# Double Dimer Covers on Snake Graphs from Super Cluster Expansions

Gregg Musiker  ,  Nicholas Ovenhouse  and  Sylvester W. Zhang

(Date: October 13, 2021)

###### Abstract.

In a recent paper, the authors gave combinatorial formulas for the Laurent
expansions of super $\lambda$-lengths in a marked disk, generalizing
Schiffler’s $T$-path formula. In the present paper, we give an alternate
combinatorial expression for these super $\lambda$-lengths in terms of double
dimer covers on snake graphs. This generalizes the dimer formulas of Musiker,
Schiffler, and Williams.

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

_Email_<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. 1 Decorated Super-Teichmüller Theory
  2. 2 Fan Decomposition, Default Orientation, and Positive Order
  3. 3 Snake Graphs
  4. 4 Double Dimer Covers
  5. 5 Double Dimer Recurrences
  6. 6 The Main Formula
  7. 7 Relationship Between Different Combinatorial Models for Super $\lambda$-Lengths
  8. 8 Lattice Paths and Dual Snake Graphs
  9. 9 Order Ideals and Distributive Lattices
  10. 10 Examples
  11. 11 Super Fibonacci Numbers from Annuli and Once-Punctured Tori

###### Contents

  1. 1 Decorated Super-Teichmüller Theory
  2. 2 Fan Decomposition, Default Orientation, and Positive Order
  3. 3 Snake Graphs
  4. 4 Double Dimer Covers
  5. 5 Double Dimer Recurrences
  6. 6 The Main Formula
  7. 7 Relationship Between Different Combinatorial Models for Super $\lambda$-Lengths
  8. 8 Lattice Paths and Dual Snake Graphs
  9. 9 Order Ideals and Distributive Lattices
  10. 10 Examples
  11. 11 Super Fibonacci Numbers from Annuli and Once-Punctured Tori

## Introduction

A _marked surface_ is a surface $S$ with boundary, with a collection of marked
points $M\subset\partial S$ (we require at least one marked point per boundary
component). Although it is common in the literature to also consider interior
marked points (i.e. “punctures”), we will only consider unpunctured surfaces.
It is well-known in the theory of cluster algebras that the _decorated
Teichmüller space_ of $S$ possesses a cluster structure [GSV05] [FG06]
[FST08]. Given a fixed triangulation, with arcs terminating at marked boundary
points, the collection of _$\lambda$ -lengths_ gives a set of cluster
coordinates. In particular, this means that the $\lambda$-length of any
geodesic arc between marked points is expressible as a Laurent polynomial in
these coordinates. The “flip” of a triangulation corresponds to cluster
mutation, realized by the hyperbolic version of Ptolemy’s relation.

There have been many combinatorial ways to enumerate the terms occurring in
these Laurent expansions. One of the earliest was the notion of _$T$ -paths_
[Sch08] [ST09]. A few years later, a dimer (perfect matching) interpretation
was given [MS10] [MSW11]. There have been other combinatorial models (e.g.
[Yur19], [LLM15], [Cla20]), but these two will be our main focus in this
article.

In recent years, there have been efforts to find supersymmetric
generalizations of both cluster algebras and decorated Teichmüller spaces. On
the one hand, there have been a couple different attempts to define super
cluster algebras formally (see [LMRS21] and [Ovs15, OS19]). There is also a
supersymmetric version of frieze patterns (which are related to cluster
combinatorics) [MGOT15]. On the other hand, in a more geometric setting,
Penner and Zeitlin have recently introduced _decorated super-Teichmüller
spaces_ [PZ19].

In a recent paper, the current authors took a first step in unifying the
algebraic, combinatorial, and geometric viewpoints mentioned above [MOZ21]. In
particular, we found an analogue of the $T$-path Laurent formula for decorated
super-Teichmüller spaces. We also showed that the collection of super
$\lambda$-lengths and $\mu$-invariants fit into a super frieze pattern, as in
[MGOT15]. Lastly, we explored the connection between our approach and the
definitions of super cluster algebras given in [OS19].

The main purpose of the present paper is to give a super analogue of the dimer
formulas from [MS10]. Surprisingly, the terms in the super Laurent polynomials
are more naturally expressed in terms of _double dimer covers_ (rather than
ordinary dimer covers). We also give double dimer formulas for some (but not
all) of the odd variables corresponding to triangles. Later sections of the
paper give other combinatorial interpretations which can be derived from the
double dimer model. The structure of the paper is as follows.

  * •

In Section 1, we review background on decorated super-Teichmüller theory of
[PZ19].

  * •

In Section 2, we recall some definitions and conventions from our previous
paper [MOZ21] regarding triangulations of polygons, and how they are labelled.

  * •

Section 3 defines _snake graphs_ , and gives examples. These graphs will be
important in our main theorem, which says that any super $\lambda$-length can
be written as a Laurent polynomial whose terms are indexed by double dimer
covers on a certain snake graph.

  * •

Section 4 introduces general terms and notations related to double dimer
covers.

  * •

Section 5 gives recurrence formulas for double dimer covers on snake graphs.
The lemmas appearing in this section are the main ingredients of the inductive
proof of the main theorem later in the paper.

  * •

Section 6 presents the main theorem (Theorem 6.2), which says that the Laurent
expansions of super $\lambda$-lengths are sums over weights of double dimer
covers on a certain snake graph.

  * •

In Section 7, we define a modified version of the super $T$-paths from
[MOZ21], and use these to give an explicit weight-preserving bijection between
super $T$-paths and double dimer covers.

  * •

In Section 8, we discuss _dual snake graphs_. Looking at these dual snake
graphs, we get yet another combinatorial description of the Laurent terms,
this time in terms of lattice paths on the snake graph.

  * •

In Section 9, we discuss how the terms in these Laurent polynomials form a
distributive lattice, and thus can be identified with order ideals of a
certain poset.

  * •

In Section 10, we give some examples and illustrations.

  * •

Lastly, in Section 11, we consider super $\lambda$-lengths on other surfaces,
including the once-punctured torus and the annulus with one marked point on
each boundary. We then discuss super analogues of Fibonacci numbers and end
with open problems.

##  1\. Decorated Super-Teichmüller Theory

In this section, we survey the theory of decorated super-Teichmüller spaces
recently developed by Penner and Zeitlin [PZ19]. We start with a brief
overview of Penner’s classical theory of decorated Teichmüller spaces. The
readers are referred to [Pen12] for a detailed reference.

Consider a surface $S$ with marked points on its boundary111In general, a
surface can also contain marked points on its interior, known as _punctures_ ,
but we will not be concerned with this case, except for a discussion of the
once-punctured torus in Section 11., such that each boundary component
contains at least one marked point.

The _Teichmüller space_ of $S$, denoted $\mathcal{T}(S)$, is the space of
(equivalence classes of) hyperbolic metrics on $S$ with constant negative
curvature, with cusps at the marked points. More formally, the Teichmüller
space of $S$ is defined to be the quotient space

$\mathcal{T}(S)=\operatorname{Hom}(\pi_{1}(S),\operatorname{PSL}(2,\operatorname{\mathbb{R}}))/\operatorname{PSL}(2,\operatorname{\mathbb{R}}).$

The _decorated Teichmüller space_ of $S$, written
$\widetilde{\mathcal{T}}(S)$, is a trivial vector bundle over
$\mathcal{T}(S)$, with fiber $\mathbb{R}_{>0}^{n}$. The fibers represent a
choice of a positive real number associated to each marked point. At each
marked point, we draw a horocycle whose size (or _height_) is determined by
the corresponding positive number. Truncating the geodesics using these
horocycles, it now makes sense to talk about their lengths. If $\ell$ is the
truncated length of one of these geodesic segments, then the $\lambda$-length
(or _Penner coordinate_) associated to that geodesic arc is defined to be

$\lambda:=\exp(\ell/2)$

Fixing a triangulation of the marked surface, the collection of
$\lambda$-lengths corresponding to the arcs in the triangulation (including
segments of the boundary) form a system of coordinates for
$\widetilde{\mathcal{T}}(S)$. Choosing a different triangulation results in a
different system of coordinates, but they are related by simple
transformations which are a hyperbolic analogue of Ptolemy’s theorem from
classical Euclidean geometry. If two triangulations differ by the flip of a
single arc as in Figure 1, then the $\lambda$-lengths are related by:
$ef=ac+bd$.

$a$$b$$c$$d$$e$

$a$$b$$c$$d$$f$

Figure 1. Ptolemy transformation

In a recent paper [PZ19], Penner and Zeitlin gave a supersymmetric analogue of
the above mentioned decorated Teichmüller theory. The _decorated super-
Teichmüller space_ of $S$ is a superalgebra, generated by even elements
corresponding to the $\lambda$-lengths and odd elements, called _$\mu$
-invariants_, associated to the triangles.

To understand the multiplication of the anti-commutative $\mu$-invariants, we
need additional combinatorial data called a _spin structure_. Connected
components of the super-Teichmüller space $ST(S)$ are indexed by the set of
spin structures on $S$, which are formulated as the set of isomorphism classes
of Kasteleyn orientations of a certain graph embedded on a deformation retract
of $S$ [CR07, CR08]. These Kasteleyn orientations correspond to equivalence
classes of orientations of a fatgraph spine of $S$. For our purpose, we use
the dual of this formalism, where the _spin structures_ will be identified
with the equivalence classes of orientations of edges of a triangulation,
under the equivalence relation $\sim$ which we describe right now.

Given a triangulation $T$ with an orientation $\tau$ on its edges. For any
triangle $t$, consider the transformation which reverses the orientation of
the three sides of $t$. We say that two orientations are equivalent
$\tau\sim\tau^{\prime}$ if they are related by a sequence of these moves.

Now we are ready to state the super version of the Ptolemy relations.

(1) $\displaystyle ef$ $\displaystyle=ac+bd+\sqrt{abcd}\,\sigma\theta$ (2)
$\displaystyle\sigma^{\prime}$
$\displaystyle=\frac{\sigma\sqrt{bd}-\theta\sqrt{ac}}{\sqrt{ac+bd}}$ (3)
$\displaystyle\theta^{\prime}$
$\displaystyle=\frac{\theta\sqrt{bd}+\sigma\sqrt{ac}}{\sqrt{ac+bd}}$

$a$$b$$c$$d$$e$$\theta$$\sigma$

$a$$b$$c$$d$$f$$\theta^{\prime}$$\sigma^{\prime}$

Figure 2. super Ptolemy transformation

Note that in Equation 1, the order of multiplying the two odd variables
$\sigma$ and $\theta$ is determined by the orientation of the edge being
flipped (see the arrow in Figure 2). In addition, it was shown in [MOZ21,
Proposition 6.3] that equations 3 and 2 can be simplified as follows.

(2*) $\displaystyle\sigma^{\prime}$
$\displaystyle=\frac{\sigma\sqrt{bd}-\theta\sqrt{ac}}{\sqrt{ef}}$ (3*)
$\displaystyle\theta^{\prime}$
$\displaystyle=\frac{\theta\sqrt{bd}+\sigma\sqrt{ac}}{\sqrt{ef}}$

In Figure 2, the orientations of the edges around the boundary of the
quadrilateral are not indicated, but the super Ptolemy transformation does
affect the boundary orientations, where three of the four edges are unchanged
and only the orientation of the edge labelled $b$ is changed (see Figure 3).

$\epsilon_{a}$$\epsilon_{b}$$\epsilon_{c}$$\epsilon_{d}$$\theta$$\sigma$

$\epsilon_{a}$$-\epsilon_{b}$$\epsilon_{c}$$\epsilon_{d}$$\theta^{\prime}$$\sigma^{\prime}$

$-\epsilon_{a}$$-\epsilon_{b}$$\epsilon_{c}$$\epsilon_{d}$$-\theta$$\sigma$

Figure 3. Flip effect on spin structures. Here $\epsilon_{x}$ denotes the
orientation of an edge $x$.

Note the super Ptolemy relation is _not_ an involution. As depicted in Figure
3, flipping an edge twice results in reversing the orientations around the top
triangle, and additionally negating the $\mu$-invariant. Hence
$\mu$-invariants are well-defined only up to sign.

In our paper, we mostly consider marked disks with a triangulation such that
every triangle has a boundary (see Section 2 for an explanation), in which
case the orientations of the boundary edges will not affect our calculation of
super $\lambda$-lengths. Therefore we can often ignore the boundary
orientations. The following lemma from [MOZ21] tells us that for any spin
structure, the corresponding equivalence class of orientations will exhaust
all possible interior orientations after ignoring the boundary.

######  Lemma 1.1 (Proposition 4.1 of [MOZ21]).

Fix a triangulation of a polygon in which every triangle has at least one
boundary edge. Then there is a unique spin structure after ignoring the
boundary edges. In particular, this means that, from any representative
orientation of a fixed spin structure, one can obtain all other orientations
on the interior diagonals, without changing the spin structure.

######  Remark 1.2 (Remarks 4.2 and 4.3 of [MOZ21]).

Note that the equivalence relation guarantees that the result after flipping
twice represents the same spin structure, but algebraically it has the effect
of negating the $\mu$-invariant of that triangle ($\theta\mapsto-\theta$ in
Figure 3). This means that the specific $\mu$-invariants are not a feature of
the triangulation and spin structure alone, but also the choice of
representative orientation. Choosing a different orientation corresponds to
changing the sign of some of the $\mu$-invariants. On the other hand, the
expressions of $\lambda$-lengths in terms of an initial triangulation are
independent of the orientation of the arc as part of a spin structure, and of
the flip sequence used to obtain a triangulation containing that arc. This is
proven in [PZ19] in the case of surfaces without boundaries. In [MOZ21], the
authors proved the well-definedness of super $\lambda$-lengths for marked
disks by deriving a super Pentagon relation (see [MOZ21, Appendix A.]).

##  2\. Fan Decomposition, Default Orientation, and Positive Order

In this section, we recall some definitions and conventions used in [MOZ21].
In particular, given a triangulation of a polygon, and an (oriented) arc $i\to
j$ not in the triangulation, we define an orientation of the arcs in the
triangulation called the _default orientation_ , and a total order on the set
of triangles (and odd generators of the super algebra) called the _positive
order_.

Let $P$ be a polygon with $n+3$ vertices, i.e. a disk with $n+3$ marked points
on the boundary. Let $a$ and $b$ be two non-adjacent vertices on the boundary
and let $(a,b)$ be the arc that connects $a$ and $b$. Without loss of
generality, we assume that $(a,b)$ crosses all internal diagonals of $T$. (For
the purpose of $\lambda$-length calculation, triangles that do not intersect
with the arc $(a,b)$ can simply be removed from the picture.)

###  2.1. Decomposing a Triangulation into Fans

A triangulation is called a _fan_ if all the internal diagonals share a common
vertex. We will introduce a canonical way to break any triangulation $T$ of
$P$ into smaller ones that are fans. For this purpose, certain vertices of $P$
will be distinguished as _fan centers_.

$\ a=c_{0}$$c_{2}$$b=c_{3}\ $$c_{1}$ Figure 4. centers of fan segments.

######  Definition 2.1.

The intersections of arc $(a,b)$ and internal diagonals of $T$ create small
triangles (colored yellow in Figure 4), whose vertices in $P$ are called _fan
centers_. We set $a=c_{0}$ and $b=c_{N+1}$ and as a convention we will name
these centers $c_{1},\cdots,c_{N}$ such that

  1. nosep

The edge $(c_{i},c_{i+1})$ is in $T$ which crosses $(a,b)$ for $1\leq i\leq
N-1$;

  2. nosep

The intersection $(c_{i},c_{i+1})\cap(a,b)$ is closer to $a$ than
$(c_{j},c_{j+1})\cap(a,b)$ if $i<j$.

Now the polygon $P$ is broken into smaller polygons by the edges
$(c_{i},c_{i+1})$, each of which comes with a fan triangulation induced from
$T$. Moreover, the sub-triangulation of $T$ bounded by $c_{i-1}$, $c_{i}$ and
$c_{i+1}$ is called the $i$-th _fan segment_ of $T$, whose _center_ is said to
be $c_{i}$.

###  2.2. Default Orientation and Positive Order

For a triangulation $T$ of $P$ and the (directed) arc $(a,b)$ which crosses
all internal diagonals of $T$, we define a _default orientation_. Such
orientation determines an ordering of the $\mu$-invariants, which we call the
_positive order_ , in which the super $\lambda$-length expansion has positive
coefficients. Note that we will omit the orientation of boundary edges because
only the orientation of interior edges affects our calculation of super
$\lambda$-lengths.

######  Definition 2.2.

When the triangulation is a single fan with $c_{1}$ being the center, every
interior edge is oriented away from $c_{1}$. When $T$ is a triangulation with
$N>1$ fans, where $c_{1},\cdots,c_{N}$ are the centers, the interior edges
within each fan segment are oriented away from its center. The edges where two
fans meet each other are oriented as $c_{1}\rightarrow
c_{2}\rightarrow\cdots\rightarrow c_{N-1}\rightarrow c_{N}$. See Figure 5.

$c_{0}$$c_{2}$$c_{4}$$c_{5}$$c_{3}$$c_{1}$$\alpha_{1}$$\alpha_{2}$$\alpha_{3}$$\beta_{1}$$\beta_{2}$$\gamma_{1}$$\gamma_{2}$$\gamma_{3}$$\delta_{1}$$\delta_{2}$
Figure 5. The default orientation of a generic triangulation where each fan
segment is colored differently. The faces are labelled by their
$\mu$-invariants.

######  Remark 2.3.

As mentioned above, the definition of default orientation depends on the
choice of direction $a\to b$. In particular, choosing the opposite direction
$b\to a$ would change the labelling so that $c_{i}$ becomes $c_{N-i}$. The
effect is that the orientation of the diagonals within a fan are unchanged,
but the diagonals connecting two fan centers would have the reverse
orientation.

######  Definition 2.4.

We define the _positive ordering_ inductively, triangle-by-triangle, as
follows: Let the triangles be labelled
$\theta_{1},\theta_{2},\dots,\theta_{n}$ in order from $a$ to $b$. For each
triangle $\theta_{k}$, we look at the edge separating $\theta_{k}$ and
$\theta_{k+1}$:

  1. (1)

If the edge is oriented so that $\theta_{k}$ is to the right, then we declare
that $\theta_{k}>\theta_{i}$ for all $i>k$.

  2. (2)

Otherwise, if $\theta_{k}$ is to the left, we declare that
$\theta_{k}<\theta_{i}$ for all $i>k$.

For example, in Figure 5, the positive ordering on the faces is

$\alpha_{1}>\alpha_{2}>\alpha_{3}>\gamma_{1}>\gamma_{2}>\gamma_{3}>\delta_{2}>\delta_{1}>\beta_{2}>\beta_{1}.$

######  Remark 2.5.

Note that inside each of the fan segments, the triangles are ordered
counterclockwise around the fan center. For a more detailed description of the
positive order, see Section 4 of [MOZ21].

From now on, we will always assume our triangulation has the default
orientation on its interior diagonals, and state our theorems under this
assumption. This is allowed because of Lemma 1.1. If we start with an
arbitrary orientation, we can first apply a sequence of equivalence relations
(reversing the arrows around a triangle and negating the $\mu$-invariant) to
get to the default orientation, and apply our theorems therein.

##  3\. Snake Graphs

######  Definition 3.1.

A _snake graph_ is a planar graph consisting of a sequence of square _tiles_ ,
each connected to either the top or right side of the previous tile. Given a
snake graph $G$, the _word_ of $G$, denoted $W(G)$, is a string in the
alphabet $\\{\text{R,U}\\}$ (standing for “ _right_ ” and “ _up_ ”) indicating
how each tile is connected to the previous. See Figure 6 for examples.

$W(G)=\varnothing$$W(G)=\text{RR}$$W(G)=\text{UR}$$W(G)=\text{RUR}$ Figure 6.
Examples of snake graphs

We will now review a construction from [MS10], which associates to each arc of
a triangulated polygon a labelled snake graph. In the remainder of this
section, we assume we are given a polygon with $n$ sides, and a chosen
triangulation. As in [MOZ21], we label the edges and triangular faces of the
triangulation with the corresponding generators of the super algebra.

######  Definition 3.2.

Let $\gamma$ be a diagonal of the polygon. We will construct a snake graph
$G_{\gamma}$. Suppose vertices $a$ and $b$ are the endpoints of $\gamma$. If
we traverse $\gamma$ from $a$ to $b$, let $x_{1},\dots,x_{k}$ be the labels of
the diagonals crossed by $\gamma$, in order. Without loss of generality, we
suppose that $\gamma$ is a _longest edge_ , in which case $k=n-3$. We will
also always draw our polygons so that $a$ is at the bottom and $b$ is at the
top (the arc $\gamma$ is oriented bottom-to-top).

There will be one tile of $G_{\gamma}$ for each diagonal $x_{i}$. For each
$i$, the diagonal $x_{i}$ is the common side of two triangles in the
triangulation, and is thus the diagonal of a quadrilateral. We make a square
tile $T_{i}$ whose four sides are labelled the same as the sides of this
quadrilateral. If $i$ is odd, then the orientation of $T_{i}$ matches that of
the triangulation, and if $i$ is even, then the orientation of $T_{i}$ is
reversed. Examples are shown in Figure 7. By convention, we draw the first
tile $T_{1}$ so that the endpoint of $\gamma$ is the bottom left corner.

Although we do not draw the diagonals in the snake graph tiles, we still speak
of $x_{i}$ as being “ _the diagonal of tile $T_{i}$_”. These diagonals,
although not drawn, should be thought of as separating the $\theta_{i}$’s
labelling the triangles. These $\theta$ labels are positioned in the corners
of the tiles corresponding to their position in the triangulation.

$x_{3}$$x_{2}$$x_{1}$$y_{1}$$y_{2}$$y_{3}$$y_{4}$$y_{5}$$y_{6}$$\theta_{1}$$\theta_{2}$$\theta_{3}$$\theta_{4}$$T_{1}$$\colon$$y_{2}$$y_{3}$$x_{2}$$y_{1}$$\theta_{1}$$\theta_{2}$$x_{3}$$x_{2}$$x_{1}$$y_{1}$$y_{2}$$y_{3}$$y_{4}$$y_{5}$$y_{6}$$\theta_{1}$$\theta_{2}$$\theta_{3}$$\theta_{4}$$T_{2}$$\colon$$x_{1}$$y_{6}$$x_{3}$$y_{3}$$\theta_{2}$$\theta_{3}$
Figure 7. Tiles corresponding to diagonals in a triangulation

By construction, tiles $T_{i}$ and $T_{i+1}$ share a triangle in common. Two
of the sides of this triangle are the diagonals $x_{i}$ and $x_{i+1}$. Let $e$
denote the label on the third side of this triangle. The tiles $T_{i}$ and
$T_{i+1}$ will both have a side labelled $e$ (which will be either the top or
right edge of $T_{i}$). We will glue the tiles along the edge $e$.

Examples of snake graphs are pictured in Figure 8.

$x_{3}$$x_{2}$$x_{1}$$y_{1}$$y_{2}$$y_{3}$$y_{4}$$y_{5}$$y_{6}$$\theta_{1}$$\theta_{2}$$\theta_{3}$$\theta_{4}$$G_{\gamma}$$\colon$$y_{2}$$y_{3}$$x_{2}$$y_{1}$$x_{1}$$y_{6}$$x_{3}$$x_{2}$$y_{4}$$y_{5}$$\theta_{1}$$\theta_{2}$$\theta_{2}$$\theta_{3}$$\theta_{3}$$\theta_{4}$$x_{2}$$x_{1}$$x_{3}$$y_{1}$$y_{2}$$y_{3}$$y_{4}$$y_{5}$$y_{6}$$\theta_{1}$$\theta_{2}$$\theta_{3}$$\theta_{4}$$G_{\gamma}$$\colon$$y_{2}$$y_{3}$$x_{2}$$y_{1}$$x_{1}$$x_{3}$$y_{4}$$y_{5}$$y_{6}$$x_{2}$$\theta_{1}$$\theta_{2}$$\theta_{2}$$\theta_{3}$$\theta_{3}$$\theta_{4}$
Figure 8. Snake graph corresponding to a diagonal $\gamma$ (drawn dashed)

######  Definition 3.3.

Let $\gamma=(i,j)$ be a diagonal of a triangulated polygon. Define
$\mathrm{cross}(\gamma)=\mathrm{cross}(i,j)$ to be the monomial
$x_{1}x_{2}\cdots x_{k}$, the product of all diagonals which cross $\gamma$.

######  Theorem 3.4.

[MS10] Let $\mathcal{A}$ be the cluster algebra coming from a polygon, with
initial seed given by a chosen triangulation, and let $\gamma=(i,j)$ be a
diagonal of the polygon. Then

$x_{ij}=\frac{1}{\mathrm{cross}(\gamma)}\sum_{M\in
D(G_{\gamma})}\mathrm{wt}(M)$

Here, $D(G_{\gamma})$ denotes the collection of dimer covers on the snake
graph $G_{\gamma}$. In particular, each $M$ is a subset of edges on
$G_{\gamma}$ such that every vertex of $G_{\gamma}$ is incident to exactly one
edge of $M$. The weight of $M$, denoted as $\mathrm{wt}(M)$, is the product of
all edge labels for all edges in $M$. In the next section, we extend these
definitions to the case of _double_ dimer covers.

##  4\. Double Dimer Covers

######  Definition 4.1.

If $G$ is a planar bipartite graph, a _double dimer cover_ of $G$ is a
multiset $M$ of edges such that each vertex of $G$ is incident to exactly two
edges from $M$ (which are allowed to be two copies of the same edge). Given a
double dimer cover $M$, we call each element of $M$ a _dimer_. Dimers will be
pictured as wavy orange lines, and two overlapping dimers (called a _double
dimer_) will be pictured as a solid blue line.

######  Definition 4.2.

Let $G$ be a snake graph, and let $T$ denote the last tile of $G$. Then we
define:

  * •

$D(G)$ is the set of all double dimer covers on $G$.

  * •

$D_{R}(G)$ is the set of $M\in D(G)$ which use two dimers on the right edge of
$T$.

  * •

$D_{T}(G)$ is the set of $M\in D(G)$ which use two dimers on the top edge of
$T$.

  * •

$D_{tr}(G)$ is the set of $M\in D(G)$ which use single dimers on the top and
right edges of $T$.

  * •

$D_{r}(G)$ is the set of $M\in D(G)$ which use at least one dimer on the right
edge of $T$.

  * •

$D_{t}(G)$ is the set of $M\in D(G)$ which use at least one dimer on the top
edge of $T$.

Looking at the top-right vertex of the last tile, we see that $D(G)$ is the
disjoint union

$D(G)=D_{R}(G)\cup D_{T}(G)\cup D_{tr}(G)$

######  Remark 4.3.

Note that $D_{r}(G)=D_{R}(G)\cup D_{tr}(G)$ and $D_{t}(G)=D_{T}(G)\cup
D_{tr}(G)$.

######  Definition 4.4.

If $G$ is a snake graph, and $M\in D(G)$, we will define the _weight_ of $M$,
denoted $\mathrm{wt}(M)$, as a monomial of the super algebra. The weight of a
dimer (an element of the multiset $M$) is the square root of the label of the
corresponding edge of $G$. Let $c(M)$ be the set of cycles formed by the edges
of $M$. For a cycle $C\in c(M)$, let $\theta_{i}$ be the odd variable
corresponding to the triangle in the bottom-left corner of $C$, and
$\theta_{j}$ for the top-right corner. We define the weight of the cycle to be
$\mathrm{wt}(C)=\theta_{i}\theta_{j}$. Finally, we define the weight of $M$ as

$\mathrm{wt}(M):=\prod_{e\in M}\mathrm{wt}(e)\prod_{C\in c(M)}\mathrm{wt}(C)$

where the products are taken under the positive order of the underlying
triangulation.

Note that we can map any dimer cover of $G$ to a double dimer cover of $G$ by
turning each dimer edge into a double dimer, i.e. a doubled edge. For such
$M$’s, the weight of $M$, as defined here, versus following the definition
from Theorem 3.4 coincide. Thus there is no abuse in notation for using
$\mathrm{wt}(M)$ in both settings.

##  5\. Double Dimer Recurrences

In this section, we give some recurrences for double dimer covers on snake
graphs. In particular, we establish bijections between each of the sets
$D_{R}(G)$, $D_{T}(G)$, and $D_{tr}(G)$ with certain subsets of double dimer
covers on smaller snake graphs. We also give explicit expressions for how the
weights transform under these bijections.

######  Remark 5.1.

Although we usually start with a triangulation, and build a snake graph from
it (as in Definition 3.2), in this section we instead start with a labelled
snake graph. It is not hard to see that the construction of Definition 3.2 can
be reversed, and the triangulation can be reconstructed from the snake graph.
As such, we will still speak of the “ _diagonal_ ” of a tile, meaning the
label on the corresponding diagonal of the triangulation.

######  Definition 5.2.

Define $G^{(-k)}$ to be the snake graph $G$ with the last $k$ tiles removed.

######  Lemma 5.3.

  * $(a)$

If $W(G)$ ends in “$R$”, then there is a bijection $f\colon D(G^{(-1)})\to
D_{R}(G)$, satisfying $\mathrm{wt}(f(M))=a\cdot\mathrm{wt}(M)$, where “$a$” is
the label on the right edge of the last tile. In particular,

$\sum_{M\in D(G^{(-1)})}\mathrm{wt}(M)=\frac{1}{a}\sum_{M\in
D_{R}(G)}\mathrm{wt}(M)$

  * $(b)$

If $W(G)$ ends in “$U$”, then there is a bijection $f\colon D(G^{(-1)})\to
D_{T}(G)$, satisfying $\mathrm{wt}(f(M))=a\cdot\mathrm{wt}(M)$, where “$a$” is
the label on the top edge of the last tile. In particular,

$\sum_{M\in D(G^{(-1)})}\mathrm{wt}(M)=\frac{1}{a}\sum_{M\in
D_{T}(G)}\mathrm{wt}(M)$

###### Proof.

A double dimer cover in $D(G^{(-1)})$ can be uniquely extended to a double
dimer cover in $D_{R}(G)$ (resp. $D_{T}(G)$) by adjoining two copies of the
edge labeled $a$. See Figure 9. ∎

######  Lemma 5.4.

  * $(a)$

If $W(G)$ ends in either “$RRURUR\cdots R$” or “$UURURU\cdots R$” (of length
$k\geq 2$), then there is a bijection $f\colon D(G^{(-k)})\to D_{T}(G)$
satisfying $\mathrm{wt}(f(M))=be_{2}\cdots e_{k}\cdot\mathrm{wt}(M)$, where
$e_{1},\dots,e_{k}$ are the diagonals of the last $k$ tiles in reverse order,
and $b$ is the label on the top side of the last tile. In particular,

$\sum_{M\in D(G^{(-k)})}\mathrm{wt}(M)=\frac{1}{be_{2}\cdots e_{k}}\sum_{M\in
D_{T}(G)}\mathrm{wt}(G)$

  * $(b)$

If $W(G)$ ends in either “$UURURU\cdots U$” or “$RRURUR\cdots U$” (of length
$k\geq 2$), then there is a bijection $f\colon D(G^{(-k)})\to D_{R}(G)$
satisfying $\mathrm{wt}(f(M))=be_{2}\cdots e_{k}\cdot\mathrm{wt}(M)$, where
$e_{1},\dots,e_{k}$ are the diagonals of the last $k$ tiles in reverse order,
and $b$ is the label on the right side of the last tile. In particular,

$\sum_{M\in D(G^{(-k)})}\mathrm{wt}(M)=\frac{1}{be_{2}\cdots e_{k}}\sum_{M\in
D_{R}(G)}\mathrm{wt}(G)$

###### Proof.

Supposing the snake graph ends in a staircase which ends with “R”, then the
last $k$ tiles of $G$ look like the picture in Figure 10. Note that because of
the snake graph, i.e. see Definition 3.2, if $e_{i}$ is the label of the
diagonal of the $i$th tile counting from the end (e.g. $e_{1}$ is the label of
the diagonal of the last tile), then the W edge or S edge (depending on
whichever is a boundary edge) of the succeeding tile is labeled $e_{i}$.
Similarly, the N edge or E edge of the previous tile is labeled $e_{i}$. It is
easy to see that if the top edge of the last tile is occupied by two dimers,
then the opposite edge (the bottom) must also have a double edge.
Consequently, the left side of the previous tile must use a double edge. Going
on, one can see that this uniquely determines a double dimer cover on the rest
of the staircase segment where one edge of each corner is used. This is
pictured in Figure 10 (Left). It is easily seen that the weight of these
dimers is the product $be_{2}\cdots e_{k}$. Removing the last $k$ tiles leads
to $G^{(-k)}$ as pictured in Figure 10 (Right). ∎

$\cdots$$e_{1}$$a$$b$$e_{2}$$e_{2}$$e_{3}$$e_{3}$$e_{4}$$e_{4}$$e_{5}$$e_{5}$$e_{6}$

$\cdots$$e_{1}$$e_{2}$$e_{3}$$e_{3}$$e_{4}$$e_{4}$$e_{5}$$e_{5}$$e_{6}$

Figure 9. Parts of double dimer covers which use the edge $a$ twice

$\cdots$$b$$a$$e_{1}$$e_{2}$$e_{2}$$e_{3}$$e_{3}$$e_{4}$$e_{4}$$e_{5}$$e_{5}$$e_{6}$

$\cdots$$e_{1}$$a$$b$$e_{2}$$e_{2}$$e_{3}$$e_{3}$$e_{4}$$e_{4}$$e_{5}$$e_{5}$$e_{6}$

Figure 10. Parts of double dimer covers ending in a “staircase” which use the
edge $b$ twice

######  Lemma 5.5.

  * $(a)$

If $W(G)=(RU)^{n}R$ or $W(G)=(UR)^{n}$, then $D_{T}(G)$ contains a unique
double dimer cover with weight $bce_{2}\cdots e_{k}$ (with $k=2n+2$ or
$k=2n+1$), where $e_{1},\dots,e_{k}$ are the diagonals of the last $k$ tiles
in reverse order, $b$ is the label on the top side of the last tile, and $c$
is the label on the left or bottom edge of the first tile.

  * $(b)$

If $W(G)=(UR)^{n}U$ or $W(G)=(RU)^{n}$, then $D_{R}(G)$ contains a unique
double dimer cover with weight $bce_{2}\cdots e_{k}$ (with $k=2n+2$ or
$k=2n+1$), where $e_{1},\dots,e_{k}$ are the diagonals of the last $k$ tiles
in reverse order, $b$ is the label on the right side of the last tile, and $c$
is the label on the left or bottom edge of the first tile.

###### Proof.

The proof is essentially the same as for Lemma 5.4, using Figure 11 which
illustrates the case of part (a). ∎

$a$$e_{1}$$b$$e_{2}$$e_{2}$$e_{3}$$e_{3}$$e_{4}$$e_{4}$$e_{5}$$e_{5}$$e_{6}$$c$
Figure 11. The unique double dimer cover on a “staircase” using the edge $b$
twice

We will now define an involution on the set of monomials of the super-algebra
$A=\mathbb{R}[x_{ij}^{\pm 1/2}\leavevmode\nobreak\ |\leavevmode\nobreak\
\theta_{k}]$, given by _toggling_ $\theta_{n}$.

######  Definition 5.6.

Let $x\in A$ be a monomial, written in the positive order. If $x$ contains
$\theta_{n}$, then $x^{*}$ is defined as $x$ with $\theta_{n}$ removed. If $x$
does not contain $\theta_{n}$, then $x^{*}$ is defined by inserting
$\theta_{n}$ such that the positive order is preserved.

######  Example 5.7.

Let $n=4$, and suppose the positive order is
$\theta_{1}>\theta_{3}>\theta_{4}>\theta_{2}$. Then
$(\theta_{1}\theta_{3}\theta_{2})^{*}=\theta_{1}\theta_{3}\theta_{4}\theta_{2}$,
and $(\theta_{1}\theta_{4})^{*}=\theta_{1}$.

######  Lemma 5.8.

Suppose that $G$ contains $(n-1)$ tiles, so that $\theta_{n-1}$ (resp.
$\theta_{n}$) labels the bottom-left (resp. top-right) triangle of the last
tile. Let $x^{\dagger}$ and $x^{\ast}$ denote the involutions corresponding to
$\theta_{n-1}$ and $\theta_{n}$, respectively.

  * $(a)$

If $W(G)$ ends in “R” , then there is a bijection $f\colon D_{r}(G^{(-1)})\to
D_{tr}(G)$ satisfying
$\mathrm{wt}(f(M))=\sqrt{\frac{abc}{d}}\,\left(\mathrm{wt}(M)^{\dagger}\right)^{\ast}$,
where “$d$” is the label on the left edge of the last tile, and $a,b,c$ are
the labels on the other three sides. In particular,

$\sum_{M\in D_{r}(G^{(-1)})}\mathrm{wt}(M)=\sqrt{\frac{d}{abc}}\sum_{M\in
D_{tr}(G)}\left(\mathrm{wt}(M)^{\ast}\right)^{\dagger}$

  * $(b)$

If $W(G)$ ends in “U” , then there is a bijection $f\colon D_{t}(G^{(-1)})\to
D_{tr}(G)$ satisfying
$\mathrm{wt}(f(M))=\sqrt{\frac{abc}{d}}\,\left(\mathrm{wt}(M)^{\dagger}\right)^{*}$,
where “$d$” is the label on the bottom edge of the last tile, and $a,b,c$ are
the labels on the other three sides. In particular,

$\sum_{M\in D_{t}(G^{(-1)})}\mathrm{wt}(M)=\sqrt{\frac{d}{abc}}\sum_{M\in
D_{tr}(G)}\left(\mathrm{wt}(M)^{\ast}\right)^{\dagger}$

###### Proof.

Suppose $W(G)$ ends in “R”. Let $M\in D_{r}(G^{(-1)})$. Then we define $f(M)$
by first erasing a dimer from the right edge of the last tile of $G^{(-1)}$,
and then add single dimers on the top, right, and bottom edges of the last
tile of $G$. This is illustrated in Figure 12. Clearly, this has the effect of
multiplying the weight of $M$ by a factor $\sqrt{\frac{abc}{d}}$. But it also
changes the odd variables, since the cycles at the end surround a different
set of tiles.

There are two cases, corresponding to the two pictures in Figure 12. First
consider the top picture, corresponding to $M\in D_{R}(G^{(-1)})$. Since $M$
does not contain a cycle surrounding the last tile, $\mathrm{wt}(M)$ does not
include the variable $\theta_{n-1}$. Looking at the figure, we see that $f(M)$
contains a cycle around just the last tile, which means we must multiply by
$\theta_{n-1}\theta_{n}$. This is the same as applying both the $\dagger$ and
$\ast$ involutions (the $\dagger$ multiplies by $\theta_{n-1}$ and the $\ast$
multiplies by $\theta_{n}$). Following Definition 5.6, the odd variables,
including $\theta_{n-1}$ and $\theta_{n}$, are written in the positive order.

In the second case, $M$ has a cycle surrounding the last tile of $G^{(-1)}$,
and so $\mathrm{wt}(M)$ includes a factor of $\theta_{n-1}$. The effect of $f$
is to extend the cycle around the last tile of $G$. So we need to exchange
$\theta_{n-1}$ for $\theta_{n}$. Again applying both involutions acheives this
effect, since $\dagger$ will remove $\theta_{n-1}$, and then $\ast$ will add
$\theta_{n}$ in the proper place relative to positive order.

$\cdots$$a$$b$$c$$d$$\theta_{n}$$\theta_{n-1}$$\theta_{n-1}$$f$$\cdots$$a$$b$$c$$d$$\theta_{n}$$\theta_{n-1}$$\theta_{n-1}$$\cdots$$a$$b$$c$$d$$\theta_{n}$$\theta_{n-1}$$\theta_{n-1}$$f$$\cdots$$a$$b$$c$$d$$\theta_{n}$$\theta_{n-1}$$\theta_{n-1}$
Figure 12. Proof of Lemma 5.8

∎

##  6\. The Main Formula

In this section we give a double dimer interpretation for the terms of the
Laurent expansion of $\lambda$-lengths and certain types of $\mu$-invariants.
The proof will induct on the number of triangles in the polygon (equivalently,
the number of tiles of the corresponding snake graph). There are several cases
to consider, with subtle differences, depending on the following factors:

  * •

Whether there are an even or odd number of triangles/tiles. This affects
whether the last tile of the snake graph has either normal or reversed
orientation.

  * •

Whether the word $W(G)$ of the snake graph ends in “R” or “U”. This affects
which versions (parts $(a)$ or $(b)$) of Lemmas 5.3 – 5.8 must be used in the
induction. This will correspond to interchanging $D_{T}(G)$ and $D_{R}(G)$ in
the recursion formulas.

  * •

Whether the top-most fan center is on the left (as in Figure 13(a)) or the
right. This affects which edge ($c$ or $d$ in Figure 13) crosses more
diagonals.

Rather than go through the proof for all possible cases, we will make
particular choices for the factors listed above, so that we are in the case
pictured in Figure 13. Namely, we assume that the top-most fan center (the
vertex labelled $j$ in Figure 13) is on the left (adjacent to edge $a$, and
not $b$), and that the triangulation has an odd number of triangles. This
actually determines the third condition (whether $W(G)$ ends with “R” or “U”),
which we now explain in the following lemma.

######  Lemma 6.1.

  1. $(a)$

Suppose a triangulated polygon is oriented as in Figure 13, so that the top-
most fan center (vertex $j$) is on the left side of the polygon. Then $W(G)$
ends in “R” if and only if there are an odd number of triangles (even number
of tiles in $G$), and $W(G)$ ends in “U” if and only if there are an even
number of triangles (odd number of tiles in $G$).

  2. $(b)$

Suppose a triangulated polygon is oriented opposite of Figure 13, so that the
top-most fan center is on the right side of the polygon. Then $W(G)$ ends in
“R” if and only if there are an even number of triangles (odd number of tiles
in $G$), and $W(G)$ ends in “U” if and only if there are an odd number of
triangles (even number of tiles in $G$).

###### Proof.

In the case of a fan triangulation (i.e. $j$ is the only fan center), this is
easy to see. By the construction outlined in Definition 3.2, the assumption
that the fan center $j$ is on the left implies that $W(G)$ always begins with
a “U”. The result then follows for the case of a single fan segment. The
argument for part $(b)$ in a single fan is the same, since $W(G)$ will always
begin with “R” in that case.

Now we will induct on the number of fan segments. Suppose parts $(a)$ and
$(b)$ are true for all triangulations with $n-1$ fan segments. We will
consider the case of $n$ fan segments. Let $G^{\prime}$ be the smaller snake
graph corresponding to the longest arc in the polygon containing only the
first $n-1$ fan segments. Decompose $W(G)$ as $W(G^{\prime})W^{\prime}$, where
$W^{\prime}$ is the remainder of $W(G)$. Note that by the construction in
Definition 3.2, the first letter of $W^{\prime}$ is the same as the last
letter of $W(G^{\prime})$. This means if $W^{\prime}$ has odd length, then
$W(G)$ and $W(G^{\prime})$ end in the same letter, and if $W^{\prime}$ has
even length, then $W(G)$ and $W(G^{\prime})$ end in the opposite letter.

Also note that if $W^{\prime}$ has even length, then the parity of the numbers
of tiles of $G$ and $G^{\prime}$ are the same, while if $W^{\prime}$ has odd
length, the parity of the numbers of tiles in $G$ and $G^{\prime}$ are
opposite. Combining these observations with the induction assumption gives the
result. ∎

$l$$k$$j$$i$$m$$h$$a$$b$$d$$c$$e$$e_{2}$$e_{k}$$(a)$$b$$a$$d$$c$$e$$\theta$$\sigma$$b$$a$$d$$c$$f$$\varphi$$(b)$
Figure 13. $(a)$ Triangulation $T$ with $n$ triangles. $(b)$ Ptolemy relation
in the quadrilateral $ijkl$.

######  Theorem 6.2.

Consider a triangulation as pictured in Figure 13 where $f=(i,k)$ is the
longest arc, with corresponding snake graph $G$. In particular, we assume that
$a$ and $b$ are boundary edges, the top-most fan center $j$ is on the left.
Then

  1. $(a)$

$\displaystyle f=\frac{1}{\mathrm{cross}(f)}\sum_{M\in D(G)}\mathrm{wt}(M)$

  2. $(b)$

If the polygon has an odd number of triangles222By Lemma 6.1, this implies
that $W(G)$ ends with “R”., then

$\sqrt{df}\;\varphi=\frac{1}{\mathrm{cross}(f)}\;\sqrt{\frac{e}{b}}\sum_{M\in
D_{t}(G)}\mathrm{wt}(M)^{*}$

where $x\mapsto x^{*}$ is the involution toggling $\theta_{n}$ (the top-most
triangle). If there are an even number of triangles, then the sum is over
$D_{r}(G)$ instead of $D_{t}(G)$333This is because, by Lemma 6.1, $W(G)$ ends
with “U” in this case..

###### Proof.

We will prove this by induction on the number of triangles.

####  Base case

$c$$d$$a$$b$$\sigma$$\theta$$ac$$c$$d$$a$$b$$\sigma$$\theta$$bd$$c$$d$$a$$b$$\sigma$$\theta$$\sqrt{abcd}\;\sigma\theta$
Figure 14. The three double dimer covers on a single-tile snake graph

The base case is a quadrilateral, in which case $W(G)$ is the empty word.
Assume the quadrilateral is labelled as in Figure 2. Then the snake graph $G$
(corresponding to the diagonal $f$) is just a single tile. It is easy to see
that there are only three double dimer covers on the square graph. These are
pictured in Figure 14, along with their weights. There is only one diagonal in
this triangulation (the edge $e$), and so $\mathrm{cross}(f)=e$. We see then
that the weights of these three double dimer covers (after being divided by
$e$) are precisely the three terms in the super Ptolemy relation (Equation 1).
This confirms part $(a)$ in the base case.

Next, we consider part $(b)$ for the base case. We are interested in what is
called $\theta^{\prime}$ in the super Ptolemy relation (Equation 3). Recall
that $D_{r}(G)=D_{R}(G)\cup D_{tr}(G)$. In the case of a single tile,
$D_{r}(G)$ consists of the second and third double dimer covers in Figure 14.
In this case the $x^{*}$ involution corresponds to $\theta$, and we have

$(bd)^{*}=bd\;\theta\quad\text{ and
}\quad(\sqrt{abcd}\;\sigma\theta)^{*}=\sqrt{abcd}\;\sigma$

As noted above, $\mathrm{cross}(f)=e$, and so the right-hand side of part
$(b)$ in the case of a single tile is

$\frac{1}{\mathrm{cross}(f)}\;\sqrt{\frac{e}{b}}\sum_{M\in
D_{r}(G)}\mathrm{wt}(M)^{*}=\frac{1}{\sqrt{be}}\left(bd\;\theta+\sqrt{abcd}\;\sigma\right)$

On the other hand, the left-hand side of part $(b)$ (using the notations of
Figure 2) is

$\displaystyle\sqrt{df}\;\theta^{\prime}$
$\displaystyle=\frac{\sqrt{df}}{\sqrt{ef}}\left(\sqrt{bd}\;\theta+\sqrt{ac}\;\sigma\right)$
$\displaystyle=\frac{1}{\sqrt{e}}\left(d\sqrt{b}\;\theta+\sqrt{acd}\;\sigma\right)$
$\displaystyle=\frac{1}{\sqrt{be}}\left(bd\;\theta+\sqrt{abcd}\;\sigma\right)$

This confirms the base case for part $(b)$.

#### Induction Assumptions

Now we assume the formulas in parts $(a)$ and $(b)$ for triangulations with
fewer than $n$ triangles, for some $n>2$.

As mentioned above, we assume the triangulation to be pictured as in Figure
13, so that the top-most fan center $j$ is on the left. The argument for the
other case (that the top-most fan center is on the right) is similar: we would
need to use part $(b)$ of Lemma 6.1 rather than part $(a)$, and the roles of
the edges labelled $c$ and $d$ in Figure 13 would be swapped.

We will also assume that the polygon has an odd number of triangles
(equivalently, the snake graph has an even number of tiles). By Lemma 6.1,
this implies that the snake graph ends with “R”. Because $G$ has an even
number of tiles, this means the last tile of $G$ has orientation opposite of
the polygon. These assumptions imply that the end of the snake graph looks as
follows:

$h$$b$$a$$e_{2}$$e$$\theta_{n}$$\theta_{n-1}$$\theta_{n-1}$

On the other hand, if the triangulation has an even number of tiles (the snake
graph has an odd number of tiles), then the snake graph ends with “U” by Lemma
6.1, and the end of the snake graph would instead look as follows:

$a$$b$$h$$e_{2}$$\theta_{n}$$\theta_{n-1}$$\theta_{n-1}$

We will assume the former case for the remainder of the proof. The argument
for the latter case is similar: occurrences of $D_{R}(G)$ and $D_{r}(G)$ would
need to be replaced by $D_{T}(G)$ and $D_{t}(G)$.

#### Induction for part ${\bf(b)}$

First we prove part $(b)$. Looking at Figure 13, the super Ptolemy relation
gives the following expression for $\varphi$:

$\varphi=\frac{1}{\sqrt{ef}}\left(\sqrt{bd}\,\theta+\sqrt{ac}\,\sigma\right)$

Multiplying by $\sqrt{df}$ gives

$\sqrt{df}\,\varphi=\sqrt{\frac{d}{e}}\left(\sqrt{bd}\,\theta+\sqrt{ac}\,\sigma\right)=d\,\sqrt{\frac{b}{e}}\,\theta+\sqrt{\frac{acd}{e}}\,\sigma$

Recall that $D_{t}(G)=D_{T}(G)\cup D_{tr}(G)$. We claim that the two terms on
the right-hand side correspond to $D_{T}(G)$ and $D_{tr}(G)$. Consider the
first term, $d\,\sqrt{\frac{b}{e}}\,\theta$. The elements $b,e,\theta$ are all
in the initial triangulation, but $d$ might not be. Since $d$ is a diagonal in
a smaller triangulation, we may use induction and say that

$d=\frac{1}{\mathrm{cross}(d)}\,\sum_{M\in D(G^{(-k)})}\mathrm{wt}(M),$

where $k$ is the number of diagonals in the top fan segment.

Note that $G^{(-k)}$ is $G$ with the last maximal “ _staircase_ ” removed. Let
$e_{1},\dots,e_{k}$ be the last $k$ internal diagonals of the last fan segment
(with $e_{1}=e$). These appear on the snake graph in the positions indicated
in Figure 10 (Left). Then $\mathrm{cross}(f)=e_{1}\cdots
e_{k}\cdot\mathrm{cross}(d)$. Substituting the expression for $d$ into the
expression we had before, we get that the first term is

$d\,\sqrt{\frac{b}{e}}\,\theta=\frac{e_{1}\cdots
e_{k}}{\mathrm{cross}(f)}\,\sqrt{\frac{b}{e}}\,\sum_{M\in
D(G^{(-k)})}\mathrm{wt}(M)\,\theta=\frac{e_{2}\cdots
e_{k}}{\mathrm{cross}(f)}\,\sqrt{be}\sum_{M\in
D(G^{(-k)})}\mathrm{wt}(M)\,\theta$

Here we need to check that the multiplication puts $\theta$ at the correct
place in the positive order. By the induction hypothesis, the $\mu$-invariants
in the monomials $\operatorname{wt}(M)$ are already written in the positive
order with respect to $T$. Depending on the orientation of edge $e$, the
$\mu$-invariant $\theta$ can possibly be at either the beginning or end of the
order (Definition 2.4). Because of the construction in Definition 4.4, the
monomial $\mathrm{wt}(M)$ contains an even number of odd variables for any
choice of $M$. Thus $\theta$ commutes with $\operatorname{wt}(M)$, and the
resulting product can always be written in the correct order without changing
its sign.

Now we use Lemma 5.4 to conclude that

$\sum_{M\in D(G^{(-k)})}\mathrm{wt}(M)=\frac{1}{be_{2}\cdots e_{k}}\sum_{M\in
D_{T}(G)}\mathrm{wt}(M)$

Making this substitution gives

$d\sqrt{\frac{b}{e}}\,\theta=\frac{1}{\mathrm{cross}(f)}\,\sqrt{\frac{e}{b}}\sum_{M\in
D_{T}(G)}\,\mathrm{wt}(M)\,\theta$

Note that since elements of $D_{T}(G)$ cannot have a cycle around the last
tile, $\mathrm{wt}(M)\,\theta=\mathrm{wt}(M)^{*}$. So we get that the first
term is the contribution from $D_{T}(G)$, as claimed.

Next, we examine the second term, $\sqrt{\frac{acd}{e}}\,\sigma$. Recall that
we assume $c$ is the second-longest edge (crosses all but the last diagonal),
as in Figure 13. Also let $h$ be the boundary side of the polygon adjacent to
the endpoint of $c$, as in the figure. Note that $\sqrt{cd}\,\sigma$ is
precisely the left-hand side of the expression in part $(b)$ for the triangle
$\sigma$ (in the role of $\varphi$). Let $x\mapsto x^{\dagger}$ denote the
toggle involution for $\theta_{n-1}$. Then by induction, since $\sigma$ is in
a smaller polygon, we will assume the formula from part $(b)$. Since there is
one less triangle in this smaller polygon, the sum is over $D_{r}$ rather than
$D_{t}$. So we get

$\displaystyle\sqrt{\frac{acd}{e}}\,\sigma$
$\displaystyle=\sqrt{\frac{a}{e}}\,\frac{1}{\mathrm{cross}(c)}\,\sqrt{\frac{e_{2}}{h}}\sum_{M\in
D_{r}(G^{(-1)})}\mathrm{wt}(M)^{\dagger}$
$\displaystyle=\frac{\sqrt{ae}}{\mathrm{cross}(f)}\,\sqrt{\frac{e_{2}}{h}}\sum_{M\in
D_{r}(G^{(-1)})}\mathrm{wt}(M)^{\dagger}$

where the products are already in the correct positive order by induction.
Finally, we use Lemma 5.8 to substitute the summation over $D_{r}(G^{(-1)})$
for a summation over $D_{tr}(G)$:

$\sum_{M\in D_{r}(G^{(-1)})}\mathrm{wt}(M)^{\dagger}=\sum_{M\in
D_{tr}(G)}\sqrt{\frac{h}{abe_{2}}}\mathrm{wt}(M)^{*}$

Making this substitution cancels the $\sqrt{\frac{ae_{2}}{h}}$ factors, and
gives the result:

$\sqrt{\frac{acd}{e}}\,\sigma=\frac{1}{\mathrm{cross}(f)}\,\sqrt{\frac{e}{b}}\sum_{M\in
D_{tr}(G)}\mathrm{wt}(M)^{*}$

#### Induction for part ${\bf(a)}$

Consider the Ptolemy relation Equation 1 on the quadrilateral $(i,j,k,l)$:

(4) $f={ac\over e}+{bd\over e}+{\sqrt{abcd}\over e}\sigma\theta$

First we examine the first term in Equation 4: $ac/e$. Since $c$ is the
longest arc in the smaller polygon with one less triangle, we have by
induction that

(5) $c=\frac{1}{\operatorname{cross}(c)}\sum_{M\in
D(G^{(-1)})}\operatorname{wt}(M)$

where $G^{(-1)}$ is the snake graph for $c$, which is obtained by removing the
last tile from $G$. Now multiply Equation 5 by $a/e$, noting that
$\operatorname{cross}(f)=e\cdot\operatorname{cross}(c)$, to get

${ac\over e}={1\over\operatorname{cross}(f)}\sum_{M\in
D(G^{(-1)})}\operatorname{wt}(M)a$

By Lemma 5.3, double dimer covers in $G^{(-1)}$ are exactly those in $D$ which
have two dimers on edge $a$ of the last tile (see Figure 10). Hence we have:

(6) ${ac\over e}=\frac{1}{\operatorname{cross}(f)}\sum_{M\in
D_{R}(G)}\operatorname{wt}(M)$

Next we examine the term $bd/e$ in Equation 4.

Recall that by induction, $d$ is given by

(7) $d={1\over\operatorname{cross}(d)}\sum_{M\in
D(G^{(-k)})}\operatorname{wt}(M)$

Multiply Equation 7 by $b/e$ to get

${bd\over e}={1\over\operatorname{cross}(d)}\cdot{b\over e}\sum_{M\in
D(G^{(-k)})}\operatorname{wt}(M)$

Since $\operatorname{cross}(f)=\operatorname{cross}(d)ee_{2}e_{3}\cdots
e_{k}$, we have

${bd\over e}={1\over\operatorname{cross}(f)}\sum_{M\in
D(G^{(-k)})}\operatorname{wt}(M)be_{2}e_{3}\cdots e_{k}$

Now by Lemma 5.4, double dimer covers in $D(G^{(-k)})$ are in bijection with
those in $D_{T}(G)$ up to multiplying by the weight of edges in the first $k$
tiles $ee_{2}e_{3}\cdots e_{k}$ (see Figure 10). Hence we have:

(8) ${bd\over e}={1\over\operatorname{cross}(f)}\sum_{M\in
D_{T}(G)}\operatorname{wt}(M)$

Note that in Equations 6 and 8, the products of $\mu$-invariants are taken
under the positive order by induction.

Lastly we examine the term ${\sqrt{abcd}\over e}\sigma\theta$ in Equation 4.
By induction hypothesis of part (b) we have

(9) $\sqrt{cd}\sigma={1\over\operatorname{cross}(c)}\sqrt{e_{2}\over
h}\sum_{M\in D_{r}(G^{(-1)})}\operatorname{wt}(M)^{\dagger}$

Here $G^{(-1)}$ has an even number of tiles, therefore the sum is over
$D_{r}(G^{(-1))}$ rather than $D_{t}(G^{(-1)})$. Right multiply Equation 9 by
${\sqrt{ab}\over e}\theta$ we get

$\displaystyle{\sqrt{abcd}\over e}\sigma\theta$
$\displaystyle={1\over\operatorname{cross}(c)}\cdot{\sqrt{ab}\over
e}\cdot\sqrt{e_{2}\over h}\sum_{M\in
D_{r}(G^{(-1)})}\operatorname{wt}(M)^{\dagger}\theta$
$\displaystyle={1\over\operatorname{cross}(f)}\sqrt{abe_{2}\over h}\sum_{M\in
D_{r}(G^{(-1)})}\operatorname{wt}(M)^{\dagger}\theta$

Since we assumed that the direction of edge $e$ is from left to right as
depicted in Figure 13, $\theta$ will appear at the end of the positive order
of $T$, hence right-multiplication of $\theta$ will result in the correct
order. In the other situation where the edge $e$ goes from right to left, the
product in Equation 4 would be $\theta\sigma$ instead, giving the positive
order in that case.

Note that for $M\in D_{r}(G^{(-1)})$, we have
$\operatorname{wt}(M)^{\dagger}\theta=(\operatorname{wt}(M)^{\dagger})^{*}$.
Hence applying the bijection in Lemma 5.8, we get

(10) ${\sqrt{abcd}\over
e}\sigma\theta={1\over\operatorname{cross}(f)}\sum_{M\in
D_{tr}(G)}\operatorname{wt}(M).$

Finally, combining Equations 6, 8 and 10 gives us

$f=\frac{1}{\mathrm{cross}(f)}\sum_{M\in D(G)}\mathrm{wt}(M),$

which completes the proof. ∎

##  7\. Relationship Between Different Combinatorial Models for Super
$\lambda$-Lengths

In this section, we relate the combinatorial interpretation of super
$\lambda$-lengths given by double dimer covers of snake graphs, as in Theorem
6.2 (a), to our previous combinatorial interpretation given in [MOZ21] by
super $T$-paths. As a by-product of our efforts towards this comparison, we
introduce an additional family of combinatorial objects, which we call
_twisted_ super $T$-paths, that yield the same elements of the super algebra
$A=\mathbb{R}[x_{ij}^{\pm 1/2}\leavevmode\nobreak\ |\leavevmode\nobreak\
\theta_{k}]$.

###  7.1. Twisted Super $T$-Paths

Recall from Section 4 of [MOZ21] that we distinguished certain vertices of a
triangulation $T$ to be fan centers and consequently defined an auxiliary
graph from this data. To match our notation to that of [MOZ21], we fix a
choice of arc $(a,b)$, noting this is the arc whose super $\lambda$-length we
wish to compute, i.e. $\lambda_{ab}$. Then recall by restricting to a sub-
triangulation $T(a,b)\subset T$ if necessary, we assume $(a,b)$ is the longest
arc in $T(a,b)$.

For a triangulation $T$ and a pair of vertices $a$ and $b$, the auxiliary
graph is the graph of the triangulation $T$ with some additional vertices and
edges.

  1. (1)

For each face of the triangulation $T$, we place an internal vertex, which
lies on the arc $(a,b)$. We denote the internal vertices using
$\theta_{1},\cdots,\theta_{n+1}$, such that $\theta_{i}$ is closer to $a$ than
$\theta_{j}$ if and only if $i<j$.

  2. (2)

For each face of $T$, we add an edge $\sigma_{i}:=(\theta_{i},c_{j})$
connecting the internal vertex $\theta_{i}$ to the center of the fan segment
which contains $\theta_{i}$. We denote by $\sigma$ the set of all such edges.

  3. (3)

For each $\theta_{i}$ and $\theta_{j}$ with $i<j$, we add an edge connecting
$\theta_{i}$ and $\theta_{j}$. We denote the collection of these edges as
$\tau=\\{\tau_{ij}:i<j\\}$. For simplicity the $\tau$-edges are drawn to be
overlapping.

See Figure 15 for an example. Note that we continue to follow the convention
of Definition 3.2 and assume that the longest edge $(a,b)$ traverses the
triangulation $T$ from bottom-to-top.

$\ \ b$$y\ \ $$c_{1}$$a\ \ $$\ \
x$$c_{2}$$\theta_{4}$$\theta_{3}$$\theta_{2}$$\theta_{1}$$\sigma_{4}$$\sigma_{3}$$\sigma_{2}$$\sigma_{1}$
Figure 15. The auxiliary graph as defined in [MOZ21].

We now define a twisted auxiliary graph $\Gamma_{T}^{a,b}$ that we use to
define twisted super $T$-paths. For our alternative twisted definition we
introduce new $\sigma$-edges by using the complements of the fan centers. More
precisely, given a triangulation $T$ admitting longest edge $(a,b)$ as above,
for every triangle we associate two $\sigma$-edges (rather than one) and note
that one of these two $\sigma$-edges is closer to the starting vertex $a$
while the other is closer to the ending vertex $b$. We color the former in
thick blue and call it a $\sigma^{A}$-edge and color the latter in cyan and
call it a $\sigma^{B}$-edge. For each $\theta_{i}$ and $\theta_{j}$ with
$i<j$, $\Gamma_{T}^{a,b}$ contains the edge $\tau_{ij}:i<j$ just as before.
See Figure 16 for an example.

Just as super $T$-paths follow edges of the auxiliary graph, as defined in
[MOZ21], we will define twisted super $T$-paths to follow edges of this newly
defined twisted auxiliary graph $\Gamma_{T}^{a,b}$.

$\ \ b$$y\ \ $$c_{1}$$a\ \ $$\ \
x$$c_{2}$$\theta_{4}$$\theta_{3}$$\theta_{2}$$\theta_{1}$$\sigma^{A}_{4}$$\sigma^{B}_{4}$$\sigma^{A}_{3}$$\sigma^{B}_{3}$$\sigma^{A}_{2}$$\sigma^{B}_{2}$$\sigma^{A}_{1}$$\sigma^{B}_{1}$
Figure 16. The twisted auxiliary graph $\Gamma_{T}^{a,b}$.

We define twisted super $T$-paths from $a$ to $b$ axiomatically as follows.

######  Definition 7.1.

A _twisted super $T$-path_ $t$ from $a$ to $b$ is a sequence

$t=(a_{0},a_{1},\cdots,a_{\ell(t)}|t_{1},t_{2},\cdots,t_{\ell(t)})$

such that

  1. (T1)

$a=a_{0},a_{1},\cdots,a_{\ell(t)}=b$ are vertices on $\Gamma_{T}^{a,b}$.

  2. (T2)

For each $1\leq i\leq\ell(t)$, $t_{i}$ is an edge in $\Gamma_{T}^{a,b}$
connecting $a_{i-1}$ and $a_{i}$.

  3. (T3)

$t_{i}\neq t_{j}$ if $i\neq j$.

  4. (T4)

$\ell(t)$ is odd.

  5. (T5’)

$t_{i}$ crosses $(a,b)$ if $i$ is even. The $\tau$-edges are considered to
cross $(a,b)$, and any step along a $\tau$-edge must end further from endpoint
$a$ and closer to endpoint $b$.

  6. (T6’)

$t_{i}\in\sigma$ only if $i$ is odd, $t_{i}\in\tau$ only if $i$ is even.

  7. (T7)

If $i<j$ and both $t_{i}$ and $t_{j}$ cross the arc $(a,b)$, then the
intersection $t_{i}\cap(a,b)$ is closer to the vertex $a$ than the
intersection $t_{j}\cap(a,b)$.

We define $\mathcal{T}_{a,b}$ to be the set of twisted super $T$-paths from
$a$ to $b$.

######  Remark 7.2.

This is a slight variant of Definition 4.2 of [MOZ21] with the only difference
being that we replaced the previous axioms

  1. (T5)

$t_{i}$ crosses $(a,b)$ if $i$ is even. The $\sigma$-edges are considered to
cross $(a,b)$.

  2. (T6)

$t_{i}\in\sigma$ only if $i$ is even, $t_{i}\in\tau$ only if $i$ is odd

with the axioms (T5’) and (T6’), thereby switching the parity of
$\sigma$-steps and $\tau$-steps.

We also define new weights for such twisted super $T$-paths. (Note that the
weights of $\sigma$-edges differ from Definition 4.8 of [MOZ21] since
$\sigma$-edges are now assumed to be odd steps rather than even steps.)

######  Definition 7.3.

Let $t\in\mathcal{T}_{ab}$ be a twisted super $T$-path which uses edges
$t_{1},t_{2},\dots$ in the twisted auxiliary graph $\Gamma^{T}_{a,b}$. We will
assign to each edge $t_{i}$ a twisted weight, which will be an element in the
super algebra
$\mathbb{R}[x_{1}^{\pm\frac{1}{2}},\cdots,x_{2n+3}^{\pm\frac{1}{2}}\leavevmode\nobreak\
|\leavevmode\nobreak\ \theta_{1},\cdots,\theta_{n+1}]$ (where $\theta_{i}$’s
are the odd generators) as follows. For the parity of edges $t_{i}\in\sigma$
or $\tau$, we recall axiom (T6’) of Definition 7.1.

$\operatorname{twt}(t_{i}):=\begin{cases}x_{j}&\text{ if }t_{i}\in T\text{,
with }\lambda\text{-length }x_{j}\text{, and }i\text{ is odd}\\\
x_{j}^{-1}&\text{ if }t_{i}\in T\text{, with }\lambda\text{-length
}x_{j}\text{, and }i\text{ is even}\\\
\sqrt{\frac{x_{k}x_{l}}{x_{j}}}\,\theta_{s}&\text{ if }t_{i}\in\sigma\text{
and the face containing }t_{i}\text{ is as pictured below}\leavevmode\nobreak\
(i\text{ must be odd})\\\ 1&\text{ if }t_{i}\in\tau\leavevmode\nobreak\
(i\text{ must be even})\end{cases}$

$\theta_{s}$$t_{i}$$x_{l}$$x_{k}$$x_{j}$

Here, $\theta_{s}$ is the $\mu$-invariant associated to the face containing
$t_{i}$. Finally, we define the (twisted) weight of a twisted super $T$-path
to be the product of the (twisted) weights of its edges

$\operatorname{twt}(t)=\prod_{t_{i}\in t}\operatorname{twt}(t_{i})$

where the product of $\mu$-invariants is taken under the positive order.

We use this to arrive at an analogue of Theorem 4.9 of [MOZ21].

######  Theorem 7.4.

Under the default orientation, the $\lambda$-length of $(a,b)$ is given by

$\lambda_{a,b}=\sum_{t\in\mathcal{T}_{a,b}}\operatorname{twt}(t)$

where we use twisted super $T$-paths on the twisted auxiliary graph, and with
twisted weights of each of the steps.

Before proving this theorem, we construct a weight-preserving bijection
between super $T$-paths, as defined in [MOZ21], and twisted super $T$-paths,
as introduced in Definition 7.1. For this bijection, it will also be useful to
define _twisted super steps_ in analogy to super steps in [MOZ21].

######  Definition 7.5.

A triple of steps along a twisted super $T$-path which consist of a
$\sigma$-edge, followed by a $\tau$-edge, and then a second $\sigma$-edge will
be called a _twisted super step_.

######  Lemma 7.6.

Given a triangulation $T$ of a polygon such that $(a,b)$ is its unique longest
arc, there is a weight-preserving bijection $\psi$ that maps every super
$T$-path $t=(t_{1},t_{2},\dots,t_{\ell(t)})$ to a _twisted_ super $T$-path
$\psi(t)=(t_{1}^{\prime},t_{2}^{\prime},\dots,t_{\ell(t)^{\prime}}^{\prime})$
through the auxiliary graph $\Gamma_{T}^{a,b}$. Furthermore, for each super
$T$-path $t$, the weight of $t$, i.e. $wt(t)=\prod_{i=1}^{\ell(t)}wt(t_{i})$,
and the twisted weight of $\psi(t)=t^{\prime}$, i.e.
$twt(t^{\prime})=\prod_{i=1}^{\ell(t^{\prime})}twt(t_{i}^{\prime})$, coincide.

###### Proof.

We construct a bijection between super $T$-paths and twisted super $T$-paths
via the following:

Recall that a non-ordinary super $T$-path, as defined in [MOZ21], involves
super steps, which consists of the three-step combination of
$\sigma_{i}\tau_{ij}\sigma_{j}$ where $\sigma_{i}$ is the incoming edge to the
face center corresponding to $\theta_{i}$, $\tau_{ij}$ is the teleportation
between the face centers for $\theta_{i}$ and $\theta_{j}$, and $\sigma_{j}$
is the outgoing edge to the face center corresponding to $\theta_{j}$.
Further, we assume that $\sigma_{i}$ and $\sigma_{j}$ are even steps while
$\tau_{ij}$ is an odd step.

In our bijection, we either replace two out of three steps of a super step
with either a single step or a three step combination. The consequence of this
map switches the parity of $\sigma$-steps as needed while adding (or deleting)
an ordinary step. The parity of the $\tau_{ij}$ steps is similarly switched
from odd to even. These switches are also local in the sense that the
remainder of the super $T$-paths may be left alone. Locally, there are four
possible cases, and they are each illustrated in Figure 17. Traversing the
twisted auxiliary graph $\Gamma_{T}^{a,b}$ from bottom-to-top, in cases (i)
and (ii), we illustrate the two ways a super $T$-path (resp. super $T$-path)
may enter the triangle with face center $\theta_{i}$ followed by a
$\tau_{ij}$-step. (Blue or cyan steps correspond to odd-indexed steps while
red steps correspond to even-indexed steps.) Cases (iii) and (iv) illustrate
the analogous options for exiting the triangle with face center $\theta_{j}$
proceeding after a $\tau_{ij}$-step.

ORIGINALTWISTED(iii)$x$$y$$z$$\sigma_{j}$$\theta_{j}$$x$$y$$z$$\sigma_{j}$$\theta_{j}$(iv)$x$$y$$z$$\sigma_{j}$$\theta_{j}$$x$$y$$z$$\sigma_{j}$$\theta_{j}$(i)$a$$b$$c$$\sigma_{i}$$\theta_{i}$$a$$b$$c$$\sigma_{i}$$\theta_{i}$(ii)$a$$b$$c$$\sigma_{i}$$\theta_{i}$$a$$b$$c$$\sigma_{i}$$\theta_{i}$
Figure 17. Illustration of the bijection between super $T$-paths and twisted
super $T$-paths; here the vertical steps through edge $c$ (or respectively
$z$) indicate the beginning (resp. ending) of a $\tau$-step, i.e.
teleportation

Furthermore, comparing weights to twisted weights, this transformation is
weight-preserving. We verify this case-by-case following Figure 17. In
particular, we see the weights of the $\sigma$-step (or corresponding two
steps) on the left-hand-side (respectively right-hand-side) are as in Table 1.

| ORIGINAL |  |  |  |  |  | TWISTED
---|---|---|---|---|---|---|---
(i) | $\displaystyle\sqrt{\frac{a}{bc}}\theta_{i}$ |  |  |  |  |  | $\displaystyle\frac{1}{b}\cdot\sqrt{\frac{ab}{c}}\theta_{i}$
(ii) | $\displaystyle b\cdot\sqrt{\frac{a}{bc}}\theta_{i}$ |  |  |  |  |  | $\displaystyle\sqrt{\frac{ab}{c}}\theta_{i}$
(iii) | $\displaystyle\sqrt{\frac{x}{yz}}\theta_{j}$ |  |  |  |  |  | $\displaystyle\frac{1}{y}\cdot\sqrt{\frac{xy}{z}}\theta_{j}$
(iv) | $\displaystyle y\cdot\sqrt{\frac{x}{yz}}\theta_{j}$ |  |  |  |  |  | $\displaystyle\sqrt{\frac{xy}{z}}\theta_{j}$
Table 1. Weights of pieces of a super $T$-path (Left) or twisted super
$T$-path (Right), respectively

Comparing these expressions, we see that these weights on the left- and right-
hand sides indeed agree. Since $\tau$-steps are unweighted, notice that
changing the parity of the $\tau$-step does not affect the weight of the super
or twisted super $T$-path. We thus first apply this transformation to a
$\sigma$-step in the triangle corresponding to the $\mu$-invariant
$\theta_{i}$, as illustrated in case (i) or (ii). Then second, we switch the
parity of the $\tau$-step $\tau_{ij}$ without changing the weight. Finally,
third, we apply the same transformation to a $\sigma$-step in the triangle
corresponding to the $\mu$-invariant $\theta_{j}$, as illustrated in case
(iii) or (iv). Thus we have replaced a super step with a twisted super step,
or vice-versa.

These transformations can be applied either from left-to-right (original to
twisted super $T$-paths) or instead from right-to-left, thus providing the
inverse map for this bijection. ∎

As a consequence of this bijection, we obtain important constraints on
$\sigma$-steps of a twisted super $T$-paths that go beyond the axioms listed
above.

######  Corollary 7.7.

In addition to axioms (T1), (T2), (T3), (T4), (T5’), (T6’), and (T7), we can
deduce that in any twisted super $T$-path, we have the following two
properties:

  * $(i)$

we only allow edges moving from a boundary vertex to a face center along a
$\sigma^{A}$-edge or allow edges moving from a face center to a boundary
vertex along a $\sigma^{B}$-edge.

  * $(ii)$

any $\tau$-edge, e.g. $\tau_{ij}$ between face center $\theta_{i}$ and face
center $\theta_{j}$, must be immediately preceded by the edge $\sigma^{A}_{i}$
and immediately followed by the edge $\sigma^{B}_{j}$.

As a consequence of this corollary, twisted super steps must specifically have
the form of the edge $\sigma^{A}_{i}$ followed by $\tau_{ij}$ and then
$\sigma^{B}_{j}$. Thus we may be more precise than in Definition 7.5.

###### Proof of Theorem 7.4.

This is now an immediate consequence of combining Lemma 7.6 and Theorem 4.9 of
[MOZ21]. ∎

###  7.2. Bijection between twisted super $T$-paths and double dimer covers

In this subsection we use the new combinatorial objects presented in the
previous subsection to give a weight-preserving bijection between double dimer
covers on snake graphs and super $T$-paths. In light of the bijection from
Lemma 7.6, it is sufficient to provide a weight-preserving bijection between
_twisted_ super $T$-paths and double dimer covers.

######  Theorem 7.8.

Given the hypotheses of Lemma 7.6, i.e. a triangulation $T$ with longest arc
$(a,b)$, we build the corresponding snake graph $G$ following the construction
in Definition 3.2. Then we have a weight-preserving bijection between twisted
super $T$-paths and double dimer covers of $G$.

###### Proof.

As indicated in Corollary 7.7, every teleportation step, i.e. $\tau$-edge, of
a twisted super $T$-path is preceded by a $\sigma^{A}$-edge and followed by a
$\sigma^{B}$-edge. We thus are able to decompose any twisted super $T$-path
into building blocks of three different possible types:

  * $(i)$

Odd-indexed edges $t_{i}$ that travel along either a boundary edge or internal
diagonal of triangulation $T$. We will refer to such edges as blue steps.

  * $(ii)$

Even indexed edges $t_{i}$ that must travel along an internal diagonal of
triangulation $T$. We will refer to such edges as red steps.

  * $(iii)$

Twisted super steps, which are triples
$(\sigma^{A}_{i},\tau_{ij},\sigma^{B}_{j})$ such that $\sigma^{A}_{i}$ (resp.
$\sigma^{B}_{j}$) is an entering (resp. exiting) $\sigma$-step, and
$\tau_{ij}$ is a teleportation step between the two associated face centers
$\theta_{i}$ and $\theta_{j}$. By Axiom (T6’) in Definition 7.1, the steps
$\sigma^{A}_{i}$ and $\sigma^{B}_{j}$ are odd-indexed while $\tau_{ij}$ is
even-indexed.

We map twisted super $T$-paths to double dimer covers in $G$, the snake graph
associated to arc $\gamma$ and triangulation $T$, via these building blocks.
First off, we map twisted super steps to pieces in double dimer covers that
correspond to a cycle around a connected sub-snake graph. More precisely,
consider a twisted super step $(\sigma^{A}_{i},\tau_{ij},\sigma^{B}_{j})$ such
that $\tau_{ij}$ teleports from the face center associated to $\theta_{i}$ to
the face center associated to $\theta_{j}$. Then there is a unique tile of the
snake graph $G$ that contains a triangle in the lower left corner labeled by
$\theta_{i}$ and a unique tile of $G$ that contains a triangle in the upper
right corner labeled by $\theta_{j}$. Further, this second tile is to the
northeast of the first tile since the $j$th face center is closer to the end
of $\gamma$ than the $i$th face center, due to axiom (T5’). The cycle
circumscribes the connected sub-snake graph beginning and ending at these two
tiles.

Since a twisted super step must begin and end with an odd-indexed step, a
twisted super $T$-path cannot contain two twisted super steps immediately
following one another. Instead, unless they lie at the beginning or end of the
twisted super $T$-path, they must be immediately preceded and followed by a
red edge. Excluding twisted super steps and these accompanying red edges, the
remaining connected pieces of a twisted super $T$-path all consist entirely of
blue and red edges, and are each of odd-length. Due to the axioms of
Definition 7.1, these connected pieces are ordinary $T$-paths of a
subtriangulation $S$ of $T$ with longest diagonal $\gamma^{\prime}$. It thus
suffices to apply the bijection from [MS10] to obtain dimer covers of sub-
snake graphs associated to such ordinary $T$-paths on subtriangulation $S$ and
longest diagonal $\gamma^{\prime}$. If $\gamma^{\prime}$ starts on the bottom
of the triangle with face center $\theta_{i}$ and ends at the top of the
triangle with face center $\theta_{j}$, we get a dimer cover on the connected
sub-snake graph between the tile with lower-left corner $\theta_{i}$ and
upper-right corner $\theta_{j}$. We duplicate each of the edges in this dimer
cover to obtain a double dimer cover.

Lastly, we note that the red edges immediately preceding or following a
twisted super step correspond to blank tiles in the snake graph $G$ such that
if the red edge is on the diagonal $\tau_{k}$, then the blank tile in $G$ is
the unique one with diagonal $x_{k}$.

Every diagonal of $T$ is either crossed by one of these ordinary $T$-paths, by
one of the red edges preceding or following a twisted super step, or by the
$\tau$-step as part of a twisted super step. Hence every tile of $G$ is
respectively either covered by doubled edges, a blank tile, or a cycle of
single edges. The inverse map combines the inverse map in [MS10] to get
ordinary $T$-paths from double dimer covers on a connected sub-snake graph
consisting exclusively of doubled edges and sending cycles to the unique
twisted super step defined by the face centers associated to the beginning and
ending tiles.

With this combinatorial bijection defined, we wish to also verify that the
weights of twisted super $T$-paths agree with weights of the associated double
dimer covers. Again, we utilize the decomposition of a twisted super $T$-path
into blue edges, red edges and twisted super steps. The weights of blue edges
correspond to the weights of the associated dimer in $G$, using the fact that
we map blue edges in a twisted super $T$-path to a doubled-edge in $G$ but
then weight it by a square-root.

Similarly, the weights of red edges do not contribute anything to the
numerator but either correspond to a piece of an ordinary $T$-path or to a
blank tile associated to the edge preceding or following a twisted super step.
Either way, the weights of such red edges agree with the contributions to the
weight of the double dimer cover given by the crossing monomial in the
denominator.

Finally, a twisted super step consists of a $\sigma^{A}_{i}$-step, an
unweighted $\tau$-step, and a $\sigma^{B}_{j}$-step. As illustrated in the
right-hand-side of Figure 17, in cases (ii) and (iv), the weight of the
$\sigma^{A}_{i}$-edge is $\sqrt{\frac{ab}{c}}\theta_{i}$ while the weight of
the $\sigma^{B}_{j}$-edge is $\sqrt{\frac{xy}{z}}\theta_{j}$.

Letting $\gamma_{AB}$ denote the arc starting from the source of the
$\sigma^{A}_{i}$-edge and ending at the target of the $\sigma^{B}_{j}$-edge,
let $G_{\gamma_{AB}}$ be the corresponding snake graph, as defined in
Definition 3.2. By construction, $a$ and $b$ are the weights of the SW edges
of the first tile in $G_{\gamma_{AB}}$ while $x$ and $y$ are the weights of
the NE edges of the last tile.

Following Theorem 3.4, we consider the denominator cross$(\gamma_{AB})$ which
is the product of the weights of all diagonals in $G_{\gamma_{AB}}$. However,
the snake graph and its edge-weighting is also constructed so that the
following is true:

(*) For every444Exluding the diagonal edges corresponding to the first and
last tiles of $G_{\gamma_{AB}}$. diagonal edge in $G_{\gamma_{AB}}$, the N or
E edge of the previous tile has the same weight, as does the S or W edge of
succeeding tile. In both cases, we pick the one edge out of the two which is a
boundary (as opposed to internal) edge of the snake graph.

As a consequence, the product of the square root of edge weights of all
boundary edges of $G_{\gamma_{AB}}$ exactly equals
$\sqrt{ab\cdot\frac{\mathrm{cross}(\gamma_{AB})^{2}}{cz}\cdot xy}$ where $c$
(respectively $z$) denotes the diagonal of the first (resp. last) tile of
$G_{\gamma_{AB}}$. We include $\theta_{i}$ and $\theta_{j}$ in this product,
corresponding to the beginning and end of the twisted super step; or the first
and last triangles of the corresponding snake graph.

In conclusion, the weight contributed to a twisted super $T$-path by a twisted
super step consisting of ($\sigma^{A}_{i},\tau_{ij},\sigma^{B}_{j}$) agrees
with the weight contributed to a double dimer cover via a cycle of single
edges around the associated sub-snake graph. ∎

##  8\. Lattice Paths and Dual Snake Graphs

###  8.1. Duality of Snake Graphs

In this section, we describe an involution on the set of all snake graphs
which was described in [Pro05], and also discussed extensively in [Cla20].
Under this involution, dimer covers are taken to lattice paths. By a _lattice
path_ on a snake graph, we mean a path from the bottom-left corner of the
first tile to the top-right corner of the last tile, such that every step goes
either right or up.

######  Definition 8.1.

Define an involution $x\mapsto\overline{x}$ on the set
$\\{\text{R},\text{U}\\}$ given by $\overline{\text{R}}=\text{U}$ and
$\overline{\text{U}}=\text{R}$.

######  Definition 8.2.

Define an involution $w\mapsto\overline{w}$ on the set of words in the
alphabet $\\{\text{R},\text{U}\\}$ as follows. If $w=w_{1}w_{2}w_{3}\cdots
w_{2n}$ or $w=w_{1}w_{2}w_{3}\cdots w_{2n+1}$, then define
$\overline{w}=\overline{w}_{1}w_{2}\overline{w}_{3}\cdots w_{2n}$ or
$\overline{w}_{1}w_{2}\overline{w}_{3}\cdots w_{2n}\overline{w}_{2n+1}$. In
other words, the involution toggles the odd-numbered letters.

######  Definition 8.3.

The involution on snake graphs is defined by applying the involution to the
word $W(G)$. In other words, for a snake graph $G$, we define the _dual snake
graph_ $\overline{G}$ so that $W(\overline{G})=\overline{W(G)}$.

If the snake graph $G$ is labelled, then $\overline{G}$ inherets a labelling
in the following way. Recall the tiles $T_{i}$ of $G$ alternate in
orientation, so that $T_{2k+1}$ has usual orientation, and $T_{2k}$ has
reversed orientation. For the dual snake graph, if $i$ is odd, then the bottom
and left sides of $\overline{T}_{i}$ are labelled the same as $T_{i}$, but the
top and right labels are swapped. If $i$ is even, then the top and right sides
of $\overline{T}_{i}$ are the same as $T_{i}$, and the bottom and left are
swapped. This is illustrated in Figure 18. An example of two dual snake graphs
is pictured in Figure 19.

$T_{2k+1}$$b$$c$$d$$a$$\overline{T}_{2k+1}$$b$$d$$c$$a$$T_{2k}$$b$$c$$d$$a$$\overline{T}_{2k}$$a$$c$$d$$b$
Figure 18. Relation between tile labeling in $G$ and $\overline{G}$
$a$$b$$x$$j$$c$$y$$i$$d$$z$$h$$e$$f$$g$$W(G)=\text{RRR}$$a$$b$$j$$x$$y$$i$$c$$d$$h$$z$$f$$g$$e$$W(G)=\text{URU}$
Figure 19. Two dual snake graphs

There is a bijection, described in [Pro05] and [Cla20], between perfect
matchings (dimer covers) of $G$ and lattice paths in $\overline{G}$ going from
the bottom left to the top right corner. With the labelling convention
described above for $\overline{G}$, this bijection is weight-preserving, where
the weight of a lattice path is the product of the weights of the edges in the
path. An example of this correspondence is pictured in Figure 20.

$a$$b$$x$$j$$c$$y$$i$$d$$z$$h$$e$$f$$g$$a$$b$$j$$x$$y$$i$$c$$d$$h$$z$$f$$g$$e$
Figure 20. Correspondence between dimer covers of $G$ and lattice paths in
$\overline{G}$

###  8.2. Double Lattice Paths

Recall that a double dimer cover is a multiset of edges such that each vertex
is incident to exactly two edges from the multiset. Another way of describing
it is as follows. Let $C(G)$ be the set of dimer covers of a snake graph $G$.
There is a natural map $\pi\colon C(G)\times C(G)\to D(G)$. For each pair of
dimer covers, $\pi$ maps the pair to the corresponding multiset. This map is
not in general injective, and different pairs can result in the same double
dimer cover. Despite this non-uniqueness, it is often convenient to think of
double dimer covers as simply pairs of dimer covers.

In the previous section, we described a bijection which takes dimer covers on
$G$ to lattice paths on $\overline{G}$. Let $\ell(\overline{G})$ be the set of
lattice paths on $\overline{G}$, and let $f\colon C(G)\to\ell(\overline{G})$
be this bijection. We then naturally get a map $f\times f\colon C(G)\times
C(G)\to\ell(\overline{G})\times\ell(\overline{G})$. As before, we have a map
$\pi^{\prime}$ from $\ell(\overline{G})\times\ell(\overline{G})$ to the set of
multisets of edges on $\overline{G}$.

######  Definition 8.4.

Given a snake graph $G$, define the set $L(\overline{G})$ to be the image of
the map $\pi^{\prime}$. Its elements are called _double lattice paths_ on
$\overline{G}$.

In other words, we want to think of a double lattice path as the superposition
of two lattice paths (despite the fact that, as with double dimer covers, this
is not always unique).

As with double dimer covers, we draw edges of a lattice path as wavy orange
lines, where a solid blue line indicates two edges superimposed.

The following is a convenient way of specifying a double lattice path.

######  Definition 8.5.

Given a snake graph $G$, consider all ways to label each tile with one of the
numbers $0$, $1$, or $2$. Let $a_{i}$ be the label of tile $T_{i}$. Let $X(G)$
be the subset of labelings such that:

  * •

If $T_{i+1}$ is to the right of $T_{i}$, then $a_{i}\leq a_{i+1}$

  * •

If $T_{i+1}$ is above $T_{i}$, then $a_{i}\geq a_{i+1}$

######  Theorem 8.6.

There is a bijection between $X(G)$ and $L(G)$.

###### Proof.

Given a double lattice path, choose a pre-image of $\pi^{\prime}$ in
$\ell(G)\times\ell(G)$. In other words, choose a pair of lattice paths which
represents the double lattice path. Label each tile according to how many of
the two lattice paths go above this tile. Because the lattice paths can only
go up and right, this ensures the inequalities, so this gives an element of
$X(G)$. Conversely, given a choice of labels $0$, $1$, or $2$ for each tile,
the double lattice path can uniquely be reconstructed. ∎

######  Definition 8.7.

The weight of a double lattice path $P$ is the product of the square roots of
all edge labels in $P$, times the product of odd variables corresponding to
the beginning and end of each cycle in $P$. The product of odd variables is
taken in the positive order.

The following is immediate from the weight-preserving bijection mentioned in
the previous section.

######  Theorem 8.8.

Let $G$ be the snake graph of a diagonal $\gamma=(i,j)$ in a triangulated
polygon. Then

$x_{ij}=\frac{1}{\mathrm{cross}(i,j)}\sum_{P\in
L(\overline{G})}\mathrm{wt}(P)$

##  9\. Order Ideals and Distributive Lattices

It was shown in [Pro02] that the set of dimer covers of a planar graph forms a
distributive lattice, whose cover relations are local moves called _twists_.
By Birkhoff’s theorem, every distributive lattice is isomorphic to the lattice
of order ideals of some poset. For a poset $P$, let $J(P)$ denote the lattice
of order ideals in $P$ (ordered by inclusion). For the example of the lattice
of dimer covers of a snake graph $G$, a description of this poset was given in
[MSW13] (and studied also in [Cla20] and [MSS21]).

Using the bijection described earlier between dimer covers of $G$ and lattice
paths of $\overline{G}$, it will be easier to use the latter description. We
will describe a bijection between lattice paths of $\overline{G}$ and lower
order ideals of a certain poset $P(\overline{G})$. The Hasse diagram of
$P(\overline{G})$ is constructed as follows. Put a vertex inside each tile of
$\overline{G}$, and connect two vertices with an edge (a cover relation in
$P(\overline{G})$) if the tiles share a side. Finally, rotate this picture
clockwise $45^{\circ}$ to get the Hasse diagram of $P(\overline{G})$. This is
pictured in Figure 21.

Figure 21. The Hasse diagram of the poset $P(\overline{G})$

We define an order relation on the set of lattice paths in $\overline{G}$ as
follows. The minimal lattice path is the one which follows the bottom-right
edges of the boundary of $\overline{G}$, and the maximal path is the one wich
uses the top-left edges of the boundary. If a lattice path uses both the
bottom and right edge of some tile, then this path is covered by the path
which swaps those for the top and left edges of the same tile.

The bijection between lattice paths and order ideals is as follows. As
described above, the elements of $P(\overline{G})$ correspond to the tiles of
$\overline{G}$. For each lattice path, there is some subset of tiles which are
underneath the path. The corresponding subset of vertices of the Hasse diagram
of $P(\overline{G})$ is an order ideal. The minimal lattice path has no tiles
underneath it, and so corresponds to the empty set. An example is pictured in
Figure 22.

Figure 22. A lattice path in $\overline{G}$ and its corresponding order ideal
in $P(\overline{G})$

We can extend this idea to the current situation of double dimer covers and
double lattice paths. Let $\mathbb{P}(\overline{G})$ be the product of
$P(\overline{G})$ with a chain of length 2 (with elements $0<1$). The
following is the double dimer analogue of the situation described above.

######  Theorem 9.1.

There is a poset isomorphism $L(G)\cong J(\mathbb{P}(G))$, between the set of
double lattice paths in $G$ and the lattice of lower order ideals in
$\mathbb{P}(G)$.

###### Proof.

Since $\mathbb{P}(G)=P(G)\times\\{0,1\\}$, we will write elements as $(x,0)$
or $(x,1)$, where $x\in P(G)$.

We already have a bijection between $L(G)$ and $X(G)$. So we will give a
bijection between $X(G)$ and $J(\mathbb{P}(G))$. The tiles in the snake graph
correspond to the elements of the poset $P(G)$. Let $p_{i}$ be the element of
the poset corresponding to the tile $T_{i}$. Given a labelling from $X(G)$, we
will construct an order ideal. If $T_{i}$ is labelled $0$, then neither
$(p_{i},0)$ nor $(p_{i},1)$ are in the order ideal. If $T_{i}$ is labelled
$1$, then $(p_{i},0)$ is in the order ideal, but $(p_{i},1)$ is not. If
$T_{i}$ is labelled $2$, then both $(p_{i},0)$ and $(p_{i},1)$ are in the
order ideal. The inequalities defining $X(G)$ guarantee that this is an order
ideal. The inverse to this map is obvious, and it is clearly a bijection.

To see that this bijection is order-preserving, note that the cover relations
in terms of $X(G)$ are simply changing the label of a tile by increasing its
value by $1$. Under the bijection described above, this corresponds to adding
a single element (either $(p_{i},0)$ or $(p_{i},1)$) to the order ideal, which
is a cover relation in the lattice of order ideals. ∎

######  Example 9.2.

An example of the poset isomorphism from Theorem 9.1 is shown in Figure 23.

Figure 23. Poset isomorphism between $L(G)$ and $J(\mathbb{P}(G))$

##  10\. Examples

######  Example 10.1.

Consider the triangulated pentagon, with edges oriented as illustrated.

$1$$2$$3$$4$$5$$a$$b$$c$$d$$e$$x_{1}$$x_{2}$$\theta_{1}$$\theta_{2}$$\theta_{3}$$\lambda_{2,5}=\frac{acx_{1}}{x_{1}x_{2}}+\frac{abd}{x_{1}x_{2}}+\frac{bex_{2}}{x_{1}x_{2}}$
$+\frac{b\sqrt{ade}}{x_{1}\sqrt{x_{2}}}\theta_{1}\theta_{2}+{a\sqrt{bcd}\over\sqrt{x_{1}}x_{2}}\theta_{2}\theta_{3}+{\sqrt{abce}\over\sqrt{x_{1}x_{2}}}\theta_{1}\theta_{3}.$

Using a snake graph consisting of two tiles, and edge labels as below, we
obtain the following double dimer interpretations corresponding to the six
terms in the super lambda length $\lambda_{2,5}$. Note that the leftmost tile
has diagonal $x_{1}=\lambda_{13}$ while the rightmost tile has diagonal
$x_{2}=\lambda_{14}$.

$23$$12$$14$$34$$13$$15$$45$$\frac{acx_{1}}{x_{1}x_{2}}$$23$$12$$14$$34$$13$$15$$45$$\frac{abd}{x_{1}x_{2}}$$23$$12$$14$$34$$13$$15$$45$$\frac{bex_{2}}{x_{1}x_{2}}$$23$$12$$14$$34$$13$$15$$45$$\theta_{1}$$\theta_{2}$$\frac{b\sqrt{adex_{2}}}{x_{1}x_{2}}\theta_{1}\theta_{2}$$23$$12$$14$$34$$13$$15$$45$$\theta_{2}$$\theta_{3}$$\frac{a\sqrt{bcdx_{1}}}{x_{1}x_{2}}\theta_{2}\theta_{3}$$23$$12$$14$$34$$13$$15$$45$$\theta_{1}$$\theta_{2}$$\theta_{2}$$\theta_{3}$$\frac{\sqrt{abcex_{1}x_{2}}}{x_{1}x_{2}}\theta_{1}\theta_{3}$

Next, we picture the same six terms as double lattice paths on the dual snake
graph.

$23$$12$$14$$34$$13$$15$$45$$\frac{acx_{1}}{x_{1}x_{2}}$$23$$12$$14$$34$$13$$15$$45$$\frac{abd}{x_{1}x_{2}}$$23$$12$$14$$34$$13$$15$$45$$\frac{bex_{2}}{x_{1}x_{2}}$$23$$12$$14$$34$$13$$15$$45$$\theta_{1}$$\theta_{2}$$\frac{b\sqrt{adex_{2}}}{x_{1}x_{2}}\theta_{1}\theta_{2}$$23$$12$$14$$34$$13$$15$$45$$\theta_{2}$$\theta_{3}$$\frac{a\sqrt{bcdx_{1}}}{x_{1}x_{2}}\theta_{2}\theta_{3}$$23$$12$$14$$34$$13$$15$$45$$\theta_{1}$$\theta_{2}$$\theta_{2}$$\theta_{3}$$\frac{\sqrt{abcex_{1}x_{2}}}{x_{1}x_{2}}\theta_{1}\theta_{3}$

Finally, we illustrate the three non-ordinary twisted super $T$-paths and
their weights:

$1$$2$$3$$4$$5$$a$$b$$c$$d$$e$$x_{1}$$x_{2}$$\theta_{1}$$\theta_{2}$$\theta_{3}$$\sqrt{\frac{ae}{x_{1}}}\cdot
1\cdot\sqrt{\frac{dx_{2}}{x_{1}}}\cdot\frac{1}{x_{2}}\cdot
b\leavevmode\nobreak\
\theta_{1}\theta_{2}$$1$$2$$3$$4$$5$$a$$b$$c$$d$$e$$x_{1}$$x_{2}$$\theta_{1}$$\theta_{2}$$\theta_{3}$$a\cdot\frac{1}{x_{1}}\cdot\sqrt{\frac{dx_{1}}{x_{2}}}\cdot
1\cdot\sqrt{\frac{bc}{x_{2}}}\leavevmode\nobreak\
\theta_{2}\theta_{3}$$1$$2$$3$$4$$5$$a$$b$$c$$d$$e$$x_{1}$$x_{2}$$\theta_{1}$$\theta_{2}$$\theta_{3}$$\sqrt{\frac{ae}{x_{1}}}\cdot
1\cdot\sqrt{\frac{bc}{x_{2}}}\leavevmode\nobreak\ \theta_{1}\theta_{3}$

######  Example 10.2.

Consider the following triangulation of a hexagon. Its snake graph and dual
snake graph are also pictured. We remind the reader of our convention of going
bottom-to-top (i.e. the first tile of the snake graph corresponds the bottom
triangles in the hexagon).

$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$x_{6}$$x_{9}$$x_{8}$$x_{7}$$\theta_{4}$$\theta_{3}$$\theta_{2}$$\theta_{1}$$x_{4}$$x_{5}$$x_{7}$$x_{8}$$x_{1}$$x_{8}$$x_{9}$$x_{2}$$x_{6}$$x_{3}$$\theta_{1}$$\theta_{2}$$\theta_{2}$$\theta_{3}$$\theta_{3}$$\theta_{4}$$x_{4}$$x_{5}$$x_{8}$$x_{6}$$x_{7}$$x_{9}$$x_{3}$$x_{8}$$x_{1}$$x_{2}$$\theta_{1}$$\theta_{2}$$\theta_{2}$$\theta_{3}$$\theta_{3}$$\theta_{4}$

The following figures show, for each non-ordinary $T$-path (i.e. those that
include super steps), a comparison of all the combinatorial models discussed
in the present paper: super $T$-paths, twisted super $T$-paths, double dimer
covers, and lattice paths.

$\frac{x_{5}\sqrt{x_{1}x_{2}x_{3}x_{8}}}{x_{7}x_{9}}\,\theta_{4}\theta_{3}$$\theta_{3}$$\theta_{4}$$\theta_{3}$$\theta_{4}$

$\frac{x_{4}x_{6}\sqrt{x_{1}x_{2}x_{3}}}{x_{7}x_{9}\sqrt{x_{8}}}\,\theta_{4}\theta_{3}$$\theta_{3}$$\theta_{4}$$\theta_{3}$$\theta_{4}$

$\frac{x_{4}\sqrt{x_{1}x_{2}x_{6}}}{\sqrt{x_{7}x_{8}x_{9}}}\,\theta_{2}\theta_{4}$$\theta_{2}$$\theta_{3}$$\theta_{3}$$\theta_{4}$$\theta_{2}$$\theta_{3}$$\theta_{3}$$\theta_{4}$

$\frac{\sqrt{x_{1}x_{2}x_{4}x_{5}}}{\sqrt{x_{7}x_{9}}}\,\theta_{1}\theta_{4}$$\theta_{1}$$\theta_{2}$$\theta_{2}$$\theta_{3}$$\theta_{3}$$\theta_{4}$$\theta_{1}$$\theta_{2}$$\theta_{2}$$\theta_{3}$$\theta_{3}$$\theta_{4}$

$\frac{x_{1}x_{4}\sqrt{x_{3}x_{6}}}{x_{8}\sqrt{x_{7}x_{9}}}\,\theta_{2}\theta_{3}$$\theta_{2}$$\theta_{3}$$\theta_{2}$$\theta_{3}$

$\frac{x_{1}\sqrt{x_{3}x_{4}x_{5}}}{\sqrt{x_{7}x_{8}x_{9}}}\,\theta_{1}\theta_{3}$$\theta_{1}$$\theta_{2}$$\theta_{2}$$\theta_{3}$$\theta_{1}$$\theta_{2}$$\theta_{2}$$\theta_{3}$

$\frac{x_{1}x_{3}\sqrt{x_{4}x_{5}x_{6}}}{x_{7}x_{9}\sqrt{x_{8}}}\,\theta_{1}\theta_{2}$$\theta_{1}$$\theta_{2}$$\theta_{1}$$\theta_{2}$

$\frac{x_{2}\sqrt{x_{4}x_{5}x_{6}x_{8}}}{x_{7}x_{9}}\,\theta_{1}\theta_{2}$$\theta_{1}$$\theta_{2}$$\theta_{1}$$\theta_{2}$

$\frac{\sqrt{x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}}}{x_{7}x_{9}}\,\theta_{1}\theta_{2}\theta_{4}\theta_{3}$$\theta_{1}$$\theta_{2}$$\theta_{3}$$\theta_{4}$$\theta_{1}$$\theta_{2}$$\theta_{3}$$\theta_{4}$

##  11\. Super Fibonacci Numbers from Annuli and Once-Punctured Tori

In this section, we generalize our formulas to two special classes of
surfaces, the annuli and the once-punctured tori, which gives rise to a
sequence of Grassmann numbers which we call _super Fibonacci numbers_.

We first look at the decorated super-Teichmüller space of a once punctured
torus, which has been systematically studied in [HPZ19]555The third author
also studied this situation in an undergraduate research project, which
motivates much of the current section.. An ideal triangulation $\Delta$ of a
once punctured torus creates two different triangles on the surface. This can
be seen by presenting a torus by a piece of its universal cover — a rectangle,
and letting the four ‘vertices’ of the rectangle be the puncture. Then an
ideal triangulation decomposes the surface into two triangles (one white, the
other gray), using three arcs:

The quadrilaterals where a flip is taking place can be realized by taking
$a=c$ and $b=d$ in Figure 2. We also choose a spin structure such that the
edges around both triangles are oriented cyclically as in Figure 24. The super
Ptolemy transformation in Equations 1, 2 and 3 now takes the following form
(see Figure 24 for the edge labels).

(11) $\displaystyle ef$ $\displaystyle=a^{2}+b^{2}+ab\,\sigma\theta$ (12)
$\displaystyle\sigma^{\prime}$
$\displaystyle={{b\sigma-a\theta}\over{a^{2}+b^{2}}}\quad\text{ and
}\quad\theta^{\prime}={{b\theta+a\sigma}\over{a^{2}+b^{2}}}$

$a$$b$$a$$b$$e$$\theta$$\sigma$

$a$$b$$a$$b$$f$$\theta^{\prime}$$\sigma^{\prime}$

Figure 24. Cyclic orientation is preserved under flips.

Notice that the ideal arcs surrounding both $\theta$ and $\theta^{\prime}$ are
counterclockwise, and those surrounding both $\sigma$ and $\sigma^{\prime}$
are clockwise. In other words, we obtain the same oriented triangulation after
the flip as before. This means flipping any of the three arcs will always
result in the same relations 11 and 12, and will always result in the same
oriented triangulation. Figure 25 illustrates the result of flipping the other
two edges ($a$ and $b$).

$a$$b$$e$$e$$b$$\theta$$\sigma$$a^{\prime}$$b$$e$$e$$b$$\theta^{\prime}$$\sigma^{\prime}$$a$$a$$e$$e$$b$$\theta$$\sigma$$a$$a$$e$$e$$b^{\prime}$$\theta^{\prime}$$\sigma^{\prime}$
Figure 25. Effect of flipping edges $a$ and $b$

Since any sequence of flips results in the same oriented triangulation, then
for any edge, $\sigma^{\prime}$ will always be on the right and
$\theta^{\prime}$ on the left. Since
$\sigma\theta=\sigma^{\prime}\theta^{\prime}$, the Ptolemy relation will
always take the form

(13) $ef=a^{2}+b^{2}+ab\,\varepsilon$

where $\varepsilon=\sigma\theta$. We can therefore forget the original
$\mu$-invariants and just use the variable $\varepsilon$.

In the classical (non-super) case, applying the Ptolemy transformation on two
of the edges alternately gives rises to odd-indexed Fibonacci numbers. In the
same spirit, starting with one of the four oriented triangulations of the once
punctured torus illustrated in Figure 25, setting $a=Z_{1}$, $b=Z_{2}$, $e=1$,
and flipping the edges $a$ and $b$ alternatively will give us a sequence of
elements of the super algebra, $\\{Z_{m}\\}$ satisfying the recurrence
relation:

(14) $Z_{m}Z_{m-2}=Z_{m-1}^{2}+Z_{m-1}\varepsilon+1$

for $m\geq 3$.

$b$$a$$\sigma$$\theta$

$\theta$$\sigma$$\theta$$\sigma$$\theta$$\sigma$$\cdots$$\cdots$$b$$a$$b$$a$$b$$a$$b$$+$$+$$+$$+$$+$$+$

Figure 26. Triangulation of a marked annulus and its universal cover.

One can derive equation 14 in an alternative way, using the decorated super-
Teichmüller space of an annulus. Consider an annulus with two marked points,
one on each boundary component, with orientation as depicted in Figure 26.
Similar to the torus case, if we set the $\lambda$-lengths of the boundary
arcs to be $1$ and set $a=Z_{1}$ and $b=Z_{2}$, then flipping the edges $a$
and $b$ alternately gives the same sequence $\\{Z_{m}\\}$ satisfying the same
recurrence equation 14.

Similar to the case of the once-punctured torus, the relative spin structure
is preserved in both orders: either flipping $a$ then $b$ or flipping $b$
first. However, the two orders work in a slightly different way. Flipping $a$
first would only alter the boundary orientations and leave the interior ones
unchanged, while flipping $b$ first does change the orientations on the
interior edges. In particular, see how the spin structure is affected by flips
in Figures 27 and 28.

$\theta^{\prime}$$\sigma^{\prime}$$\theta^{\prime}$$\sigma^{\prime}$$\theta^{\prime}$$\sigma^{\prime}$$\cdots$$\cdots$$b$$a^{\prime}$$b$$a^{\prime}$$b$$a^{\prime}$$b$$-$$-$$-$$+$$+$$+$

$\theta^{\prime\prime}$$\sigma^{\prime\prime}$$\theta^{\prime\prime}$$\sigma^{\prime\prime}$$\theta^{\prime\prime}$$\sigma^{\prime\prime}$$\cdots$$\cdots$$b^{\prime}$$a^{\prime}$$b^{\prime}$$a^{\prime}$$b^{\prime}$$a^{\prime}$$b^{\prime}$$-$$-$$-$$-$$-$$-$$\theta^{\prime\prime}$$\sigma^{\prime\prime}$

Figure 27. Flipping $a$ (Top) then $b$ (Bottom) of a marked annulus, labeled
as above, only changes orientation on the boundaries.

$\theta^{\prime}$$\sigma^{\prime}$$\theta^{\prime}$$\sigma^{\prime}$$\theta^{\prime}$$\sigma^{\prime}$$\cdots$$\cdots$$b^{\prime}$$a$$b^{\prime}$$a$$b^{\prime}$$a$$b^{\prime}$$+$$+$$+$$+$$+$$+$$\sigma^{\prime}$$\theta^{\prime}$

$\theta^{\prime\prime}$$\sigma^{\prime\prime}$$\theta^{\prime\prime}$$\sigma^{\prime\prime}$$\theta^{\prime\prime}$$\sigma^{\prime\prime}$$\cdots$$\cdots$$b^{\prime}$$a^{\prime}$$b^{\prime}$$a^{\prime}$$b^{\prime}$$a^{\prime}$$b^{\prime}$$+$$+$$+$$+$$+$$+$$\sigma^{\prime\prime}$$\theta^{\prime\prime}$$a^{\prime}$

Figure 28. On the other hand, flipping $b$ (Top) then $a$ (Bottom) of a marked
annulus, labeled as above, changes orientation of internal arcs.

For the remainder of this section, we will give an explicit solution of
recurrence (14) in the special case that $Z_{1}=a=1$ and $Z_{2}=b=1$. The
solution is given by the double dimer partition functions of snake graphs as
discussed earlier. Although we have only defined snake graphs for arcs in a
polygon, the construction of Definition 3.2 can easily be generalized to arcs
in surfaces with nontrivial topology as in [MS10]. The arcs obtained from the
flip sequence described above yield snake graphs with words
$W(G)=R^{2n+1}=RR\cdots R$. In other words, the resulting snake graphs are
horizontal rows of tiles (with an odd number of tiles). Since the arcs cross
the same triangles multiple times, the $\mu$-invariants in the corners of the
tiles will repeat. With this construction in mind, we define the following.

######  Definition 11.1.

Let $G_{m}$ be the snake graph with $m$ tiles in a horizontal row, where all
edges have weight $1$, and all tiles have the same two $\mu$-invariants
$\sigma$ and $\theta$:

$\cdots$$\sigma$$\theta$$\theta$$\sigma$$\sigma$$\theta$

######  Remark 11.2.

This snake graph $G_{m}$ from Definition 11.1 can also be obtained from a
triangulated polygon by specializing some of the variables. Specifically, take
a polygon with a “ _zig-zag_ ” triangulation (see e.g. Example 10.2 for the
$m=3$ case). Then the construction from Definition 3.2 will give a snake graph
with the word $W(G)=RR\cdots R$, and the snake graph will be a horizontal row
of boxes. Finally, setting $\theta_{1}=\theta_{3}=\cdots=\theta_{2n-1}$ and
$\theta_{2}=\theta_{4}=\cdots=\theta_{2n}$ will give the correct
$\mu$-invariants.

Let $x_{m}$ be the number of double dimer covers of $G_{m}$ which do not
include cycles. These are in bijection with ordinary (i.e. not double) dimer
covers. Hence the sequence of $x_{m}$’s coincide with every-other Fibonacci
number (see e.g. Section 2 of [MP07]). More precisely, if we index the
Fibonacci numbers $F_{n}$ by $F_{1}=F_{2}=1$ and $F_{n}=F_{n-1}+F_{n-2}$ for
$n\geq 3$, then $x_{m}=F_{m+2}$ for $m\geq 1$. Let $y_{m}$ be the number of
double dimer covers of $G_{m}$ which include a single cycle of odd length.
Finally, we define $p_{m}=x_{m}+y_{m}\varepsilon$, where
$\varepsilon=\sigma\theta$. We will denote the odd-indexed $p_{m}$’s by
$z_{n}=p_{2n-5}$ and denote the even-indexed $p_{m}$’s by $w_{n}=p_{2n-4}$.
See Remark 11.18 below.

######  Remark 11.3.

Note that $y_{m}$ is not defined as the total number of double dimer covers
containing a cycle, but only those with a single cycle of odd length. Because
every tile of $G_{m}$ contains the same $\mu$-invariants $\sigma$ and
$\theta$, a cycle of even length would contain a factor $\sigma^{2}$ or
$\theta^{2}$, and so such a double dimer cover would have weight zero.
Similarly, a double dimer cover with more than one cycle would also contain
multiple factors of $\sigma\theta$. So indeed $p_{m}=x_{m}+y_{m}\varepsilon$
is the double dimer partition function for $G_{m}$.

Recall that the super $\lambda$-lengths in an annulus (for the flip sequence
described earlier) are denoted by the $Z_{m}$’s.

######  Proposition 11.4.

The super algebra elements $Z_{m}$’s defined above agree with the $z_{m}$’s,
which, as explained in Remark 11.3, are the double dimer partition functions
for $G_{2m-5}$. Here, $G_{2m-5}$ is the snake graph with $\mu$-invariants as
defined in Definition 11.1.

We will prove this proposition in steps using a sequence of lemmas.

######  Lemma 11.5.

If $m$ is even, then

$p_{m}=p_{m-1}+p_{m-2}+\left(p_{m-2}+p_{m-4}+\cdots+p_{2}+1\right)\varepsilon$

If $m$ is odd, then

$p_{m}=p_{m-1}+p_{m-2}+\left(p_{m-2}+p_{m-4}+\cdots+p_{1}+1\right)\varepsilon$

###### Proof.

This is easily seen by the fact that $D(G_{m})=D_{R}(G_{m})\cup
D_{T}(G_{m})\cup D_{tr}(G_{m})$. The fact that the terms in $p_{m-1}$
correspond to the elements of $D_{R}(G_{m})$ follows from Lemma 5.3. The fact
that the terms in $p_{m-2}$ correspond to the elements of $D_{T}(G_{m})$
follows from Lemma 5.4. Finally, consider the elements of $D_{tr}(G_{m})$.
They must have a cycle which surrounds the last tile. As mentioned earlier,
there can only be one cycle, which must have odd length. The weight
contributed by the cycle is $\varepsilon$. If the cycle surrounds only one
tile, then there can by any double dimer cover from $D(G_{m-2})$ on the
remaining part of the snake graph. If the cycle surrounds three tiles, the
remaining part can have any double dimer cover from $D(G_{m-4})$, etc. When
$m$ is odd, these contributions continue until the case of a cycle surrounding
all of $G$. On the other hand, if $m$ is even, the last contribution is the
case of a cycle of surrounding $(m-1)$ tiles and the remaining part must be a
double dimer on the leftmost vertical edge of $G$. In either case, there is
exactly one such way to complete the double dimer cover beyond the cycle of
length $m$ (respectively $(m-1)$). ∎

######  Lemma 11.6.

If $m$ is odd, then

$p_{m}=p_{m-1}+p_{m-2}+p_{m-1}\varepsilon=(1+\varepsilon)p_{m-1}+p_{m-2}$

If $m$ is even, then

$p_{m}=p_{m-1}+p_{m-2}+(p_{m-1}-1)\varepsilon=(1+\varepsilon)p_{m-1}+p_{m-2}-\varepsilon$

###### Proof.

In light of the previous lemma, we only need to check that
$(p_{m-2}+p_{m-4}+\cdots+p_{2}+1)\varepsilon$ is equal to $p_{m-1}\varepsilon$
(resp. $(p_{m-2}+p_{m-4}+\cdots+p_{1}+1)\varepsilon$ is equal to
$(p_{m-1}-1)\varepsilon$) when $m$ is odd (resp. even). By multiplying by
$\varepsilon$, all terms in either of these identities already containing an
$\varepsilon$ vanish. Hence, we only need to check that
$x_{m-2}+x_{m-4}+\cdots+x_{2}+1$ is equal to $x_{m-1}$ (resp.
$x_{m-2}+x_{m-4}+\cdots+x_{1}+1$ is equal to $x_{m-1}-1$. As observed above,
the $x_{m}$’s coincide with the Fibonacci numbers. We thus prove our desired
result via induction using the Fibonacci recurrence.

Consider the base case when $m=3$. We have $x_{1}=F_{3}=2$, hence
$x_{1}+1=3=F_{4}=x_{2}$. Inductively for $m$ odd,
$x_{m-2}+(x_{m-4}+x_{m-6}+\cdots+x_{1}+1)=x_{m-2}+x_{m-3}=F_{m}+F_{m-1}=F_{m+1}=x_{m-1}$.

For the $m$ even case, we consider the base case when $m=4$. We have
$x_{2}=F_{4}=3$, hence $x_{2}+1=4=F_{5}-1=x_{3}-1$. Inductively for $m$ even,
$x_{m-2}+(x_{m-4}+x_{m-6}+\cdots+x_{2}+1)=x_{m-2}+(x_{m-3}-1)=F_{m}+(F_{m-1}-1)=F_{m+1}-1=x_{m-1}-1$.
∎

######  Lemma 11.7.

The $z_{n}$’s satisfy the recurrence

$z_{n}=(3+2\varepsilon)z_{n-1}-z_{n-2}-\varepsilon$

Note that in the absence of the $\varepsilon$’s, this is the usual linear
recurrence satisfied by every-other Fibonacci number.

###### Proof.

Written in terms of $p$’s, this becomes

$p_{2n-5}=(3+2\varepsilon)p_{2n-7}-p_{2n-9}-\varepsilon$

First, since $2n-5$ is odd, we have

$p_{2n-5}=(1+\varepsilon)p_{2n-6}+p_{2n-7}$

Since $2n-6$ is even, we have

$p_{2n-6}=(1+\varepsilon)p_{2n-7}+p_{2n-8}-\varepsilon$

Substitutuing this in the previous equation yields

$\displaystyle p_{2n-5}$
$\displaystyle=(1+\varepsilon)^{2}p_{2n-7}+(1+\varepsilon)p_{2n-8}-(1+\varepsilon)\varepsilon+p_{2n-7}$
$\displaystyle=(2+2\varepsilon)p_{2n-7}+(1+\varepsilon)p_{2n-8}-\varepsilon$

Finally, we note that since $2n-7$ is odd, we have the relation

$(1+\varepsilon)p_{2n-8}=p_{2n-7}-p_{2n-9}$

Substituting this in the equation above gives the desired result. ∎

######  Lemma 11.8.

The $z_{n}$’s satisfy the relation

$z_{n}z_{n-2}=z_{n-1}^{2}+z_{n-1}\varepsilon+1$

###### Proof.

We follow the methodology used to prove Proposition 1 of [MP07] in the
ordinary (non-super) case. Using Lemma 11.7, we can rewrite the product
$z_{n}z_{n-2}$ as

$\displaystyle z_{n}z_{n-2}$
$\displaystyle=\bigg{(}(3+2\varepsilon)z_{n-1}-z_{n-2}-\varepsilon\bigg{)}z_{n-2}$
$\displaystyle=(3+2\varepsilon)z_{n-1}z_{n-2}-\bigg{(}z_{n-2}^{2}+\varepsilon
z_{n-2}\bigg{)}.$

By induction, we may rewrite

$z_{n-1}z_{n-3}=z_{n-2}^{2}+\varepsilon z_{n-2}+1$

and hence

$\displaystyle z_{n}z_{n-2}$
$\displaystyle=(3+2\varepsilon)z_{n-1}z_{n-2}-\bigg{(}z_{n-1}z_{n-3}-1\bigg{)}$
$\displaystyle=z_{n-1}\bigg{(}(3+2\varepsilon)z_{n-2}-z_{n-3}\bigg{)}+1.$

We use Lemma 11.7 again, noting that
$z_{n-1}=(3+2\varepsilon)z_{n-2}-z_{n-3}-\varepsilon$, and conclude that

$\displaystyle z_{n}z_{n-2}$
$\displaystyle=z_{n-1}\bigg{(}z_{n-1}+\varepsilon\bigg{)}+1$
$\displaystyle=z_{n-1}^{2}+\varepsilon z_{n-1}+1$

as desired. ∎

###### Proof of 11.4.

Comparing Equation 14 and Lemma 11.8, we see that the $z_{n}$’s satisfy the
same recurrence as the $\lambda$-lengths. We need only verify that they have
the same initial conditions. We observe that $z_{3}$ and $z_{4}$ are defined
combinatorially as the double dimer partition functions for $G_{1}$ and
$G_{3}$, respectively. Comparing this with $Z_{3}$ and $Z_{4}$, as defined via
recurrence (14) and $Z_{1}=Z_{2}=1$, we indeed see that
$Z_{3}=(Z_{2}^{2}+\varepsilon Z_{2}+1)/Z_{1}=2+\varepsilon=z_{3}$ and
$Z_{4}=(Z_{3}^{2}+\varepsilon Z_{3}+1)/Z_{2}=5+6\varepsilon=z_{4}$. ∎

######  Remark 11.9.

In the more general case where $Z_{1}=a$ and $Z_{2}=b$ are not set to be $1$,
we conjecture that the solution to recurrence (14) is still given as the
double dimer partition function of the snake graphs $G_{2m-5}$’s, while taking
into account edge weights.

For example, letting $z_{1}=a$, $z_{2}=b$, we calculate double dimer partition
functions for $G_{1}$, $G_{2}$, and $G_{3}$ respectively as

$p_{1}=z_{3}=\frac{b^{2}+1}{a}+\frac{b}{a}\varepsilon,$
$p_{2}=\frac{b^{2}+a^{2}+1}{ab}+\frac{b+a}{ab}\varepsilon\mathrm{\leavevmode\nobreak\
and\leavevmode\nobreak\ }$
$p_{3}=z_{4}=\frac{b^{4}+2b^{2}+a^{2}+1}{a^{2}b}+\frac{2b^{3}+2b+ab^{2}+a}{a^{2}b}\varepsilon.$

In particular, $z_{2}z_{4}=z_{3}^{2}+\varepsilon z_{3}+1$ in this case, hence
$z_{m}=Z_{m}$ is still a solution to recurrence (14) for $1\leq m\leq 4$ in
this more general case.

Above we defined $p_{m}=x_{m}+y_{m}\varepsilon$ based on counting the number
of double dimer covers on the $2\times(m+1)$ grid graph $G_{m}$ with $m$ tiles
with $\mu$-invariants repeating $2$-periodically. Recall that $x_{m}=F_{m+2}$
counts the double dimer covers with no cycles and hence is in bijection with
dimer covers of $G_{m}$. The $y_{m}$’s count the number of double dimer covers
containing exactly one cycle, and the cycle must be of odd length. We now give
several different compact algebraic formulae for the $y_{m}$’s. First we
express them in terms of Fibonacci numbers. Second, as a double-sum over
binomial coefficients. Finally, as a single sum utilizing binomial
coefficients.

######  Lemma 11.10.

Define $g_{m}$ as the self-convolution of the Fibonacci sequence666$g_{m}$ is
the OEIS sequence A001629.:

$g_{m}=\sum_{k=1}^{m}F_{k}F_{m-k+1}$

Then $y_{m}$ can be expressed as the sum777$y_{m}$ is the OEIS sequence
A054454 as well as the third column of the triangular array A054453.

$y_{m}=\sum_{j=0}^{\lfloor m/2\rfloor}g_{m-2j}$

###### Proof.

We will first show that $g_{m}$ as the number of double dimer covers of the
graph $G_{m}$ which contain exactly one cycle and that cycle is of exactly
length one. The placement in the graph $G_{m}$ of the unique cycle bisects the
graph. Consequently, a double dimer cover is completed by picking the
equivalent of a dimer cover on the left side of $G_{m}$ and a dimer cover on
the right side of $G_{m}$. Summing over the possibilities gives us
$g_{m}=\sum_{k=1}^{m}F_{k}F_{m-k+1}$ as desired.

Since $y_{m}$ counts all double dimers containing a cycle, rather than only
those of length one, we next count the double dimer covers containing a cycle
of length three, five, etc. In each of these cases, counting the number of
double dimer covers with a cycle of length $(2j+1)$ on $G_{m}$ is equivalent
to counting the number of double dimer covers with a cycle of length $1$ on
$G_{m-2j}$. Consequently, we get the formula
$y_{m}=\sum_{j=0}^{\lfloor\frac{m}{2}\rfloor}g_{m-2j}$. ∎

######  Lemma 11.11.

The values of $g_{m}$ satisfy the recurrence

$g_{m}=g_{m-2}+g_{m-1}+x_{m-2}$

for $m\geq 3$.

###### Proof.

We use $F_{1}=1$, $x_{m-2}=F_{m}$, and Lemma 11.10 to expand the right-hand-
side as

$\displaystyle g_{m-2}+g_{m-1}+x_{m-2}$
$\displaystyle=\sum_{k=1}^{m-2}F_{k}F_{m-1-k}+\sum_{k=1}^{m-1}F_{k}F_{m-k}+F_{m}F_{1}$
$\displaystyle=\sum_{k=1}^{m-2}F_{k}F_{m-1-k}$
$\displaystyle+\left(\sum_{k=1}^{m-2}F_{k}F_{m-k}+F_{m-1}F_{1}\right)+F_{m}F_{1}.$

Note that the two sums involving $(m-2)$ terms each can now be combined using
the Fibonacci recurrence as

$\sum_{k=1}^{m-2}F_{k}(F_{m-1-k}+F_{m-k})=\sum_{k=1}^{m-2}F_{k}F_{m+1-k}.$

Consequently, we see

$g_{m-2}+g_{m-1}+x_{m-2}=\left(\sum_{k=1}^{m-2}F_{k}F_{m+1-k}\right)+F_{m-1}F_{1}+F_{m}F_{1}$

However, using the equality $F_{1}=F_{2}$, we can rewrite this as

$g_{m-2}+g_{m-1}+x_{m-2}=\sum_{k=1}^{m-2}F_{k}F_{m+1-k}+F_{m-1}F_{2}+F_{m}F_{1}=\sum_{k=1}^{m}F_{k}F_{m+1-k}=g_{m}$

where the last equality follows from Lemma 11.10. ∎

######  Lemma 11.12.

The Fibonacci numbers can also be expanded in terms of binomial coefficients
as

$x_{m}=F_{m+2}=\sum_{j=0}^{\lfloor\frac{m+1}{2}\rfloor}{m+1-j\choose j}$

###### Proof.

This identity goes back to Lucas [Luc78] and appears explicitly in Hoggatt-
Lind [HJL69]. Here we give a combinatorial proof analogous to that of Lemma
11.10. We recall that a double dimer cover with no cycles is in bijection with
a dimer cover, and utilize this throughout.

Observe that the graph $G_{m}$ has $m$ possible tiles on which to place a pair
of horizontal dimers but $(m+1)$ possible vertical edges in which to place a
vertical dimer. Hence a dimer cover of $G_{m}$ with exactly $2j$ horizontal
dimers consists of $j$ tiles which contain a pair of horizontal dimers as well
as $m+1-2j$ vertical dimers. Writing out this dimer cover as a word of
$j+(m+1-2j)=m+1-j$ letters consisting of $j$ instances of $H$ and $(m+1-2j)$
instances of $V$, we see such dimer covers are indeed counted by the binomial
coefficient ${m+1-j\choose j}$. ∎

######  Proposition 11.13.

The values of $g_{n}$ can be expressed compactly via the algebraic formula

$g_{n}=\sum_{j=1}^{\lfloor\frac{n+1}{2}\rfloor}j{n+1-j\choose j}.$

###### Proof.

This formula appears in the notes on the page for OEIS sequence A001629. See
also [CFJ15]. However, here we give a self-contained proof.

Induction and Lemmas 11.11 and 11.12 allow us to rewrite

$g_{m}=g_{m-2}+g_{m-1}+x_{m-2}=\sum_{j=1}^{\lfloor\frac{m-1}{2}\rfloor}j{m-1-j\choose
j}+\sum_{j=1}^{\lfloor\frac{m}{2}\rfloor}j{m-j\choose
j}+\sum_{j=0}^{\lfloor\frac{m-1}{2}\rfloor}{m-1-j\choose j}$ $={m-1\choose
0}+\sum_{j=1}^{\lfloor\frac{m-1}{2}\rfloor}(j+1){m-1-j\choose
j}+\sum_{j=1}^{\lfloor\frac{m}{2}\rfloor}j{m-j\choose j}.$

At this point, we divide into cases, based on the parity of $m$. First
consider the case when $m=2k$ is even. Then $\lfloor\frac{m-1}{2}\rfloor=k-1$
and $\lfloor\frac{m}{2}\rfloor=k$, and so we have

$\displaystyle g_{m}$
$\displaystyle=1+\sum_{j=1}^{k-1}(j+1)\binom{m-1-j}{j}+\sum_{j=1}^{k}j\binom{m-j}{j}$
$\displaystyle=1+\sum_{j=2}^{k}j\binom{m-j}{j-1}+\sum_{j=1}^{k}j\binom{m-j}{j}$
$\displaystyle=1+\binom{m-1}{1}+\sum_{j=2}^{k}j\left[\binom{m-j}{j-1}+\binom{m-j}{j}\right]$
$\displaystyle=m+\sum_{j=2}^{k}j\binom{m-j+1}{j}$

Finally, note that $m=1\cdot\binom{m-1+1}{1}$, and also
$k=\lfloor\frac{m+1}{2}\rfloor$, so this agrees with the desired formula in
the case $m=2k$.

Next, we consider the case where $m=2k+1$ is odd. In this case,
$\lfloor\frac{m-1}{2}\rfloor=\lfloor\frac{m}{2}\rfloor=k$, so we have

$\displaystyle g_{m}$
$\displaystyle=1+\sum_{j=1}^{k}(j+1)\binom{m-1-j}{j}+\sum_{j=1}^{k}j\binom{m-j}{j}$
$\displaystyle=1+\sum_{j=2}^{k+1}j\binom{m-j}{j-1}+\sum_{j=1}^{k}j\binom{m-j}{j}$
$\displaystyle=1+\binom{m-1}{1}+(k+1)\binom{m-k-1}{k}+\sum_{j=2}^{k}j\left[\binom{m-j}{j-1}+\binom{m-j}{j}\right]$
$\displaystyle=m+(k+1)+\sum_{j=2}^{k}j\binom{m-j+1}{j}$

As before, $m=1\cdot\binom{m-1+1}{1}$ gives the $j=1$ term, and since $m=2k+1$
we see additionally $k+1=(k+1)\binom{m-(k+1)+1}{k+1}$ is the $j=k+1$ term.
Since $\lfloor\frac{m+1}{2}\rfloor=k+1$, this gives the result. ∎

######  Corollary 11.14.

Hence the parameter $y_{m}$ can also be expressed compactly as

$y_{m}=\sum_{n=0}^{\lfloor\frac{m}{2}\rfloor}\sum_{j=1}^{\lfloor\frac{n+1}{2}\rfloor}j{n+1-j\choose
j}$

######  Proposition 11.15.

We can express

$x_{n}+y_{n}=\begin{cases}g_{n+1}&\text{ if $n$ is even}\\\\[4.30554pt]
g_{n+1}+1&\text{ if $n$ is odd}\end{cases}$

###### Proof.

We will prove this result by induction on $n$. For $n=1$, we have
$x_{n}+y_{n}=3$, and $g_{2}+1=F_{1}F_{2}+F_{2}F_{1}+1=1\cdot 1+1\cdot 1+1=3$.
For $n=2$, $x_{n}+y_{2}=5$, and $g_{3}=F_{1}F_{3}+F_{2}^{2}+F_{3}F_{1}=1\cdot
2+1^{2}+2\cdot 1=5$. So the base cases are confirmed.

For the inductive step we will make use of the fact that
$D(G_{n})=D_{T}(G_{n})\cup G_{R}(G_{n})\cup D_{tr}(G_{n})$. However, we
require a slight modification. Let $\widehat{D}(G)$ denote the set of double
dimer covers with non-zero weight (and similarly for $\widehat{D}_{R}(G)$,
$\widehat{D}_{T}(G)$, and $\widehat{D}_{tr}(G)$). Note that with this
notation, we have $x_{n}+y_{n}=\left|\widehat{D}(G_{n})\right|$.

Lemma 5.3 and Lemma 5.4 easily generalize to give bijections
$\widehat{D}_{R}(G_{n})\to\widehat{D}(G_{n-1})$ and
$\widehat{D}_{T}(G_{n})\to\widehat{D}(G_{n-2})$. This means

(15)
$\left|\widehat{D}_{R}(G_{n})\cup\widehat{D}_{T}(G_{n})\right|=x_{n-1}+y_{n-1}+x_{n-2}+y_{n-2}=1+g_{n-1}+g_{n},$

where the last equality follows by induction.

Finally, we consider $\widehat{D}_{tr}(G_{n})$. Note that elements of
$D_{tr}(G)$ have a cycle around the last tile. But since elements of
$\widehat{D}(G_{n})$ can only have a single cycle (of odd length), this means
the cycle around the last tile is the _only_ cycle. If the cycle at the end of
$G_{n}$ surrounds $2k+1$ tiles, then the remaining part of the double dimer
cover is an element of $D(G_{n-2-2k})$ which contains no cycles. Recall that
these are counted by $x_{n-2-2k}$, which by Lemma 11.12 is equal to
$F_{n-2k}$. So we have

$\left|\widehat{D}_{tr}(G_{n})\right|=\sum_{k=1}^{\lfloor\frac{n+1}{2}\rfloor}F_{n-2k}$

Using the identities $\sum_{i=1}^{k}F_{2i-1}=F_{2k}$ and
$\sum_{i=1}^{k}F_{2i}=F_{2k+1}-1$, just like in the proof of Lemma 11.6, we
get two cases depending on the parity:

(16)
$\left|\widehat{D}_{tr}(G_{n})\right|=\begin{cases}F_{n+1}-1=x_{n-1}-1&\text{
if $n$ is even}\\\\[4.30554pt] F_{n+1}=x_{n-1}&\text{ if $n$ is
odd}\end{cases}$

Finally, recall that
$x_{n}+y_{n}=\left|\widehat{D}(G_{n})\right|=\left|\widehat{D}_{R}(G_{n})\cup\widehat{D}_{T}(G_{n})\right|+\left|\widehat{D}_{tr}(G_{n})\right|$.
Combining Equation 15, Equation 16, and Lemma 11.11, gives the result. ∎

The following result was conjectured by the third author in an unpublished
work (see Footnote 5).

######  Proposition 11.16.

$y_{2m}=\sum_{k=0}^{m}(2k)\binom{m+k+1}{2k+1}$
$y_{2m+1}=\sum_{k=0}^{m}(2k+1)\binom{m+k+2}{2k+2}$

Note that these two expressions can be combined to give a formula for all
$y_{n}$:

$y_{n}=\sum_{k=0}^{\lfloor n/2\rfloor}(n-2k)\binom{n-k+1}{n-2k+1}$

###### Proof.

We will prove the formula for $y_{2m+1}$. The calculation for $y_{2m}$ is
completely analogous.

Using Propositions 11.13 and 11.15, respectively, we write

$g_{2m+2}=\sum_{j=1}^{m+1}j{2m+3-j\choose j}=x_{2m+1}+y_{2m+1}-1.$

Then, using Lemma 11.12, we can rearrange the above equation to obtain

$y_{2m+1}=1+\sum_{j=1}^{m+1}j{2m+3-j\choose j}-\sum_{j=0}^{m+1}{2m+2-j\choose
j}.$

By peeling off the last term, and then shifting the index in the second sum,
we get

$y_{2m+1}=1+\left(\sum_{j=1}^{m+1}\left(j{2m+3-j\choose j}-{2m+3-j\choose
j-1}\right)\right)-{m+1\choose m+1}.$

By applying the classical binomial coefficient identity
$\binom{n}{k}=\frac{n+1-k}{k}\binom{n}{k-1}$ to the expressions above, we can
see that $j\binom{2m+3-j}{j}=(2m+4-2j)\binom{2m+3-j}{j-1}$. Subtracting
$\binom{2m+3-j}{j-1}$ from both sides, we see that the summands are each equal
to $(2m+3-2j)\binom{2m+3-j}{j-1}$. We can thus rewrite the equation above as

$y_{2m+1}=\sum_{j=1}^{m+1}(2m+3-2j){2m+3-j\choose j-1}.$

Finally, to get the desired result, we let $k=m+1-j$ and run through the sum
backwards. ∎

######  Remark 11.17.

Recall that for $n\geq 3$, the $x_{n}$’s are the Fibonacci numbers when
$x_{1}=x_{2}=1$. In the cluster algebra associated to an annulus with a marked
point on each boundary, where $\\{x_{1},x_{2}\\}$ is the initial cluster
consisting of two formal variables, then the values of $z_{n}=x_{2n-5}$ are
the corresponding cluster variables associated to other possible arcs around
the annulus.

One can also focus on the even-indexed entries, but instead of corresponding
to cluster variables, the $x_{2n-4}$’s correspond to the lambda lengths of
peripheral arcs which start at the marked point on the inner boundary, wind
around $(n-1)$ times, and end at the marked point on the inner boundary. Such
peripheral arcs have self-crossings, in particular $(n-2)$ such self-
intersections, when $n\geq 3$. See [GMV19].

######  Remark 11.18.

As explained above after Remark 11.2, for $n\geq 3$, $x_{2n-4}$’s have a
combinatorial interpretation as the (weighted) number of dimer covers in the
snake graph $G_{2n-4}$. Additionally, as we just saw in Remark 11.17, such
numbers (resp. expressions) have interpretations as lambda lengths of
peripheral arcs in the annulus.

Defining $w_{n}=p_{2n-4}=x_{2n-4}+y_{2n-4}$, we see that for $n\geq 3$,
$w_{n}=p_{2n-4}$ counts the (weighted) number of double dimer covers in
$G_{2n-4}$. For example, as computed in Remark 11.9, we see that
$w_{3}=p_{2}=\frac{b^{2}+a^{2}+1}{ab}+\frac{b+a}{ab}\varepsilon$. Comparison
with the classical case motivates the following conjecture.

######  Conjecture 11.19.

If we let $w_{1}=w_{2}=1$ (or if we let $w_{1}=a$ and $w_{2}=b$), the values
of $w_{n}=p_{2n-4}$ correspond to the super $\lambda$-lengths of a peripheral
arc in an annulus, as described in Remark 11.17, except in the context of the
decorated super-Teichmüller space.

The main obstruction to proving 11.19 and such a geometric significance is
that unlike the ordinary case, we only have $\mu$-invariants attached to
triangles in a triangulation. To be able to compute the super $\lambda$-length
of an arc with self-intersections would require a super analogue of the skein
relations for resolving a crossing rather than simply a super analogue of the
Ptolemy exchange relation for flipping a diagonal in an oriented
quadrilateral.

Nonetheless, since we have the candidate of $w_{n}$’s for the peripheral arcs
in an annulus, along with a conjectured combinatorial interpretation that
would allow us to calculate the corresponding expressions, perhaps it is
possible to reverse engineer how to properly define super skein relations in
the context of decorated super-Teichmüller space.

Another natural follow-up question to the above work involves super-Markov
numbers.

######  Remark 11.20.

The super $\lambda$-lengths of a once-punctured torus are studied by Huang,
Penner, and Zeitlin in [HPZ19] where triples of arcs in a triangulation have
super $\lambda$-lengths satisfying the super analogue of the Markoff equation
(c.f. [HPZ19, equation 26])

$x^{2}+y^{2}+z^{2}+(xy+yz+xz)\varepsilon=3(1+\varepsilon)xyz$

where $\varepsilon=\sigma\theta$. This motivates the following conjecture.

######  Conjecture 11.21.

Begin with an oriented triangulation of the once-punctured torus like in
Figure 25. However, in this case, keep $e$ as a formal variable instead of
setting it to be $1$. Allowing flips in all three directions iteratively
yields all possible arcs in a once punctured torus. We claim that the super
$\lambda$-lengths of such arcs are super analogues of the Markoff numbers, and
have combinatorial interpretations using double dimer covers of the snake
graphs appearing in Section 7 of [Pro05] in the presence of appropriately
specialized $\mu$-invariants.

We end with the following questions.

######  Question 11.22.

Assuming the validity of 11.21, do the coefficients of the $\varepsilon$’s in
super Markoff numbers have compact algebraic formulae analogous to the formula
for the $y_{m}$’s as coefficients of $\varepsilon$’s in super Fibonacci
numbers (see 11.16)?

######  Question 11.23.

Going beyond the annlus with one marked point on each boundary, or the once
punctured torus, can we find formulae for super $\lambda$-lengths of arcs on
unpunctured surfaces (or certain punctured surfaces) that have combinatorial
interpretations as double dimer covers of snake graphs, using the same snake
graphs interpreting ordinary $\lambda$-lengths of arcs as in [MSW11]?

Before approaching 11.23, we note that it is not obvious for every
triangulation and every arc that there is a corresponding spin structure (i.e.
oriented triangulation relative to the boundary and equivalence on triangles)
and flip sequence that preserves positive expansion formulas as one iterates
the computation. Nonetheless, the above combinatorial interpretation would at
least give a candidate for what algebraic expressions may be reasonable to
try.

## Acknowledgements

The authors would like to thank the support of the NSF grants DMS-1745638 and
DMS-1854162, as well as the University of Minnesota UROP program. We would
also like to thank Bruce Sagan for many helpful conversations and Ralf
Schiffler for his questions motivating some of this work.

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