta)$,

$wt_{L}(\beta\alpha\beta)=U_{1}(\lambda_{|\beta|})U_{1}(\lambda_{|\alpha|})U_{1}(\lambda_{|\beta|})=\sqrt{2}\cdot
1\cdot\sqrt{2}=2$

with how we compute $wt_{T}(\beta\alpha\beta)$

$wt_{T}(\beta\alpha\beta)=U_{2}(\lambda_{|\beta|})U_{1}(\lambda_{|\alpha|})=(2-1)\cdot
1=1$

Note that, while individual matchings have different weights,
$\sum_{w\in\mathbf{P}_{0,4}}wt_{L}(w)=\sum_{w\in\mathbf{P}_{0,4}}wt_{T}(w)$.
We prove in Theorem 9 that this will always occur.

For an example where $wt_{T}(w)$ and $wt_{A}(w)$ are distinct, consider the
matching $w=\delta\alpha\delta$. We see that
$wt_{T}(\delta\alpha\delta)=U_{1}(\lambda_{|\delta|})U_{1}(\lambda_{|\delta|})U_{1}(\lambda_{|\alpha|})=1$
since $v_{1}^{0}$ and $v_{3}^{0}$ are incident to different lifts of $\delta$
while $wt_{A}(\delta\alpha\delta)=0$.

$v_{3}^{-1}$$v_{1}^{0}$$v_{2}^{0}$$v_{3}^{0}$$v_{1}^{1}$$\alpha_{0}$$\beta_{0}$$\delta_{0}$$\delta_{1}$$\gamma_{0}$$\gamma_{-1}$$\cdots$$\cdots$
$w$ $wt_{T}(w)$ $wt_{L}(w)$ $wt_{A}(w)$ $\beta\alpha\beta$ 1 2 1
$\beta\alpha\delta$ $\sqrt{2}$ $\sqrt{2}$ $\sqrt{2}$ $\gamma\alpha\beta$ 2 2 2
$\gamma\alpha\delta$ $\sqrt{2}$ $\sqrt{2}$ $\sqrt{2}$ $\delta\alpha\beta$
$\sqrt{2}$ $\sqrt{2}$ $\sqrt{2}$ $\delta\alpha\delta$ 1 1 0
$\alpha\alpha\alpha$ 0 -1 0 $4+3\sqrt{2}$ $4+3\sqrt{2}$ $3+3\sqrt{2}$

Figure 8: Exhibiting the three weightings of matchings for the set of
matchings between $v_{0}$ and $v_{4}$

The following well-known identity of Chebyshev polynomials will be useful for
proving our next main result.

######  Lemma 11.

For all $m,n\geq-1$, $U_{m+1}(x)U_{n+1}(x)-U_{m}(x)U_{n}(x)=U_{m+n+2}(x)$

We frame Theorem 9 in the context of dissected polygons. If we are working
with an annulus or once-punctured disc, the set of subgons between any two
vertices in the universal cover forms a dissected polygon.

######  Theorem 9.

Let $P$ be a polygon with dissection $\mathcal{D}$. Let $v_{i},v_{j}$ be two
vertices of $P$, with $v_{j}$ following $v_{i}$ in counterclockwise order.
Then,

$\sum_{w\in\mathbf{P}_{i,j}}wt_{L}(w)=\sum_{w\in\mathbf{P}_{i,j}}wt_{T}(w).$

###### Proof.

Let $v_{i},v_{j}$ be vertices of our polygon, $P$, with $i<j$ so that $v_{j}$
follows $v_{i}$ in counterclockwise order. We will induct on the number of
arcs with both end points strictly between $v_{i}$ and $v_{j}$ when traveling
counterclockwise. Call the number of such arcs $k$.

First, consider the case where $k=0$. Then, we claim that for all matchings
$w\in\mathbf{P}_{i,j}$, $wt_{L}(w)=wt_{T}(w)$. In this case, it is not
possible that we use a $p$-subgon $p$ times since there are no subgons with
all vertices strictly between $i$ and $j$. It is possible we could choose a
$p$-subgon $p-1$ times, but such a matching would have weight 0 under both
weightings. Moreover, since there is no subgon contained strictly between $i$
and $j$, we cannot have a matching which uses the same subgon non-
contiguously. Thus, for any matching $w\in\mathbf{P}_{i,j}$,
$wt_{L}(w)=wt_{T}(w)$.

Now suppose we have $k>0$ arcs which have both end points between $i$ and $j$.
In this case, there will be at least one ear between $i$ and $j$. Pick an ear
and call it $Q_{0}$. Let $(a_{0},b_{0})$ be the unique non-boundary edge from
$\mathcal{D}$ which is a side of $Q_{0}$. Let $Q_{1}$ be the other subgon
which has $(a_{0},b_{0})$ as one of its sides.

$Q_{0}$$Q_{1}$$v_{i}=a_{1}$$v_{j}$$a_{0}$$b_{0}=b_{1}$$Q_{1}$$v_{i}$$a_{0}$$b_{0}$$v_{j}$$\xrightarrow[]{\text{cut
}Q_{0}}$

We partition $\mathbf{P}_{i,j}$ based on the choices at $a_{0}$ and $b_{0}$.
Let $\mathbf{P}_{i,j}^{x,y}$ be the subset of $\mathbf{P}_{i,j}$ with
$\mathcal{P}_{a_{0}}=Q_{x}$ and $\mathcal{P}_{b_{0}}=Q_{y}$ for
$x,y\in\\{0,1\\}$. We let $\mathbf{P}_{i,j}^{\prime}$ be the complement of
$\mathbf{P}_{i,j}^{0,0}\cup\mathbf{P}_{i,j}^{1,0}\cup\mathbf{P}_{i,j}^{0,1}\cup\mathbf{P}_{i,j}^{1,1}$
in $\mathbf{P}_{i,j}$; in some cases, $\mathbf{P}_{i,j}^{\prime}=\emptyset$.

Let $\overline{\mathbf{P}_{i,J}}$ be the set of matchings in $\overline{P}$,
the dissected polygon resulting from cutting $Q_{0}$ from $P$.

First, we claim that

$\sum_{w\in\mathbf{P}_{i,j}}wt_{T}(w)=\sum_{w\in\overline{\mathbf{P}_{i,j}}}wt_{T}(w)$
(4)

Since $Q_{0}$ has $|Q_{0}|-2$ vertices which are not incident to any other
subgon, a matching
$w=\mathcal{P}_{i+1},\ldots,\mathcal{P}_{j-1}\in\mathbf{P}_{i,j}$ has
$wt_{T}(w)\neq 0$ only if $w\in\mathbf{P}_{i,j}^{1,1}$ or
$w\in\mathbf{P}_{i,j}^{\prime}$. Thus, the contribution of $Q_{0}$ to any
nonzero matching is $U_{|Q_{0}|-2}(\lambda_{|Q_{0}|})=1$. This observation
provides a weighted (under $wt_{T}$) bijection between matchings in
$\mathbf{P}_{i,j}$ which use $Q_{0}$ exactly $|Q_{0}|-2$ times and matchings
in $\overline{\mathbf{P}_{i,j}}$.

Second, we claim that

$\sum_{w\in\mathbf{P}_{i,j}}wt_{L}(w)=\sum_{w\in\overline{\mathbf{P}_{i,j}}}wt_{L}(w)$
(5)

In the following, we will conflate a vertex $v_{k}$ with its index $k$. For
further ease of notation, we will assume that $1\leq i<j\leq|P|$. Let
$a_{1}^{\prime}$ (respectively $b_{1}^{\prime}$) be the first vertex clockwise
(counterclockwise) of $a_{0}$ ($b_{0}$) which is incident to an arc from
$\mathcal{D}$ besides the arc $(a_{0},b_{0})$. Note that if $a_{0}$ is
incident to multiple arcs in $\mathcal{D}$, then $a_{0}=a_{1}^{\prime}$, and
similarly for $b_{1}^{\prime}$. We set $a_{1}=a_{1}^{\prime}$ if
$a_{1}^{\prime}$ counterclockwise from $i$ (that is, $a_{1}^{\prime}\geq i$);
otherwise we set $a_{1}=i$. Let $a=a_{1}-a_{0}$. Similarly, set
$b_{1}=b_{1}^{\prime}$ if $b_{1}^{\prime}$ is clockwise from $j$
($b_{1}^{\prime}\leq j$) and set $b_{1}=j$ otherwise. Let $b=b_{1}-b_{0}$.
Note that we must choose $Q_{1}$ at all vertices strictly between $a_{1}$ and
$a_{0}$ and at all vertices strictly between $b_{0}$ and $b_{1}$, if any such
vertices exist.

In order for $w\in\mathbf{P}_{i,j}$ to have $wt_{L}(w)\neq 0$, it must be that
$w\in\mathbf{P}_{i,j}^{0,0}\cup\mathbf{P}_{i,j}^{1,1}\cup\mathbf{P}_{i,j}^{\prime}$.
We sum over matchings in each of these subsets. First we just consider
matchings in $\mathbf{P}_{i,j}^{0,0}$ and $\mathbf{P}_{i,j}^{1,1}$.

$\displaystyle\sum_{w\in\mathbf{P}_{i,j}^{0,0}}wt_{L}(w)=U_{|Q_{0}|}(\lambda_{|Q_{0}|})U_{a}(\lambda_{|Q_{1}|})U_{b}(\lambda_{|Q_{1}|})\sum_{w\in\mathbf{P}_{i,a_{1}+1}}wt_{L}(w)\sum_{w\in\mathbf{P}_{b_{1}-1,j}}wt_{L}(w)$
$\displaystyle=-U_{a}(\lambda_{|Q_{1}|})U_{b}(\lambda_{|Q_{1}|})\sum_{w\in\mathbf{P}_{i,a_{1}+1}}wt_{L}(w)\sum_{w\in\mathbf{P}_{b_{1}-1,j}}wt_{L}(w)$

$\displaystyle\sum_{w\in\mathbf{P}_{i,j}^{1,1}}wt_{L}(w)=U_{|Q_{0}|-2}(\lambda_{|Q_{0}|})U_{a+1}(\lambda_{|Q_{1}|})U_{b+1}(\lambda_{|Q_{1}|})\sum_{w\in\mathbf{P}_{i,a_{1}+1}}wt_{L}(w)\sum_{w\in\mathbf{P}_{b_{1}-1,j}}wt_{L}(w)$
$\displaystyle=U_{a+1}(\lambda_{|Q_{1}|})U_{b+1}(\lambda_{|Q_{1}|})\sum_{w\in\mathbf{P}_{i,a_{1}+1}}wt_{L}(w)\sum_{w\in\mathbf{P}_{b_{1}-1,j}}wt_{L}(w)$

From Lemma 11, we conclude that

$\sum_{w\in\mathbf{P}_{i,j}^{0,0}\cup\mathbf{P}_{i,j}^{1,1}}wt_{L}(w)=U_{a+b+2}(\lambda_{|Q_{1}|})\sum_{w\in\mathbf{P}_{i,a_{1}+1}}wt_{L}(w)\sum_{w\in\mathbf{P}_{b_{1}-1,j}}wt_{L}(w)=\sum_{w\in\overline{\mathbf{P}_{i,j}^{1,1}}}wt_{L}(w)$

where $\overline{\mathbf{P}_{i,j}^{1,1}}$ is the set of matchings between
$v_{i}$ and $v_{j}$ in $\overline{P}$ which have
$\mathcal{P}_{a_{0}}=\mathcal{P}_{b_{0}}=Q_{1}$.

If $w\in\mathbf{P}_{i,j}^{\prime}$, then $w[a_{0}:a_{0}]\neq Q_{0}$ and
$w[b_{0}:b_{0}]\neq Q_{0}$, so $k_{Q_{0}}=|Q_{0}|-2$, and $Q_{0}$ contributes
$1$ to $wt_{L}(w)$. Moreover, since $w[a_{0}:a_{0}]\neq Q_{1}$ and
$w[b_{0}:b_{0}]\neq Q_{1}$, $w[a_{0}:a_{0}]$ and $w[b_{0}:b_{0}]$ cannot be
the same subgon (when viewing the matching in the infinite strip). Thus, the
local weight of the image of $w$ after cutting $Q_{0}$ will be the same as the
local weight of $w$. We can conclude that

$\sum_{w\in\mathbf{P}_{i,j}^{\prime}}wt_{L}(w)=\sum_{w\in\overline{\mathbf{P}_{i,j}^{\prime}}}wt_{L}(w)$

By proving equations 4 and 5, we have shown that summing over all matchings
with either weighting is unaffected by cutting ears. Thus, we can always
reduce to the case where there is no ear between $v_{i}$ and $v_{j}$, and in
this case, we know the sums of the two weightings are equal. ∎

An immediate corollary of Theorem 9 is that our frieze pattern entries are
given by summing over matchings using the traditional weight.

######  Corollary 2.

  1. 1.

Let $\mathcal{F}=\\{m_{i,j}\\}$ be a finite frieze pattern of Type
$\Lambda_{p_{1},\ldots,p_{s}}$ and of width $n+3$ which is realizable by a
dissected polygon, $P$. Let $v_{1},\ldots,v_{n}$ be the vertices of $P$. Then,

$m_{i,j}=\sum_{w\in\mathbf{P}_{i,j}}wt_{T}(w)$

  2. 2.

Let $\mathcal{F}=\\{m_{i,j}\\}$ be an infinite, $n$-periodic frieze pattern of
Type $\Lambda_{p_{1},\ldots,p_{s}}$ which is realizable by a dissection of
$S_{n}$ or $A_{n,m}$. Let $I$ be the dissected infinite strip which is the
universal cover. Let $\\{v_{i}^{k}\\}$ be the infinite family of vertices of
$I$ which map to the vertices on the outer boundary of $S$ under the covering
map. Then,

$m_{i,j}=\sum_{w\in\mathbf{P}_{i,j}}wt_{T}(w)$

where $\mathbf{P}_{i,j}$ is the set of matchings between
$v_{i_{n}}^{\lfloor\frac{i}{n}\rfloor}$ and
$v_{j_{n}}^{\lfloor\frac{j}{n}\rfloor}$.

##  7 Growth Coefficients

Baur, Fellner, Parsons, and Tschabold studied the rate of growth of entries in
an infinite frieze pattern [2]. Specifically, they found that in an infinite
frieze pattern with rows of period $n$, for every $k\in\mathbb{Z}_{\geq 1}$
the difference between row $kn$ and row $kn-2$ along the columns is constant.

######  Theorem 10 (Growth Coefficients [2]).

Given an infinite frieze pattern $\mathcal{F}=\\{m_{i,j}\\}$ with period $n$,
for any $k\geq 1$, $m_{i,i+kn+1}-m_{i+1,i+kn}$ is constant for all $i$.

Let $s_{k}=m_{0,kn-1}-m_{1,kn}$ be the _$k$ -th growth coefficient_. For
example, the first growth coefficient of the frieze pattern in Figure 2 is
$8-1=11-4=7$ and the first growth coefficient for the frieze pattern in
Example 1 is $3+3\sqrt{2}$. By the following result, all $s_{k}$ are
determined by $s_{1}$; hence, we sometimes refer to $s_{1}$ as the _principal
growth coefficient_.

######  Theorem 11 ([2]).

Let $s_{0}=2$. The growth coefficients, $\\{s_{k}\\}$, of an infinite frieze
pattern $\mathcal{F}$ satisfy

$s_{k+1}=s_{1}s_{k}-s_{k-1}$

We define a third weighting for matchings on $A_{n,m}$ or $S_{n}$, working
specifically in the case of matchings of length $n$ ( in
$\mathbf{P}_{i,i+n+2}$); such matchings choose one subgon incident to each
vertex on the outer boundary of the surface.

######  Definition 20 (Annulus Weighting).

Let $S$ be an annulus or once-punctured disc with dissection $\mathcal{D}$.
Let $I$ be the dissected infinite strip with dissection
$\widetilde{\mathcal{D}}$ which is the universal cover of $S$. Let
$w=\mathcal{P}_{i+1},\ldots,\mathcal{P}_{i+n}\in\mathbf{P}_{i,i+n+1}$. For
each subgon $\mathcal{P}$ in $\mathcal{D}$, let $N_{\mathcal{P}}$ be the
number of times $\mathcal{P}_{k}$, $i+1\leq k\leq i+n$, is a lift of
$\mathcal{P}$. If any $N_{\mathcal{P}}>|\mathcal{P}|-2$, we set $wt_{A}(w)=0$.
Otherwise, set

$wt_{A}(w)=\prod_{\mathcal{P}\in\mathcal{D}}U_{N_{\mathcal{P}}}(\lambda_{|\mathcal{P}|})$

See Example 6 for sample calculations of $wt_{A}$. We call this weighting the
“annulus” weighting since it is equivalent to using the rules of the
Traditional weighting but working with respect to the surface $S$
corresponding to the frieze pattern instead of the infinite strip. Even though
we name this the annulus weighting, it is also valid in a once-punctured disc.
The following result shows that the sum of matchings of length $n$ under
$wt_{A}$ is invariant under cyclically shifting the starting point. For the
following proof, recall our notation,

$i_{n}=\begin{cases}i\pmod{n}&n\nmid i\\\ n&n\mid i\end{cases}.$

######  Lemma 12.

Let $\mathcal{F}=\\{m_{i,j}\\}$ be an $n$-periodic infinite frieze pattern of
Type $\Lambda_{p_{1},\ldots,p_{s}}$ from a dissection $\mathcal{D}$ of
$A_{n,m}$ or $S_{n}$. Let
$w=\mathcal{P}_{i+1},\ldots,\mathcal{P}_{i+n}\in\mathbf{P}_{i,i+n+1}$, and let
$u=\mathcal{P}_{j+1},\ldots,\mathcal{P}_{j+n}\in\mathbf{P}_{j,j+n+1}$ be the
result of cyclically shifting $w$. Then, $wt_{A}(w)=wt_{A}(u)$.

In particular, for any $i,j\in\mathbb{Z}$,

$\sum_{w\in\mathbf{P}_{i,i+n+1}}wt_{A}(w)=\sum_{u\in\mathbf{P}_{j,j+n+1}}wt_{A}(u)$

###### Proof.

Recall that we define matchings for $A_{n,m}$ and $S_{n}$ on the infinite
strip. Let
$w=\mathcal{P}_{i+1},\ldots,\mathcal{P}_{i+n}\in\mathbf{P}_{i,i+n+1}$. For
each $i+1\leq k\leq i+n$, $\mathcal{P}_{k}$ is a lift of a polygon incident to
$v_{k_{n}}$ in $\mathcal{D}$; call this subgon
$\overline{\mathcal{P}_{k_{n}}}$. From $w$, we can construct a matching
$u=Q_{j+1},\ldots,Q_{j+n}\in\mathbf{P}_{j,j+n+1}$, by, for all
$j+1\leq\ell\leq j+n$, setting $Q_{\ell}$ to be the lift of
$\overline{\mathcal{P}_{\ell_{n}}}$ which is incident to $v_{\ell}$ in the
infinite strip. Then, by construction, for each $\mathcal{P}$ in the
dissection $\mathcal{D}$, the number of times we use a lift of $\mathcal{P}$
is the same in $w$ and $u$, so that $wt_{A}(w)=wt_{A}(u)$.

Moreover, this map between $w\in\mathbf{P}_{i,i+n+1}$ and
$u\in\mathbf{P}_{j,j+n+1}$ is a bijection. It immediately follows that the sum
over these two sets using $wt_{A}$ is equal. ∎

Next, we verify that $wt_{A}$ is unaffected by cutting ears.

######  Lemma 13.

Let $\mathcal{F}=\\{m_{i,j}\\}$ be an $n$-periodic infinite frieze pattern of
Type $\Lambda_{p_{1},\ldots,p_{s}}$ from a dissection $\mathcal{D}$ of
$A_{n,m}$ or $S_{n}$ and suppose the dissection has an ear of size $p$. Let
$\mathcal{F}^{\prime}=\\{m^{\prime}_{i,j}\\}$ be the $(n-(p-2))$-periodic
frieze pattern from the surface after cutting this ear. Then, for all
$i\in\mathbb{Z}$,

$\sum_{w\in\mathbf{P}_{i,i+n+1}}wt_{A}(w)=\sum_{w\in\overline{\mathbf{P}_{i,i+n-(p-2)+1}}}wt_{A}(w)$

where $\overline{\mathbf{P}_{i,i+n-(p-2)+1}}$ is the set of matchings between
$v_{i}$ and $v_{i+n-(p-2)}$ in the surface after cutting this ear.

###### Proof.

Let $S$ be the surface with dissection $\mathcal{D}$ which is the realization
of $\mathcal{F}$. Without loss of generality, label the vertices on the outer
boundary of the surface $v_{1},\ldots,v_{n}$ such that the ear of size $p$ has
vertices $v_{\ell},\ldots,v_{\ell+p-1}$ with $1\leq\ell<\ell+p\leq n$. Call
this ear $\widehat{P}$. Then, every matching
$w=\mathcal{P}_{1},\ldots,\mathcal{P}_{n}\in\mathbf{P}_{0,n+1}$ necessarily
has $\mathcal{P}_{\ell+1}=\cdots=\mathcal{P}_{\ell+p-2}=\widehat{P}$.
Moreover, in order for $w$ to have $wt_{A}(w)>0$, it must be that
$\mathcal{P}_{\ell}\neq\widehat{P}$ and
$\mathcal{P}_{\ell+p-1}\neq\widehat{P}$.

Let $\overline{\mathbf{P}_{0,n+(p-2)+1}}$ be the set of matchings between
$v_{1}$ and $v_{n+(p-2)}$ in the surface $\overline{S}$ which is the result of
cutting the ear $\widehat{P}$. Note that this surface now has $n-(p-2)$
vertices on the outer boundary. We have a bijection between
$w=\mathcal{P}_{1},\ldots,\mathcal{P}_{n}\in\mathbf{P}_{0,n+1}$ with
$\mathcal{P}_{\ell}\neq\widehat{P}$ and
$\mathcal{P}_{\ell+p-1}\neq\widehat{P}$, and
$u=Q_{1},\ldots,Q_{n-(p-2)}\in\overline{\mathbf{P}_{0,n+(p-2)+1}}$. Given such
a $w$, we set $u=Q_{1},\ldots,Q_{n-(p-2)}$ as

$Q_{i}=\begin{cases}\mathcal{P}_{i}&i\leq\ell\\\
\mathcal{P}_{i+(p-2)}&i>\ell.\end{cases}$

For any subgon $R\neq\widehat{P}$ in $\mathcal{D}$, the number of times $R$
appears in is the same as the number of times $R$ appears in $u$. In $w$,
$N_{\widehat{P}}=p-2$ and $U_{p-2}(\lambda_{p})=1=U_{0}(\lambda_{p})$. Thus,
$wt_{A}(w)=wt_{A}(u)$. Since we bijectively map every $w\in\mathbf{P}_{0,n+1}$
with $wt_{A}(w)>0$ to $u\in\overline{\mathbf{P}_{0,n+(p-2)+1}}$, the claim
follows for $i=1$. By Lemma 12, the claim is true for arbitrary $i$. ∎

With the preceding two results, we can conclude that the annulus weighting
gives a combinatorial interpretation of the principal growth coefficient.

######  Theorem 12.

Let $\mathcal{F}=\\{m_{i,j}\\}$ be an $n$-periodic infinite frieze pattern of
Type $\Lambda_{p_{1},\ldots,p_{s}}$ from a dissection $\mathcal{D}$ of
$A_{n,m}$ or $S_{n}$. Let $s_{1}$ be the principal growth coefficient of
$\mathcal{F}$. Then, for any $i\in\mathbb{Z}$,

$s_{1}=\sum_{w\in\mathbf{P}_{i,i+n+1}}wt_{A}(w)$

###### Proof.

For shorthand, let $S$ be the surface corresponding to $\mathcal{F}$. Since
$S$ has $n$ vertices on the outer boundary, the rows of $\mathcal{F}$ are
necessarily $n$-periodic. Theorem 10 guarantees that $m_{i,i+n+1}-m_{i+1,i+n}$
is constant for all $i\in\mathbb{Z}$; by definition, $s_{1}$ is this constant
difference. From Corollary 2, we know that
$m_{i,i+n+1}=\sum_{w\in\mathbf{P}_{i,i+n+1}}wt_{T}(w)$ and similarly
$m_{i+1,i+n}=\sum_{w\in\mathbf{P}_{i+1,i+n}}wt_{T}(w)$. Thus, it is sufficient
to prove the following relation amongst the weights of matchings,

$\sum_{w\in\mathbf{P}_{i,i+n+1}}wt_{A}(w)=\sum_{w\in\mathbf{P}_{i,i+n+1}}wt_{T}(w)-\sum_{w\in\mathbf{P}_{i+1,i+n}}wt_{T}(w).$
(6)

First, we claim that it is sufficient to consider the case of skeletal
dissections. From Lemma 13, we know that if $\mathcal{D}$ has an ear of size
$p$, then
$\sum_{w\in\mathbf{P}_{i,i+n+1}}wt_{A}(w)=\sum_{w\in\overline{\mathbf{P}_{i,i+n-(p-2)+1}}}wt_{A}(w)$
where $\overline{\mathbf{P}_{i,i+n-(p-2)+1}}$ is the set of matchings between
$v_{i}$ and $v_{i+n-(p-2)+1}$ in $\overline{S}$, the resulting surface with
dissection after cutting this ear.

Now, we claim that the right-hand side of Equation 6 is also invariant under
cutting ears,

$\sum_{w\in\mathbf{P}_{i,i+n+1}}wt_{T}(w)-\sum_{w\in\mathbf{P}_{i+1,i+n}}wt_{T}(w)=\sum_{w\in\overline{\mathbf{P}_{i,i+n-(p-2)+1}}}wt_{T}(w)-\sum_{w\in\overline{\mathbf{P}_{i+1,i+n-(p-2)}}}wt_{T}(w).$
(7)

For convenience, we relabel the vertices of $S$ so that $i=0$. Let $Q_{0}$ be
an ear of size $p$ in $\mathcal{D}$. If the boundary edge $(v_{n},v_{1})$ (the
edge starting at $v_{n}$ and traveling counterclockwise to $v_{1}$) is not an
edge of $Q_{0}$, then from Equation 4,
$\sum_{w\in\mathbf{P}_{0,n+1}}wt_{T}(w)=\sum_{w\in\overline{\mathbf{P}_{0,n-(p-2)+1}}}wt_{T}(w)$.
This same equation shows
$\sum_{w\in\mathbf{P}_{1,n}}wt_{T}(w)=\sum_{w\in\overline{\mathbf{P}_{1,n-(p-2)}}}wt_{T}(w)$
if neither $v_{1}$ nor $v_{n}$ are vertices of $Q_{0}$.

Next, consider $\sum_{w\in\mathbf{P}_{1,n}}wt_{T}(w)$ where $(v_{n},v_{1})$ is
not an edge of $Q_{0}$, but $Q_{0}$ has $v_{1}$ or $v_{n}$ as a vertex, or
possibly both. Then, the vertices of $Q_{0}$ are $a,\ldots,a+p$ with $1\leq
a\leq n-p$. Every
$w=\mathcal{P}_{2},\ldots,\mathcal{P}_{n-1}\in\mathbf{P}_{1,n}$ with
$wt_{T}(w)$ necessarily has
$\mathcal{P}_{a+1}=\cdots=\mathcal{P}_{a+p-1}=Q_{0}$. In order for
$wt_{T}(w)\neq 0$, if $a>1$, $\mathcal{P}_{a}\neq Q_{0}$ and if $a+p<n$,
$\mathcal{P}_{a+p}\neq Q_{0}$. Each such matching can be mapped with a weight-
preserving bijection to a matching in $\overline{\mathbf{P}_{1,n-(p-2)}}$ in
$S$ after removing the ear $Q_{0}$. Note that if $v_{1}$ and $v_{n}$ are both
vertices of $Q_{0}$, there is exactly one matching in $\mathbf{P}_{1,n}$,
$w=Q_{0},\ldots,Q_{0}$ which is length $n-2=|Q_{0}|-2$ times. It follows that
$wt_{L}(w)=1$. Moreover, since in this case $n=p$, we have that
$\overline{\mathbf{P}_{1,n-(p-2)}}=\overline{\mathbf{P}_{1,n}}=\\{\emptyset\\}$.
Since $wt_{L}(\emptyset)=1$, the claim still holds.

Now, assume that one of the edges of $Q_{0}$ is the boundary edge
$(v_{n},v_{1})$. Let $(v_{b},v_{a})$ be the unique non-boundary edge from
$\mathcal{D}$ which is an edge of $Q_{0}$. Note that the vertices of $Q_{0}$
are, in counterclockwise order,
$v_{b},v_{b+1},\ldots,v_{n},v_{1},\ldots,v_{a}$ for $b\leq n$ and $a\geq 1$;
in particular, $Q_{0}$ is a $(a+(n-b+1))$-gon. The vertices of $S$ after
cutting $Q_{0}$ are $v_{a},v_{a+1},\ldots,v_{b}$.

First, let $a>1$ and $b<n$. Necessarily, all
$w=\mathcal{P}_{1},\ldots,\mathcal{P}_{n}\in\mathbf{P}_{0,n+1}$ have
$\mathcal{P}_{1}=\cdots\mathcal{P}_{a-1}=Q_{0}$ and
$\mathcal{P}_{b+1}=\cdots=\mathcal{P}_{n}=Q_{0}$. The subgons in these two
sequences are distinct in the infinite strip $I$ but not distinct in the
surface $S$.

$Q_{0}$$v_{n}$$v_{1}$$v_{b}$$v_{a}$$v_{1}$$v_{a}$$v_{b}$$v_{n}$$Q_{0}$$Q_{0}$

Let $\mathbf{P}_{0,n+1}^{0,0}$ be the set of matchings with
$\mathcal{P}_{a}=Q_{0}$ and $\mathcal{P}_{b}=Q_{0}$. Let
$\mathbf{P}_{0,n+1}^{0,1}$ be the set of matchings with
$\mathcal{P}_{a}=Q_{0}$ and $\mathcal{P}_{b}\neq Q_{0}$. Define
$\mathbf{P}_{0,n+1}^{1,0}$ and $\mathbf{P}_{0,n+1}^{1,1}$ similarly. We define
$\mathbf{P}_{1,n}^{x,y}$ similarly. (Note this is slightly different than how
these indices were used in the proof of Theorem 9.)

Now we can expand the lefthand side of Equation 7 using these partitions of
each set of matchings and use Lemma 11 on like terms. First, we look at the
terms which have $\mathcal{P}_{a}=\mathcal{P}_{b}=Q_{0}$.

$\displaystyle\sum_{w\in\mathbf{P}_{0,n+1}^{0,0}}wt_{T}(w)-\sum_{w\in\mathbf{P}_{1,n}^{0,0}}wt_{T}(w)$
$\displaystyle=U_{a}(\lambda_{|Q_{0}|})U_{n-b+1}(\lambda_{|Q_{0}|})\sum_{w\in\mathbf{P}_{a,b}}wt_{T}(w)-U_{a-1}(\lambda_{|Q_{0}|})U_{n-b}(\lambda_{|Q_{0}|})\sum_{w\in\mathbf{P}_{a,b}}wt_{T}(w)$
$\displaystyle=U_{a+(n-b+1)}(\lambda_{|Q_{0}|})\sum_{w\in\mathbf{P}_{a,b}}wt_{T}(w)=-\sum_{w\in\mathbf{P}_{a,b}}wt_{T}(w)$

We do similar computations for the other sets $\mathbf{P}_{0,n+1}^{x,y}$ and
$\mathbf{P}_{1,n}^{x,y}$. Let $\mathbf{P}_{a,b+1}^{-}$ be the set of matchings
which do not use $Q_{0}$ at $v_{b}$.

$\displaystyle\sum_{w\in\mathbf{P}_{0,n+1}^{0,1}}wt_{T}(w)-\sum_{w\in\mathbf{P}_{1,n}^{0,1}}wt_{T}(w)$
$\displaystyle=U_{a}(\lambda_{|Q_{0}|})U_{n-b}(\lambda_{|Q_{0}|})\sum_{w\in\mathbf{P}_{a,b+1}^{-}}wt_{T}(w)-U_{a-1}(\lambda_{|Q_{0}|})U_{n-b-1}(\lambda_{|Q_{0}|})\sum_{w\in\mathbf{P}_{a,b+1}^{-}}wt_{T}(w)$
$\displaystyle=U_{a+(n-b)}(\lambda_{|Q_{0}|})\sum_{w\in
w\in\mathbf{P}_{a,b+1}^{-}}wt_{T}(w)=0$

We have $U_{a+(n-b)}(\lambda_{|Q_{0}|})=0$ since $|Q_{0}|=a+(n-b+1)$. We can
analogously conclude

$\displaystyle\sum_{w\in\mathbf{P}_{0,n+1}^{1,0}}wt_{T}(w)-\sum_{w\in\mathbf{P}_{1,n}^{1,0}}wt_{T}(w)=0$

Finally, we consider $\mathbf{P}_{0,n+1}^{1,1}$ and $\mathbf{P}_{1,n}^{1,1}$.
Note that the set of matchings in $\mathbf{P}_{a-1,b+1}$ which do not use
$Q_{0}$ at $v_{a}$ or $v_{b}$ is equivalent to
$\overline{\mathbf{P}_{b,a+n-(p-2)}}$, the set of matchings between $v_{b}$
and $v_{a}$ in $\overline{S}$ after cutting $Q_{0}$. This is true since in
$\overline{S}$, $v_{a}$ and $v_{b}$ are neighbors. Then,

$\displaystyle\sum_{w\in\mathbf{P}_{0,n+1}^{1,1}}wt_{T}(w)-\sum_{w\in\mathbf{P}_{1,n}^{1,1}}wt_{T}(w)$
$\displaystyle=U_{a-1}(\lambda_{|Q_{0}|})U_{n-b}(\lambda_{|Q_{0}|})\sum_{w\in\overline{\mathbf{P}_{b,a}}}wt_{T}(w)-U_{a-1}(\lambda_{|Q_{0}|})U_{n-b-1}(\lambda_{|Q_{0}|})\sum_{w\in\overline{\mathbf{P}_{b,a}}}wt_{T}(w)$
$\displaystyle=U_{a+(n-b)-1}(\lambda_{|Q_{0}|})\sum_{w\in\overline{\mathbf{P}_{b,a+n-(p-2)}}}wt_{T}(w)=\sum_{w\in\overline{\mathbf{P}_{b,a+n-(p-2)}}}wt_{T}(w).$

Since $\mathbf{P}_{0,n+1}$ is partitioned by the sets
$\mathbf{P}_{0,n+1}^{x,y}$ for $x,y\in\\{0,1\\}$ and similarly for
$\mathbf{P}_{1,n}$, we conclude that

$\displaystyle\sum_{w\in\mathbf{P}_{0,n+1}}wt_{T}(w)-\sum_{w\in\mathbf{P}_{1,n}}wt_{T}(w)=\sum_{w\in\overline{\mathbf{P}_{b,a+n-(p-2)}}}wt_{T}(w)-\sum_{w\in\mathbf{P}_{a,b}}wt_{T}(w)$
$\displaystyle=\sum_{w\in\overline{\mathbf{P}_{b,a+n-(p-2)}}}wt_{T}(w)-\sum_{w\in\overline{\mathbf{P}_{a,b}}}wt_{T}(w).$

where $\mathbf{P}_{a,b}=\overline{\mathbf{P}_{a,b}}$ since $Q_{0}$ is not
incident to any vertices strictly between $v_{a}$ and $v_{b}$ when traveling
counterclockwise. This is the desired result since the vertices remaining
after cutting $Q_{0}$ are $v_{a},v_{a+1},\ldots,v_{b-1},v_{b}$.

Now we turn to the $a=1$ or $b=n$ case. Suppose that $a=1$. We cannot also
have $b=n$ simultaneously since $(v_{b},v_{a})$ is a non-boundary arc incident
to $Q_{0}$. Note that in this case, $|Q_{0}|=n-b+2$.

$Q_{0}$$v_{n}$$v_{1}=v_{a}$$v_{b}$$v_{1}=v_{a}$$v_{b}$$v_{n}$$v_{n+1}$$v_{b-n}$$Q_{0}$$Q_{0}$

In this case, since $v_{b+1},\ldots,v_{n}$ are only incident to $Q_{0}$, any
$w=\mathcal{P}_{1},\ldots,\mathcal{P}_{n}\in\mathbf{P}_{0,n+1}$ has
$\mathcal{P}_{b+1}=\cdots=\mathcal{P}_{n}=Q_{0}$. In order for $wt_{T}(w)>0$,
it must be that $\mathcal{P}_{b}\neq Q_{0}$. Thus, using the same notation as
above, the only subsets of $\mathbf{P}_{0,n+1}$ which contain matchings with
nonzero weight are $\mathbf{P}_{0,n+1}^{0,1}$ and $\mathbf{P}_{0,n+1}^{1,1}$.
We sum over all matchings with nonzero traditional weight in
$\mathbf{P}_{0,n+1}$, again letting $\mathbf{P}_{1,b+1}^{-}$ be the subset of
$\mathbf{P}_{1,b+1}$ which does not choose $Q_{0}$ at $b$, and similarly for
$\mathbf{P}_{0,b+1}^{-}$.

$\displaystyle\sum_{w\in\mathbf{P}_{0,n+1}}wt_{T}(w)=\sum_{w\in\mathbf{P}_{0,n+1}^{0,1}}wt_{T}(w)+\sum_{w\in\mathbf{P}_{0,n+1}^{1,1}}wt_{T}(w)$
$\displaystyle=U_{1}(\lambda_{|Q_{0}|})U_{(n-b)}(\lambda_{|Q_{0}|})\sum_{v\in\mathbf{P}_{1,b+1}^{-}}wt_{T}(v)+U_{(n-b)}(\lambda_{|Q_{0}|})\sum_{v\in\mathbf{P}_{0,b+1}^{-}}wt_{T}(v)$
$\displaystyle=\lambda_{|Q_{0}|}\sum_{v\in\mathbf{P}_{1,b+1}^{-}}wt_{T}(v)+\sum_{v\in\mathbf{P}_{0,b+1}^{-}}wt_{T}(v)$

Since $a=1$, matchings in $\mathbf{P}_{1,n}$ no longer choose a subgon at
$v_{a}$. Thus, we will alter our notation. Let $\mathbf{P}_{1,n}^{0}$ be the
subset of matchings $w=\mathcal{P}_{2},\ldots,\mathcal{P}_{n-1}$ with
$\mathcal{P}_{b}=Q_{0}$ and we define $\mathbf{P}_{1,n}^{1}$ to be the subset
of matchings with $\mathcal{P}_{b}\neq Q_{0}$.

$\displaystyle\sum_{w\in\mathbf{P}_{1,n}}wt_{T}(w)=\sum_{w\in\mathbf{P}_{1,n}^{0}}wt_{T}(w)+\sum_{w\in\mathbf{P}_{1,n}^{1}}wt_{T}(w)$
$\displaystyle=U_{(n-b)}(\lambda_{|Q_{0}|})\sum_{v\in\mathbf{P}_{1,b}}wt_{T}(v)+U_{(n-b)-1}(\lambda_{|Q_{0}|})\sum_{v\in\mathbf{P}_{1,b+1}^{-}}wt_{T}(v)$
$\displaystyle=\sum_{v\in\mathbf{P}_{1,b}}wt_{T}(v)+\lambda_{|Q_{0}|}\sum_{v\in\mathbf{P}_{1,b+1}^{-}}wt_{T}(v)$

where by Lemma 9,
$U_{(n-b)-1}(\lambda_{|Q_{0}|})=U_{1}(\lambda_{|Q_{0}|})=\lambda_{|Q_{0}|}$.

Thus, in the $a=1$ case, we conclude

$\displaystyle\sum_{w\in\mathbf{P}_{0,n+1}}wt_{T}(w)-\sum_{w\in\mathbf{P}_{1,n}}wt_{T}(w)=\sum_{v\in\mathbf{P}_{0,b+1}^{-}}wt_{T}(v)-\sum_{v\in\mathbf{P}_{1,b}}wt_{T}(v)$
$\displaystyle=\sum_{w\in\overline{\mathbf{P}_{b,1}}}wt_{T}(w)-\sum_{w\in\overline{\mathbf{P}_{1,b}}}wt_{T}(w).$

The $b=n$ case is identical.

Now we know both sides of the desired equality in Equation 6 are invariant
under removing ears from the dissection. Hence, it suffices to consider how
$wt_{T}$ and $wt_{A}$ compare in a skeletal dissection.

We will keep our vertex labels so that we consider
$\sum_{w\in\mathbf{P}_{0,n+1}}wt_{A}(w)$. Let $\hat{P}$ be the subgon which
contains the boundary edge $(v_{n},v_{1})$. Given a matching
$w=\mathcal{P}_{1},\ldots,\mathcal{P}_{n}\in\mathbf{P}_{0,n+1}$, we only have
$wt_{T}(w)\neq wt_{A}(w)$ if $\mathcal{P}_{1}=\mathcal{P}_{n}=\hat{P}$. This
is the case since, for any other subgon $Q$ in $\mathcal{D}$, there is only
one lift of $Q$ in the infinite strip between $v_{1}$ and $v_{n}$; in the
notation of Definitions 19 and 20, $k_{Q}=N_{Q}$. However, in the infinite
strip, $v_{1}$ and $v_{n}$ are adjacent to separate lifts of $\hat{P}$.

Let
$\widehat{\mathbf{P}_{0,n+1}}=\\{w=\mathcal{P}_{1},\ldots,\mathcal{P}_{n}\in\mathbf{P}_{0,n+1}:\mathcal{P}_{1}=\mathcal{P}_{n}=\hat{P}\\}$.
Note that in some dissections
$\widehat{\mathbf{P}_{0,n+1}}=\mathbf{P}_{0,n+1}$. We have a bijection,
$\varphi$, between $\widehat{\mathbf{P}_{0,n+1}}$ and $\mathbf{P}_{1,n}$,
given by sending $w\in\widehat{\mathbf{P}_{0,n+1}}$ to $w[2:n-1]$. This
bijection is not necessarily weight-preserving under any weighting.

Recall that, given $w=\mathcal{P}_{i+1},\ldots,\mathcal{P}_{j-1}$, for
$i+1\leq a\leq b\leq j-1$, $w[a:b]=\mathcal{P}_{a},\ldots,\mathcal{P}_{b}$.
For each $w\in\widehat{\mathbf{P}_{0,n+1}}$, let $a_{w}\geq 1$ be the largest
possible integer such that all subgons in $w[1:a_{w}]$ are $\hat{P}$.
Similarly, define $b_{w}\leq n$ to be the smallest possible integer such that
all subgons in $w[b_{w}:n]$ are $\hat{P}$. Then, for each
$w\in\widehat{\mathbf{P}_{0,n+1}}$,
$wt_{T}(w)=U_{a_{w}}(\lambda_{|\hat{P}|})U_{n-b_{w}+1}(\lambda_{|\hat{P}|})wt_{T}(w[a_{w}+1,b_{w}-1])$
and
$wt_{T}(\varphi(w))=U_{a_{w}-1}(\lambda_{|\hat{P}|})U_{n-b_{w}}(\lambda_{|\hat{P}|})wt_{T}(w[a_{w}+1,b_{w}-1])$.
This allows us to reduce the following, using Lemma 11,

$\displaystyle\sum_{w\in\widehat{\mathbf{P}_{0,n+1}}}wt_{T}(w)-\sum_{w\in\mathbf{P}_{1,n}}wt_{T}(w)$
$\displaystyle=\sum_{w\in\widehat{\mathbf{P}_{0,n+1}}}\bigg{(}U_{a_{w}}(\lambda_{|\hat{P}|})U_{n-b_{w}+1}(\lambda_{|\hat{P}|})-U_{a_{w}-1}(\lambda_{|\hat{P}|})U_{n-b_{w}}(\lambda_{|\hat{P}|})\bigg{)}wt_{T}(w[a_{w}+1,b_{w}-1])$
$\displaystyle=\sum_{w\in\widehat{\mathbf{P}_{0,n+1}}}U_{a_{w}+(n-b_{w}+1)}(\lambda_{|\hat{P}|})wt_{T}(w[a_{w}+1,b_{w}-1])$

Since $\hat{P}$ is not an ear, $U_{a_{w}+(n-b_{w}+1)}(\lambda_{|\hat{P}|})\geq
0$.

Let $\mathbf{P}_{0,n+1}^{\prime}$ be the complement of
$\widehat{\mathbf{P}_{0,n+1}}$ in $\mathbf{P}_{0,n+1}$. Then, we can conclude
that

$\displaystyle\sum_{w\in\mathbf{P}_{0,n+1}}wt_{T}(w)-\sum_{w\in\mathbf{P}_{1,n}}wt_{T}(w)$
$\displaystyle=\sum_{w\in\mathbf{P}_{0,n+1}^{\prime}}wt_{T}(w)+\sum_{w\in\widehat{\mathbf{P}_{0,n+1}}}U_{a_{w}+(n-b_{w}+1)}(\lambda_{|\hat{P}|})wt_{T}(w[a_{w}+1,b_{w}-1])$

As discussed, if $w\in\mathbf{P}_{0,n+1}^{\prime}$, $wt_{T}(w)=wt_{A}(w)$. If
$w\in\widehat{\mathbf{P}_{0,n+1}}$, so that for some integers $a_{w}\geq
1,b_{w}\leq n$, the submatchings $w[1,a_{w}]$ and $w[b_{w},n]$ both only use
the subgon $\hat{P}$, then
$wt_{A}(w)=U_{a_{w}+(n-b_{w}+1)}(\lambda_{|\hat{P}|})wt_{T}(w[a_{w}+1,b_{w}-1])$.
Thus, we have shown Equation 6. ∎

Knowledge of the growth coefficient allows us to refine Step 5 of our
algorithm in Section 5.4. If a skeletal frieze pattern satisfies Proposition
2, then it is either realizable by a dissection of $A_{n,m}$ or $S_{n}$. In
order to determine which is the correct surface, one needed to check the
conditions listed in Lemma 6. Checking the first growth coefficient is an
alternative way to differentiate between frieze patterns of these two
surfaces.

######  Proposition 3.

An infinite frieze pattern $\mathcal{F}$ of Type
$\Lambda_{p_{1},\ldots,p_{s}}$ which satisfies Proposition 2 is realizable by
a dissection of $S_{n}$ if and only if its first growth coefficient is 2.

###### Proof.

Note that if $wt_{A}(w)\neq 0$, then by Lemma 9, $wt_{A}(w)\geq 1$. Therefore,
if a frieze pattern has growth coefficient 2, then it has one or two matchings
which have nonzero weight under $wt_{A}$. We claim that every skeletal
dissection has at least two matchings which satisfy $wt_{A}(w)>0$. We
establish $w_{CW}$ to be the matching where each vertex uses the clockwise-
most subgon it is incident to. In a skeletal dissection, a subgon of size $p$
can have up to $p-1$ vertices incident to the outer boundary. However, no
subgon of size $p$ would appear $p-1$ times in $w_{CW}$. Thus,
$wt_{A}(w_{CW})>0$. We can similarly define $w_{CCW}$ by choosing the
counterclockwise-most subgon at each vertex. Since $w_{CW}$ and $w_{CCW}$ are
distinct matchings, we know that for any infinite, realizable, and skeletal
frieze pattern $\mathcal{F}$, $s_{1}\geq 2$.

In order for $s_{1}=2$, it must be that $wt_{A}(w_{CW})=wt_{A}(w_{CCW})=1$ and
for any other matching $w\in\mathbf{P}_{i,i+n+1},wt_{A}(w)=0$. This would
require that, if $P$ is a subgon of the dissection, then $P$ has $|P|-1$
vertices on the outer boundary. Otherwise, we would have more options for
nonzero matchings. Such a dissection is only possible on $S_{n}$.

For the converse direction, note that every skeletal dissection of $S_{n}$
looks as above; in the triangulation case, this is sometimes called a “wheel”.
When forming a matching of a skeletal dissection of $S_{n}$, once we choose
whether to use the clockwise or counterclockwise most subgon at a vertex of
degree 2 in a matching, the rest of the matching is forced since we must avoid
using $p-1$ vertices of a size $p$ subgon. Such a matching uses a size $p$
subgon $p-2$ times, and $U_{p-2}(\lambda_{p})=1$; therefore, this matching
will have weight 1. ∎

By Lemma 13, the growth coefficient of a realizable frieze pattern is
unaffected by cutting or gluing ears. Thus, we conclude with a more general
Corollary.

######  Corollary 3.

Let $\mathcal{F}$ be a frieze pattern of Type $\Lambda_{p_{1},\ldots,p_{s}}$
which terminates in step 5 of the Realizability Algorithm. Then, $\mathcal{F}$
is realizable by a dissection of $S_{n}$ if and only if first growth
coefficient of $\mathcal{F}$ is 2.

We have thus far only provided a combinatorial interpretation of the first
growth coefficient for a frieze pattern $\mathcal{F}$ realizable by a
dissection $\mathcal{D}$ of $S_{n}$ or $A_{n,m}$. For higher growth
coefficients, one could consider $wt_{A}$ on matchings in the “$k$-th power of
$\mathcal{D}$”.

######  Definition 21.

Let $\mathcal{D}$ be a dissection of $A_{n,m}$ or $S_{n}$, and let $I$ be the
infinite strip which is the universal cover of the surface, with fundamental
domain $F$. For $k\geq 1$, we construct _the $k^{th}$ power of $\mathcal{D}$
on $A_{kn,km}$ or $S_{kn}$_ by considering the dissected surface resulting in
taking $k$-consecutive copies of $F$ as a fundamental domain.

######  Example 7.

The righthand side is the second power of the dissection on the lefthand side.

$a$$c$$b$ | $a_{1}$$a_{2}$$b_{1}$$c_{1}$$b_{2}$$c_{2}$
---|---

If we instead consider $\mathcal{F}$ to have period $kn$, then the first
growth coefficient is the same as the $k$-th growth coefficient of
$\mathcal{F}$ when viewing it as $n$-periodic. Thus, we can recover higher
growth coefficients of $\mathcal{F}$, realizable by $\mathcal{D}$, by using
the $k$-th power of $\mathcal{D}$ and Theorem 12.

######  Corollary 4.

Let $\mathcal{F}$ be an infinite frieze pattern of Type
$\Lambda_{p_{1},\ldots,p_{s}}$ which is realizable by a dissection
$\mathcal{D}$ of $A_{n,m}$ or $S_{n}$. Then for any $i\in\mathbb{Z}$,

$s_{k}=\sum_{w\in\mathbf{P}_{i,i+kn+1}}wt_{A}(w)$

where we sum over matchings in the $k$-th power of $\mathcal{D}$.

###  7.1 Comparing Inner and Outer Frieze Patterns

Given a dissected annulus, we could consider the frieze pattern determined
from the inner boundary instead of the outer boundary. To differentiate these,
given a dissected annulus $A_{n,m}$, let $\mathcal{F}_{out}$ be the frieze
pattern determined with respect to the outer boundary and let
$\mathcal{F}_{in}$ be the frieze pattern determined with respect to the inner
boundary. For example, for the dissection below, $\mathcal{F}_{out}$ is
generated by the quiddity cycle $(1+2\sqrt{2},2+2\sqrt{2})$ and
$\mathcal{F}_{in}$ is generated by
$(1+2\sqrt{2},\sqrt{2},2\sqrt{2},2\sqrt{2},\sqrt{2},1+\sqrt{2})$.

The goal in this section is to use our matching formula to give a
combinatorial proof that, given a dissected annulus, the first growth
coefficients of $\mathcal{F}_{in}$ and $\mathcal{F}_{out}$ are equal. By
Theorem 11, it will follow that all growth coefficients of these two frieze
patterns are equal. See [3] for a module-theoretic approach to this problem
for triangulated annuli.

First, we record a useful lemma. This lemma is a consequence of the fact that
all finite frieze patterns have glide symmetry, shown by Coxeter in [11].

######  Lemma 14.

Let $\mathcal{F}=\\{m_{i,j}\\}$ be a finite frieze pattern of Type
$\Lambda_{p_{1},\ldots,p_{s}}$ which comes from a dissected polygon. Then,
$m_{i,j}=m_{j,i+n}$, implying that

$\sum_{w\in\mathbf{P}_{i,j}}wt_{T}(w)=\sum_{w\in\mathbf{P}_{j,i+n}}wt_{T}(w).$

######  Remark 6.

  1. 1.

Another way to think about Lemma 14 is that the weighted sum of matchings
between vertices $v_{i}$ and $v_{j}$ is the same going clockwise or
counterclockwise.

  2. 2.

We can strengthen Lemma 14 to a weighted bijection between matchings in
$\mathbf{P}_{i,j}$ and in $\mathbf{P}_{i,j}$ with nonzero $wt_{T}$. Given a
dissection of a polygon $P$ into subgons
$\mathcal{P}_{1},\ldots,\mathcal{P}_{k}$, form a multiset $S$ by taking
$|\mathcal{P}_{i}|-2$ copies of $\mathcal{P}_{i}$ for all $1\leq i\leq k$.
Then, the multiset of subgons used in any matching $w\in\mathbf{P}_{i,j}$ such
that $wt_{T}(w)>0$ is a subset of $S$. Call this subset $T_{w}$. One can show
there is a unique matching $\bar{w}\in\mathbf{P}_{j,i}$ which uses the subgons
in $S-T_{w}$. By repeated application of Lemma 9, these two matchings will
always have the same weight.

We are ready for the main result in this subsection.

######  Theorem 13.

Let $A_{n,m}$ be an annulus with dissection $\mathcal{D}$. Let
$\mathcal{F}_{in}$ and $\mathcal{F}_{out}$ be the infinite frieze patterns
with respect to the inner and outer boundaries respectively. Let $s_{1}^{in}$
and $s_{1}^{out}$ be the first growth coefficients of $\mathcal{F}_{in}$ and
$\mathcal{F}_{out}$ respectively. Then $s_{1}^{in}=s_{1}^{out}$.

###### Proof.

From Lemma 13, the sum $\sum_{w\in\mathbf{P}_{i,i+n+1}}wt_{A}(w)$ is unchanged
by cutting ears from the surface. Therefore, we can consider only the case
where $\mathcal{D}$ is skeletal.

Recall that in our construction for the infinite cover of a dissection of
$A_{n,m}$ in Section 4.1, we picked a bridging arc $\tau$, cut along this arc
to form a dissected $(n+m+2)$-gon, then glued copies of this subgon together
along $\tau$ to form a dissected infinite strip. Call the dissected polygon
$F$. Let $v_{1},w_{1}$ be the end points of $\tau$ on the outer and inner
boundary respectively, and recall our labeling of vertices of $F$ as below.

$v_{n}$$v_{1}$$v_{2}$$w_{1}$$w_{2}$$\tau$ |  |  | $\tau$$\tau$$v_{1}^{1}$$v_{2}^{1}$$v_{n}^{0}$$v_{1}^{2}$$v_{2}^{2}$$v_{n}^{1}$$w_{1}^{1}$$w_{2}^{1}$$w_{1}^{2}$$w_{2}^{2}$$\cdots$
---|---|---|---

By Theorem 12, $s_{1}^{out}=\sum_{w\in\mathbf{P}_{0,n+1}^{out}}wt_{A}(w)$ and
$s_{1}^{in}=\sum_{w\in\mathbf{P}_{0,m+1}^{in}}wt_{A}(w)$, where the second sum
is over matchings on the inner boundary. We will see that the set of matchings
$\mathbf{P}_{0,n+1}^{out}$ and $\mathbf{P}_{0,m+1}^{in}$ come from several
sets of matchings on $F$.

Consider first the sets $\mathbf{P}_{v_{1}^{1},w_{1}^{2}}$ and
$\mathbf{P}_{w_{1}^{2},v_{1}^{1}}$, where these matchings occur on $F$. A
matching $\bar{w}\in\mathbf{P}_{v_{1}^{1},w_{1}^{2}}$ lifts to a matching on
the outer boundary of the annulus, $w\in\mathbf{P}_{1,n+2}^{out}$ such that
$w[n+1:n+1]$ is a subgon incident to $v_{1}$ counterclockwise of $\tau$. Note
that $wt_{T}(\bar{w})=wt_{A}(w)$. A matching
$\bar{u}\in\mathbf{P}_{w_{1}^{2},v_{1}^{1}}$ lifts to a matching in clockwise
order on the inner boundary of the annulus,
$u=\mathcal{P}_{m},\ldots,\mathcal{P}_{1}$ such that $\mathcal{P}_{1}$ is a
subgon counterclockwise of $\tau$. Since reversing the order of a matching
will not affect its weight, we see that the matchings $\bar{u}$ lift to the
subset of $w\in\mathbf{P}_{0,m}^{in}$ where $w[1:1]$ is a subgon
counterclockwise of $\tau$, and for each $\bar{u}$,
$wt_{T}(\bar{u})=wt_{A}(u)$.

We can similarly compare $\mathbf{P}_{w_{1}^{1},v_{1}^{2}}$ and
$\mathbf{P}_{v_{1}^{2},w_{1}^{1}}$. A matching
$\bar{w}\in\mathbf{P}_{v_{1}^{2},w_{1}^{1}}$ will lift to a matching on the
outer boundary in clockwise order. When we reverse the order, we have a
matching in $w\in\mathbf{P}_{0,n+1}^{out}$ where $w[1:1]$ is a subgon
clockwise of $\tau$. Moreover, by Lemma 12, we can cyclically shift $w$ so
that $w\in\mathbf{P}_{1,n+2}^{out}$; now, $w[n+1,n+1]$ is a subgon clockwise
of $\tau$. We have again $wt_{T}(\bar{w})=wt_{A}(w)$.

A matching $\bar{u}\in\mathbf{P}_{v_{1}^{2},w_{1}^{1}}$ will lift to a
matching $u\in\mathbf{P}_{1,n+2}^{in}$ such that $u[n+1:n+1]$ is a subgon
counterclockwise of $\tau$. We shift $u$ so that $u\in\mathbf{P}_{0,m}^{in}$,
now with $u[1:1]$ a subgon clockwise of $\tau$. We have that
$wt_{T}(\bar{u})=wt_{A}(u)$.

We can moreover see that, by following this procedure, every matching
$w\in\mathbf{P}_{1,n+2}^{out}$ is a lift of a unique matching in either
$\mathbf{P}_{v_{1}^{2},w_{1}^{1}}$ or $\mathbf{P}_{v_{1}^{1},w_{1}^{2}}$,
depending on which side of $\tau$ the subgon $w[n+1:n+1]$ lies. Since our
lifts also preserved weight, we have that

$\sum_{w\in\mathbf{P}_{v_{1}^{1},w_{1}^{2}}}wt_{T}(w)+\sum_{w\in\mathbf{P}_{v_{2}^{1},w_{1}^{1}}}wt_{T}(w)=\sum_{w\in\mathbf{P}_{1,n+2}^{out}}wt_{A}(w)$

We can similarly conclude

$\sum_{w\in\mathbf{P}_{w_{1}^{2},v_{1}^{1}}}wt_{T}(w)+\sum_{w\in\mathbf{P}_{w_{1}^{1},v_{2}^{1}}}wt_{T}(w)=\sum_{w\in\mathbf{P}_{0,m}^{in}}wt_{A}(w)$

By Lemma 14, we have

$\sum_{w\in\mathbf{P}_{v_{1}^{1},w_{1}^{2}}}wt_{T}(w)+\sum_{w\in\mathbf{P}_{v_{2}^{1},w_{1}^{1}}}wt_{T}(w)=\sum_{w\in\mathbf{P}_{w_{1}^{2},v_{1}^{1}}}wt_{T}(w)+\sum_{w\in\mathbf{P}_{w_{1}^{1},v_{2}^{1}}}wt_{T}(w)$

and by our combinatorial interpretation of $s_{1}$ in Theorem 12, the claim of
the Theorem follows.

∎

##  8 Connection To T-Paths

In the case of triangulated surfaces, it is known that matchings are in
bijection with several other combinatorial objects. One such set of objects is
_T-paths_.

T-paths on polygons were first developed in [18] to give a combinatorial proof
of positive of cluster variables in type $A$. See [12] for the definition of a
cluster algebra. Carroll and Price show a bijection between T-paths, with all
edges weight 1, and BCI tuples in the case of a polygon [9], [17]. The
definition of T-path was expanded to arbitrary triangulated surfaces in [19]
and [16].

Recently, Çanakçi and Jørgensen extended the definition of T-paths to the case
of dissected polygon. We give their definition below.

######  Definition 22.

[Weak T-path in a Dissected Polygon [8]] Given an $n$-gon with vertices
$v_{1},\ldots,v_{n}$ and dissection $\mathcal{D}$, let $v_{i},v_{j}$ be a
distinct pair of vertices. A _weak T-path_ , $\alpha$, from $v_{i}$ to $v_{j}$
is a walk $\alpha_{1},\ldots,\alpha_{2k+1}$, with the following properties.

  1. 1.

No step $\alpha_{i}$ crosses an arc in $\mathcal{D}$.

  2. 2.

Each $\alpha_{i}$ is equipped with an orientation, such that

     * •

$s(\alpha_{1})=v_{i}$

     * •

$t(\alpha_{2k+1})=v_{j}$

     * •

For all $1\leq i<2k+1$, $t(\alpha_{i})=s(\alpha_{i+1})$.

  3. 3.

The even steps, $\alpha_{2\ell}$, each cross $(v_{i},v_{j})$.

  4. 4.

For all $\ell$, the crossing point of $(v_{i},v_{j})$ and $\alpha_{2\ell}$ is
closer to $v_{i}$ then the crossing point of $(v_{i},v_{j})$ and
$\alpha_{2(i+1)}$.

  5. 5.

The steps $\alpha_{i}$, considered without orientation, are pairwise distinct.

The third condition of Definition 22 explains why a weak T-path must have odd
length while the fourth and fifth prevent backtracking. From now on, we will
simply refer to these as T-paths,

######  Remark 7.

One can equivalently define _complete T-paths_. In a polygon $P$ with vertices
$v_{1},\ldots,v_{n}$ and dissection $\mathcal{D}$, consider the diagonal
$(v_{i},v_{j})$. Orienting this diagonal from $v_{i}$ to $v_{j}$, let
$\tau_{1},\ldots,\tau_{d}$ be the ordered list of arcs of $\mathcal{D}$ that
this diagonal crosses. Then a complete T-path $\alpha$ from $v_{i}$ to $v_{j}$
is a walk $\alpha_{1},\ldots,\alpha_{2d+1}$ which satisfies items 1 and 2 in
Definition 22 as well as

  1. 3’

The step $\alpha_{2\ell}$ goes along $\tau_{\ell}$.

A complete T-path will often violate condition 5 in Definition 22 as two
consecutive steps may go along the same edge in $\mathcal{D}$.

If we set a weight on each diagonal of a polygon (not just those in the
dissection), we can weight T-paths by taking a ratio of weights on the
diagonals making up the T-path.

######  Definition 23 (Weight of a T-path).

Given an $n$-gon with vertices $v_{1},\ldots,v_{n}$ and dissection
$\mathcal{D}$, and a T-path $\alpha$ from $v_{i}$ to $v_{j}$,
$\alpha=\alpha_{1},\ldots,\alpha_{2k+1}$, we define the weight of $\alpha$ as

$wt(\alpha)=\frac{\displaystyle\prod_{j=0}^{k}wt(\alpha_{2j+1})}{\displaystyle\prod_{j=1}^{k}wt(\alpha_{2j})}$

Given a polygon $P$ with dissection $\mathcal{D}$, we will set
$wt((v,w))=U_{k}(\lambda_{p})$ whenever $(v,w)$ is a $k$-diagonal of a subgon
of size $p$. Note that $U_{0}(x)=1$, so this weighting sets all diagonals in
$\mathcal{D}$ and all boundary edges to weight 1.

######  Remark 8.

In [8], the authors define a map $f$ on diagonals of a polygon to be a
_frieze_ if it satisfies the Ptolemy relation: given vertices on a polygon
$a,b,c,d,$ which appear in this cyclic order, so that $(a,c)$ and $(b,d)$ are
crossing, $f$ satisfies,

$f((a,c))f((b,d))=f((a,b))f((c,d))+f((a,d))f((b,c))$

They define a _weak frieze_ to be a map which satisfies the Ptolemy relation
when $(b,d)\in D$.

One of their main results (Theorem A) is that a map $f$ on diagonals of a
polygon $P$ is a weak frieze if and only if it satisfies the $T$-path formula.
However, our choice of weighting will in fact be a _frieze_ , as it is
equivalent to the frieze pattern discussed in [14].

Our main result of this section is that there is a bijection between T-paths
in a dissected polygon and matchings with nonzero traditional weight. Denote
the set of T-paths from $v_{i}$ to $v_{j}$, where these vertices are distinct,
as $\textbf{T}_{i,j}$.

######  Proposition 4.

Let $P$ be a polygon with vertices $v_{1},\ldots,v_{n}$. Let $v_{i},v_{j}$ be
any pair of distinct vertices. Then, there is a bijection,
$\Phi:\\{w\in\mathbf{P}_{i,j}:wt_{T}(w)\neq 0\\}\to\textbf{T}_{i,j}$ such
that, if $\Phi(w)=\pi$, then $wt_{T}(w)=wt(\pi)$.

###### Proof.

Throughout this proof, we will assume that $(v_{i},v_{j})$ crosses all arcs in
$\mathcal{D}$. For both a matching and a T-path, if this was not the case, we
could work with the portion of $P$ which is only dissected by arcs which cross
$(v_{i},v_{j})$, then paste the remaining subgons. Accordingly, let
$\tau_{1},\ldots,\tau_{d}$ be the arcs in $\mathcal{D}$, with order imposed by
the order in which the arc $(v_{i},v_{j})$, oriented from $v_{i}$ to $v_{j}$,
crosses them. By the same procedure we also index the subgons of the
dissection in order, $Q_{0},\ldots,Q_{d}$. By our assumption that
$(v_{i},v_{j})$ crosses all arcs in $\mathcal{D}$, $Q_{0}$ and $Q_{d}$ are
ears.

$v_{i}$$v_{j}$$v_{a}$$v_{b}$$Q_{0}$$Q_{1}$$Q_{2}$$Q_{3}$$Q_{d}$$\cdots$$Q_{d-1}$
Figure 9: A typical configuration for a diagonal in a dissected polygon. We
only consider the subgons which the diagonal crosses.

We define the image $\Phi(w)$ with the following claim.

Claim: Given a matching $w\in\mathbf{P}_{i,j}$, there exists a complete T-path
$\alpha=\alpha_{1},\ldots,\alpha_{2d+1}$ such that the number of vertices of
$Q_{i}$ to the left of $\alpha_{2i+1}$ is equal to the number of occurrences
of $Q_{i}$ in $w$.

###### Proof of Claim.

We prove the claim by induction on $d$. First, let $d=0$. Then, our dissection
is an empty dissection, with one subgon $P$. There is only one matching in
$\mathbf{P}_{i,j}$, which uses $P$ $j-i-1$ times. Similarly, there is one
T-path between $v_{i}$ and $v_{j}$, which simply consists of the diagonal
$(v_{i},v_{j})$. This arc has $j-i-1$ vertices of $P$ to its left.

For the inductive step, assume the claim is true for any arc which crosses
less than $d$ arcs, and let $(v_{i},v_{j})$ cross $d$ arcs. Let $Q_{d}$ be the
subgon which has vertex $v_{j}$. Let $v_{a},v_{b}$ be the vertices of
$\tau_{d}$, so that $v_{a}$ appears when traveling counterclockwise from
$v_{i}$ to $v_{j}$. For convenience, relabel the vertices so that $i<a<j<b$.

Given any $w\in\mathbf{P}_{i,j}$, either $w[a:a]=Q_{d}$ or $w[a:a]\neq Q_{d}$.
In the former case, we can consider $w[i+1:a-1]$ as a matching in
$\mathbf{P}_{i,a}$. The arc $(v_{i},v_{a})$ crosses $k<d$ arcs. Thus, there is
a T-path $\alpha^{\prime}=\alpha_{1},\ldots,\alpha_{2k+1}$ with the property
that for all $1\leq\ell\leq k$, the number of vertices of $Q_{\ell}$ to the
left of $\alpha_{2\ell+1}$ is equal to the number of occurrences of $Q_{\ell}$
in $w[i+1:a-1]$. We extend $\alpha^{\prime}$ to a T-path
$\alpha\in\textbf{T}_{i,j}$. For all $k<\ell\leq d$, we set $\alpha_{2\ell}$
to be the arc $\tau_{\ell}$ oriented away from $a$. If $\ell<d$, we set