# Cross-ratio dynamics and the dimer cluster integrable system

Niklas Affolter  Terrence George  Sanjay Ramassamy

###### Abstract

Cross-ratio dynamics, allowing to construct 2D discrete conformal maps from 1D
initial data, is a well-known discrete integrable system in discrete
differential geometry. We relate it to the dimer integrable system from
statistical mechanics by identifying its invariant Poisson structure and
integrals of motion recently found by Arnold et al. to the Goncharov-Kenyon
counterparts for the dimer model on a specific class of graphs. This solves
the open question of finding a cluster algebra structure describing cross-
ratio dynamics. The main tool relating geometry to the dimer model is the
definition of triple crossing diagram maps associated to bipartite graphs on
the cylinder. In passing we write the bivariate polynomial defining the dimer
spectral curve for arbitrary bipartite graphs on the torus as the
characteristic polynomial of a one-parameter family of matrices, a result
which may be of independent interest.

##  1 Introduction

Consider maps $f:{\mathbb{Z}}^{2}\rightarrow{\mathbb{C}\mathrm{P}}^{1}$ such
that

$\displaystyle\frac{(f_{i,j}-f_{i+1,j})(f_{i+1,j+1}-f_{i,j+1})}{(f_{i+1,j}-f_{i+1,j+1})(f_{i,j+1}-f_{i,j})}=\alpha_{j},$
(1)

where $\alpha_{j}\in{\mathbb{C}}\setminus\\{0\\}$ and all
$i,j\in{\mathbb{Z}}$. Such a map is a special case of a discrete version of
the Schwarzian KdV equation [NC], or a special case of a _discrete isothermic
surface_ [BP] restricted to the sphere $S^{2}$. In both cases it was shown
that these maps are a discrete integrable system in the sense that they admit
a discrete Lax representation. An interesting question is: Given all
$\alpha_{j}$, what is the space of solutions of (1)? To answer this, consider
each column $i$ of ${\mathbb{Z}}^{2}$ as a discrete curve
$f_{i}:{\mathbb{Z}}\rightarrow{\mathbb{C}\mathrm{P}}^{1}$. The discrete curves
corresponding to two adjacent columns $f_{i}$ and $f_{i+1}$ are called
$\vec{\alpha}$-related, where $\vec{\alpha}$ is the vector of all
$\alpha_{j}$. In the case when $\alpha_{j}$ is independent of $j$, the curve
$f_{i+1}$ is called a _Darboux transform_ [hjbook] of $f_{i}$ in the discrete
differential geometry community. If we know $f_{i}$ and one point of
$f_{i+1}$, then equation (1) determines all of $f_{i+1}$. As a consequence,
there is a complex one parameter freedom for each additional column of
${\mathbb{Z}}^{2}$. However, this changes if one considers _periodic_ maps,
that is maps
${\mathbb{Z}}\times{\mathbb{Z}}/m{\mathbb{Z}}\rightarrow{\mathbb{C}\mathrm{P}}^{1}$
for some $m\in{\mathbb{N}}$ that satisfy equation (1). In this case, if we
know $f_{i}$ then there are only two possible solutions for $f_{i+1}$, because
$f_{i+1}$ has to be periodic as well. Thus if we know both $f_{i-1}$ as well
as $f_{i}$ and assume that $f_{i+1}\neq f_{i-1}$, then $f_{i+1}$ is determined
_uniquely_. Special attention has been payed to the case that $\alpha_{j}$
does not depend on $j$. In this case, periodic solutions to equation (1) have
been studied as _periodic discrete conformal maps_ [HMNP] with respect to
algebro-geometric integrability. Also in this case, the map
$(f_{i-1},f_{i})\mapsto(f_{i},f_{i+1})$ is called _cross-ratio dynamics_
[AFIT]. Cross-ratio dynamics can also be generalized from discrete periodic
curves to discrete _twisted_ curves, that is curves
$f_{i}:{\mathbb{Z}}\rightarrow{\mathbb{C}\mathrm{P}}^{1}$ such that
$f_{i}(j+n)=M\circ f_{i}(n)$ for all $j\in{\mathbb{Z}}$ and for some
$n\in{\mathbb{N}}$ and $M\in\mathrm{PGL}_{2}$. For cross-ratio dynamics the
integrability in the sense of Liouville was proved by Arnold, Fuchs, Izmestiev
and Tabachnikov [AFIT]. Indeed, they provide Poisson brackets that are
preserved by the dynamics as well as integrals of motion that are Casimirs and
Hamiltonians. We will henceforth call them the AFIT Poisson structures,
Casimirs and Hamiltonians.

$p_{1}$$q_{1}$$p_{3}$$q_{3}$$p_{5}$$q_{5}$$q_{2}$$q_{2}$$p_{2}$$q_{4}$$q_{4}$$p_{4}$$q_{6}$$q_{6}$$p_{6}$$p_{1}$$q_{1}$

$p_{1}$$q_{1}$$p_{3}$$q_{3}$$p_{5}$$q_{5}$$q_{2}$$q_{2}$$p_{2}$$q_{4}$$q_{4}$$p_{4}$$p_{1}$$q_{1}$$p_{1}$

Figure 1: The relevant bipartite graphs to describe a pair of curves of length
$n=6$ (left) and length $n=5$ (right). Here the top and the bottom sides of
each graph are identified, yielding the cylinder graphs
$\Gamma_{6,{\mathbb{A}}}$ and $\Gamma_{5,{\mathbb{A}}}$. If we additionally
identify the left and right sides of each graph, we obtain the torus graphs
$\Gamma_{6}$ and $\Gamma_{5}$.

Another widely studied discrete system which is integrable both in the
algebro-geometric sense and in the Liouville sense is the dimer model on
bipartite graphs on the torus [GK13] coming from statistical mechanics.
Goncharov and Kenyon identified a Poisson structure on the space of face
weights associated to a given graph and obtained the Casimirs and Hamiltonians
of the underlying integrable system as coefficients of the polynomial defining
the spectral curve of the model. Our first main result consists in identifying
the AFIT Liouville integrable structure [AFIT] with the Goncharov-Kenyon
Liouville integrable structure [GK13] for a special class of graphs. It is
stated loosely in Theorem 1.1 below and stated in full detail as Theorem 6.7
and Theorem 7.1 later. Let $G$ (resp. $G^{\text{o}dd}$) be the cylinder graph
represented on the left (resp. right) of Figure 8. In both pictures, the top
and the bottom sides of the rectangle are identified. If $n$ is even, let
$\Gamma_{n}$ be the torus graph obtained by gluing in a cyclic chain
$\tfrac{n}{2}$ copies of $G$. If $n$ is odd let $\Gamma_{n}$ be the torus
graph obtained by gluing in a cyclic chain $\tfrac{n-1}{2}$ copies of $G$ and
one copy of $G^{\text{o}dd}$. See Figure 1 for pictures of $\Gamma_{6}$ and
$\Gamma_{5}$. In this article we consider a more general setup than [AFIT],
namely we assume that the $\alpha_{j}$ are periodic of period $n$, but not
necessarily constant.

######  Theorem 1.1.

Let $n\geq 2$. The AFIT Poisson structure, Casimirs and Hamiltonians for
cross-ratios dynamics for twisted discrete curves coincide with the Goncharov-
Kenyon Poisson structure, Casimirs and Hamiltonians for the dimer integrable
system on the torus graph $\Gamma_{n}$.

In order to prove Theorem 1.1, we find an expression for the bivariate
polynomial defining the dimer spectral curve for general bipartite graphs on
the torus as the characteristic polynomial of a matrix depending on a single
parameter. More precisely, given an edge-weighted bipartite graph $\Gamma$ on
the torus, we construct a matrix $\Pi(w)$ depending on a parameter $w$ and on
the choice of a zigzag path in $\Gamma$. Then we have

######  Theorem 1.2.

The spectral curve of the dimer model on $\Gamma$ is given by

$\\{(z,w)\in({\mathbb{C}}^{*})^{2},\ \det(zI-\Pi(w))=0\\}$

Theorem 1.2 holds for general graphs and not just for those related to cross-
ratio dynamics, hence it may be of independent interest. The matrix $\Pi(w)$
is reminiscent of the boundary measurement matrix for networks on cylinders of
[GSV, GSTV]. In a recent preprint of which we learned during the completion of
this work, Izosimov [Izosimov] made precise the connection between the
integrable systems of [GK13] and [GSTV]. In particular his main result
provides a representation of the dimer spectral curve similar to Theorem 1.2.
We note that analogous representations have appeared in physics in the special
case of the periodic Toda chain [EFS] and in mathematical physics via
representation-theoretic arguments [FM16].

Theorem 1.1 realizes cross-ratio dynamics as a special case of the dimer
integrable system by adopting an analytic point of view using coordinates on
the spaces on which the two dynamics occur. One can actually provide a more
geometric identification by making use of the recently introduced [AGR] triple
crossing diagram maps (TCD maps). These maps, a version of the vector-relation
configurations of [AGPR] better adapted to geometric dynamics, associate a
geometric configuration in some projective space ${\mathbb{C}\mathrm{P}}^{m}$
to a bipartite graph on a disk with trivalent black vertices. To each white
vertex is associated a point in ${\mathbb{C}\mathrm{P}}^{m}$, with the
constraint that any three points associated to the neighbors of a black vertex
must be aligned. One can perform local changes of the geometric configuration
of a TCD map which are associated with the dimer local moves for bipartite
graphs. Using appropriate coordinates, these moves carry an interpretation in
terms of cluster algebra mutations.

In this article, we introduce the notion of twisted TCD maps on the cylinder.
Let $\Gamma_{n,{\mathbb{A}}}$ be the cylinder graph obtained by gluing
together the same cylinder graphs as for $\Gamma_{n}$ but without closing up
cyclically (see Figure 1 for two examples). An explicit bridge between cross-
ratio dynamics and the dimer model is given by the following result:

######  Theorem 1.3.

Pairs of $\vec{\alpha}$-related twisted discrete curves arise as twisted TCD
maps on $\Gamma_{n,{\mathbb{A}}}$ and cross-ratio dynamics arises as an
explicit sequence of local moves on these twisted TCD maps.

An important note regarding Theorem 1.3 is that the first local move in the
sequence depends on a parameter and that parameter depends globally on the
initial pair of $\vec{\alpha}$-related twisted discrete curves. Furthermore,
combining Theorem 1.1 with Theorem 1.3, we obtain an alternative proof of the
conservation of the AFIT Hamiltonians, since the dynamics on TCD maps is
conjugated to the dimer integrable dynamics of [GK13]. A more explicit
statement of Theorem 1.3 is given by Theorem 8.2. As a corollary, we answer an
open question of [AFIT] asking for an interpretation of cross-ratio dynamics
in terms of cluster algebras. Indeed, all but the first and the last
operations for TCD maps of Theorem 1.3 has a cluster algebra interpretation
[AGPR, AGR].

######  Corollary 1.4.

The evolution of the some coordinates under cross-ratio dynamics can be
written as an explicit composition of operations, with all but the first and
the last one being cluster algebra mutations.

Here we mean mutations of cluster variables known as $y$ [FZ] or as $X$ [FG]
depending on the authors.

As noted by [AFIT], cross-ratio dynamics bears a lot of resemblances with the
pentagram map, another dynamical system on discrete curves for which several
similar properties are known: Liouville integrability [OST1, OST2], algebro-
geometric integrability [Soloviev], cluster algebra interpretation [Glick],
dimer model interpretation [FM16, GR] and realization in terms of TCD maps
[AGPR, AGR]. Similar properties are known or expected to hold for many
generalizations of the pentagram map, see e.g. [GP].

There is however a notable difference with cross-ratio dynamics. For the
pentagram map and its generalizations, the dynamics is local in the sense than
one can construct a point of the curve $f_{i+1}$ knowing only a bounded number
of points of the curve $f_{i}$. For cross-ratio dynamics the dynamics is
global, one needs to know all the points of the curve $f_{i}$ to construct any
given point of $f_{i+1}$. This is related to the R-matrix constructions of
[ILP1, ILP2, Chepuri].

### Organization of the paper

In Section 2 we provide the necessary background on cross-ratio dynamics and
its integrability following mostly [AFIT], but generalizing to $n$-periodic
$\vec{\alpha}$. In Section 3 we recall the Goncharov-Kenyon integrable system
[GK13] associated with the dimer model on the torus. In Section 4 we consider
the dimer model on the cylinder, construct the matrix $\Pi(w)$ and prove
Theorem 1.2. We introduce in Section 5 the notion of twisted TCD maps
associated to a bipartite graph on the cylinder and use it in Section 6 to
prove Theorem 1.1 when $n$ is even. In Section 7 we explain how to modify the
computations to derive this result when $n$ is odd. Section 8 describes the
sequence of local moves for twisted TCD maps that realize cross-dynamics, as
stated in Theorem 1.3. Finally Appendix A presents some results used in
Section 4 related to the classical Schur complement.

##  2 The cross-ratio dynamics integrable system

###  2.1 Multi-ratios

Given points $a_{1},\dots,a_{2m}\in{\mathbb{C}\mathrm{P}}^{1}$, let
$\widetilde{a}_{1},\dots,\widetilde{a}_{{2m}}$ denote a choice of lifts to
${\mathbb{C}}^{2}$. Their multi-ratio is defined as

$\displaystyle\operatorname{mr}(a_{1},a_{2},\dots,a_{2m})=\prod_{i=1}^{m}\frac{\det(\widetilde{a}_{2i-1},\widetilde{a}_{2i})}{\det(\widetilde{a}_{2i},\widetilde{a}_{2i+1})},$
(2)

where $\text{det}(a,b)$ denotes the determinant of the matrix with columns
$a,b\in{\mathbb{C}}^{2}$ and indices are taken modulo $2m$. When $m=2$, the
multi-ratio specializes to the cross-ratio

$\displaystyle\operatorname{cr}(a_{1},a_{2},a_{3},a_{4})=\frac{\det(\widetilde{a}_{1},\widetilde{a}_{2})\det(\widetilde{a}_{3},\widetilde{a}_{4})}{\det(\widetilde{a}_{2},\widetilde{a}_{3})\det(\widetilde{a}_{4},\widetilde{a}_{1})}.$
(3)

The multi-ratio is $\operatorname{PGL}_{2}$ invariant. If the points $a_{i}$
are all in an affine chart ${\mathbb{C}}\subset{\mathbb{C}\mathrm{P}}^{1}$,
then we can choose lifts of the form
$\widetilde{a}_{i}=\begin{bmatrix}a_{i}\\\ 1\end{bmatrix}$, and then we have

$\displaystyle\operatorname{mr}(a_{1},a_{2},\dots,a_{2m})=\prod_{i=1}^{m}\frac{a_{2i-1}-a_{2i}}{a_{2i}-a_{2i+1}}.$
(4)

###  2.2 Discrete projective curves in ${\mathbb{C}}{\mathbb{P}}^{1}$

A _discrete curve_ is a map
$p:{\mathbb{Z}}\rightarrow{\mathbb{C}\mathrm{P}}^{1}$. A _twisted discrete
curve_ of length $n$ is a pair $(p,M)$ where
$p:{\mathbb{Z}}\rightarrow{\mathbb{C}\mathrm{P}}^{1}$ is a discrete curve and
$M\in\operatorname{PGL}_{2}$ is a projective transformation called monodromy
such that $p_{i+n}=M(p_{i})$ for all $i\in{\mathbb{Z}}$. A twisted discrete
curve $(p,M)$ is nondegenerate if for all $i\in{\mathbb{Z}}$, we have
$p_{i}\notin\\{p_{i+1},p_{i+2}\\}$, and closed if $M$ is the identity.

Let $\widetilde{\mathcal{P}}_{n}$ denote the space of nondegenerate twisted
discrete curves of length $n$. $\widetilde{\mathcal{P}}_{n}$ is an open
subvariety of $({\mathbb{P}}^{1})^{n}\times\operatorname{PGL}_{2}$, and
$\operatorname{PGL}_{2}$ acts on it by

$A\cdot(p_{1},p_{2},\dots,p_{n},M)=(A(p_{1}),A(p_{2}),\dots,A(p_{n}),AMA^{-1}).$

The quotient
$\mathcal{P}_{n}:=\widetilde{\mathcal{P}}_{n}/\operatorname{PGL}_{2}$ is the
moduli space of nondegenerate twisted curves of length $n$. Given
$p\in\widetilde{\mathcal{P}}_{n}$, we define the $c$-variables

$\displaystyle c_{i}:=\operatorname{cr}(p_{i-1},p_{i},p_{i+2},p_{i+1})\text{
for all }i\in{\mathbb{Z}}.$ (5)

Notice that $c_{i+n}=c_{i}$ for all $i\in{\mathbb{Z}}$. If $p$ is
nondegenerate, $c_{i}(p)\notin\\{0,\infty\\}$, so the $c$-variables give us a
morphism

$\displaystyle c:\widetilde{\mathcal{P}}_{n}$
$\displaystyle\rightarrow({\mathbb{C}}^{\times})^{n}$ (6) $\displaystyle p$
$\displaystyle\mapsto(c_{1},c_{2},\dots,c_{n}).$

Since each $c_{i}$ is a cross-ratio, this morphism is $\operatorname{PGL}_{2}$
invariant, and therefore (6) descends to a morphism
$c:\mathcal{P}_{n}\rightarrow({\mathbb{C}}^{\times})^{n}$. Given the
$c$-variables and three initial points $p_{1},p_{2},p_{3}$, the whole discrete
curve is recovered from (5). Since any three points can be mapped to any other
three points by a projective transformation, the $c$-variables characterize a
discrete curve up to projective transformations and so $c$ is an isomorphism.
The inverse morphism is given explicitly in [AFIT]*Section 3.2.

The following lemma gives an explicit representative for the
$\operatorname{PGL}_{2}$ conjugacy class of the monodromy matrix $M$ in terms
of the $c$-variables.

######  Theorem 2.1 ([AFIT]*Lemma 3.2).

$M=\begin{bmatrix}0&c_{1}\\\ -1&1\end{bmatrix}\begin{bmatrix}0&c_{2}\\\
-1&1\end{bmatrix}\cdots\begin{bmatrix}0&c_{n}\\\ -1&1\end{bmatrix}$

represents the monodromy matrix.

###  2.3 The Poisson variety $\mathcal{U}_{n}$.

Let $\overrightarrow{\alpha}=(\alpha_{i})_{i\in{\mathbb{Z}}}$ with
$\alpha_{i}\in{\mathbb{C}}\setminus\\{0,1\\}$ such that
$\alpha_{i+n}=\alpha_{i}$. Two discrete curves $p,q\in\widetilde{C}_{n}$ are
said to be $\overrightarrow{\alpha}$-related, and denoted
$p\stackrel{{\scriptstyle\overrightarrow{\alpha}}}{{\sim}}q$ if

$\operatorname{cr}(p_{i},q_{i},p_{i+1},q_{i+1})=\alpha_{i}\text{ for all
$i\in{\mathbb{Z}}$},$ (7)

and $p$ and $q$ have the same monodromy. Note that condition (7) alone does
not imply that $p$ and $q$ have the same monodromy. The relation
$\stackrel{{\scriptstyle\overrightarrow{\alpha}}}{{\sim}}$ is
$\operatorname{PGL}_{2}$ invariant and therefore descends to a relation on
$\mathcal{P}_{n}$. If $\alpha_{i}=\alpha$ for all $i$, then we say that $p$
and $q$ are $\alpha$-related and write
$p\stackrel{{\scriptstyle\overrightarrow{\alpha}}}{{\sim}}q$.

Let $\widetilde{\mathcal{U}}_{n}$ denote the space of pairs of
$\overrightarrow{\alpha}$-related nondegenerate twisted curves of length $n$.
It is a subvariety of $({\mathbb{P}}^{1})^{2n}\times\operatorname{PGL}_{2}$
and $\operatorname{PGL}_{2}$ acts on it by

$A\cdot(p_{1},\dots,p_{n},q_{1},\dots,q_{n},M)=(A(p_{1}),\dots,A(p_{n}),A(q_{1}),\dots,A(q_{n}),AMA^{-1}).$

We define the $u$-variables

$\displaystyle\quad
u_{i}=-\operatorname{cr}(p_{i},p_{i+1},q_{i},p_{i-1})\text{ for all
}i\in{\mathbb{Z}}.$ (8)

Since $p$ and $q$ are nondegenerate and $\overrightarrow{\alpha}$-related,
$u_{i}\notin\\{0,-1,\infty\\}$. Therefore the $u$-variables define a
$\operatorname{PGL}_{2}$ invariant morphism
$u:\widetilde{\mathcal{U}}_{n}\rightarrow({\mathbb{C}}\setminus\\{0,-1\\})^{n}$
which descends to a morphism

$u:\mathcal{U}_{n}:=\widetilde{\mathcal{U}}_{n}/\operatorname{PGL}_{2}\rightarrow({\mathbb{C}}\setminus\\{0,-1\\})^{n}.$

Consider the morphism

$\displaystyle\rho_{\overrightarrow{\alpha}}:\mathcal{U}_{n}$
$\displaystyle\rightarrow\mathcal{P}_{n}$ $\displaystyle(p,q,M)$
$\displaystyle\mapsto(p,M).$

#### Remark.

$\widetilde{\mathcal{U}}_{n}$ and $\mathcal{U}_{n}$ also depend on
$\overrightarrow{\alpha}$, but we suppress this dependence in the notation.

In [AFIT]*Section 4.8 it is shown that

${\mathcal{U}_{n}}$${\mathcal{P}_{n}}$${({\mathbb{C}}\setminus\\{0,-1\\})^{n}}$${({\mathbb{C}}^{\times})^{n}}$$\scriptstyle{\rho_{\overrightarrow{\alpha}}}$$\scriptstyle{u}$$\scriptstyle{c}$$\scriptstyle{\Lambda_{\overrightarrow{\alpha}}}$

commutes, where the map $\Lambda_{\overrightarrow{\alpha}}$ is determined by
$c_{i}=\frac{\alpha_{i}}{(1+u_{i})(1+\frac{1}{u_{i+1}})}$. If we are given
$u$-variables, we can recover $p$ up to projective transformations as
$c^{-1}\circ\Lambda_{\overrightarrow{\alpha}}(u_{1},\dots,u_{n})$, and $q$ is
then determined by (8). Therefore the morphism $u$ induces an isomorphism
between $\mathcal{U}_{n}$ and $({\mathbb{C}}\setminus\\{0,-1\\})^{n}$. The set
$\mathcal{U}_{n}$ is the moduli space of pairs of
$\overrightarrow{\alpha}$-related twisted discrete curves of length $n$.

The Poisson bracket

$\\{u_{i},u_{i+1}\\}_{\mathcal{U}_{n}}:=u_{i}u_{i+1},$

makes $(\mathcal{U}_{n},\\{\cdot,\cdot\\}_{\mathcal{U}_{n}})$ a Poisson
variety while the Poisson bracket

$\\{c_{i},c_{i+1}\\}_{\overrightarrow{\alpha}}:=c_{i}c_{i+1}\left(1-\frac{c_{i}}{\alpha_{i}}-\frac{c_{i+1}}{\alpha_{i+1}}\right),\quad\\{c_{i},c_{i+2}\\}_{\overrightarrow{\alpha}}:=-\frac{1}{\alpha_{i+1}}c_{i}c_{i+1}c_{i+2},$
(9)

makes $(\mathcal{P}_{n},\\{\cdot,\cdot\\}_{\overrightarrow{\alpha}})$ a
Poisson variety. For both Poisson brackets, we only give the nonzero values
obtained by pairing two coordinate functions. We also define the rescaled
coordinates $\overline{c}_{i}:=\frac{c_{i}}{\alpha_{i}}$. In these coordinates
the Poisson bracket $\\{\cdot,\cdot\\}_{\overrightarrow{\alpha}}$ takes the
simpler form

$\\{\overline{c}_{i},\overline{c}_{i+1}\\}=\overline{c}_{i}\overline{c}_{i+1}\left(1-\overline{c}_{i}-\overline{c}_{i+1}\right),\quad\\{\overline{c}_{i},\overline{c}_{i+2}\\}=-\overline{c}_{i}\overline{c}_{i+1}\overline{c}_{i+2}.$

A computation similar to [AFIT]*Lemma 3.9 shows that
$\rho_{\overrightarrow{\alpha}}$ is Poisson. [AFIT]*Corollary 2.7 shows that
$\rho_{\overrightarrow{\alpha}}$ is generically finite of degree $2$, that is
for a generic discrete curve $p\in\mathcal{P}_{n}$ there are two discrete
curves $q,r\in\mathcal{P}_{n}$ that are ${\overrightarrow{\alpha}}$-related to
$p$. Therefore the map $T_{\overrightarrow{\alpha}}:(p,q,M)\mapsto(p,r,M)$ is
a birational involution of $\mathcal{U}_{n}$. $\mathcal{U}_{n}$ also has
another involution $j:(p,q,M)\mapsto(q,p,M)$ given in coordinates by

$u_{i}\mapsto
v_{i}:=\frac{1}{u_{i}}\frac{\alpha_{i}-1}{\alpha_{i-1}-1}=-\operatorname{cr}(q_{i},q_{i+1},p_{i},q_{i-1}).$
(10)

###  2.4 Cross-ratio dynamics and integrability

Suppose $(q,M)\in\mathcal{P}_{n}$ such that
$|\rho_{\overrightarrow{\alpha}}^{-1}(q,M)|=2$, and suppose $p,r$ are the two
discrete curves $\overrightarrow{\alpha}$-related to $q$. The birational
automorphism

$\displaystyle\nu_{\overrightarrow{\alpha}}=T_{\overrightarrow{\alpha}}\circ
j$ $\displaystyle:\mathcal{U}_{n}\rightarrow\mathcal{U}_{n}$
$\displaystyle(p,q,M)$ $\displaystyle\mapsto(q,r,M)$

is called cross-ratio dynamics.

For $I\subset[n]$, let $c_{I}:=\prod_{i\in I}{c_{i}}$. Similarly we define
$\overline{c}_{I},\alpha_{I},u_{I}$ etc. Let
$c_{\text{even}}:=c_{2}c_{4}\dots$ and $c_{\text{odd}}:=c_{1}c_{3}\dots$
denote the product of the even and odd $c$ variables respectively. In the same
way, we define $\alpha_{\text{even}}$ etc.

$I\subset[i,j]$ is said to be cyclically sparse if it contains no pair of
consecutive indices where the indices are taken periodic modulo $n$. Define

$\displaystyle F_{k}(c)$ $\displaystyle=\sum_{I\text{ cyclically
sparse}:|I|=k}c_{I}\text{ for $k=0,\dots,\lfloor\frac{n}{2}\rfloor$},$ (11)
$\displaystyle E_{\overrightarrow{\alpha}}$
$\displaystyle=\frac{1}{\overline{c}_{[n]}}\left(\sum_{k=0}^{\frac{n}{2}}(-1)^{k}F_{k}\left(\overline{c}\right)\right)^{2}.$
(12)

######  Theorem 2.2 ([AFIT]*Main Theorem 1).

We have:

  1. 1.

For $n$ even, $\\{\cdot,\cdot\\}_{\overrightarrow{\alpha}}$ has corank $2$ and
the subalgebra of Casimirs is generated by $E_{\overrightarrow{\alpha}}$ and
$\frac{c_{\text{even}}}{c_{\text{odd}}}$.

  2. 2.

For $n$ odd, $\\{\cdot,\cdot\\}_{\overrightarrow{\alpha}}$ has corank $1$ and
the Casimir is $E_{\overrightarrow{\alpha}}$.

  3. 3.

For $k=1,2,\dots,\lfloor\frac{n+1}{2}\rfloor-1$, the functions
$\frac{F_{k}(c)^{2}}{c_{[n]}}$ mutually commute and form a maximal set of
functionally independent Hamiltonians, making the Poisson variety
$\mathcal{P}_{n}$ a Liouville integrable system.

Moreover, cross-ratio dynamics is discrete integrable in the following sense:

  1. 1.

$\nu_{\overrightarrow{\alpha}}$ is Poisson.

  2. 2.

The pullbacks of the Hamiltonians and the Casimirs to $\mathcal{U}_{n}$ by
$\rho_{\overrightarrow{\alpha}}$ are invariant under
$\nu_{\overrightarrow{\alpha}}$.

The following theorem shows that the Hamiltonians can be obtained from the
monodromy matrix.

######  Theorem 2.3 ([AFIT]*Theorem 1).

We have

$\frac{1}{\mathrm{det}M}\mathrm{tr}^{2}M=\frac{1}{c_{[n]}}\left(\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor}(-1)^{k}F_{k}(c)\right)^{2}.$

#### Remark.

Note that $\mathrm{tr}M$ is not $\operatorname{PGL}_{2}$ invariant, but the
normalized trace $\frac{\mathrm{tr}M}{{\sqrt{\mathrm{det}M}}}$ is. However the
normalized trace is not a regular function on $\mathcal{P}_{n}$, so we need to
square everything to make it so.

To get the Hamiltonians from Theorem 2.3, notice that $F_{k}(c)$ is the
homogeneous degree $k$ component of
$\sqrt{c_{[n]}}\frac{\text{tr}M}{\sqrt{\text{det }M}}$. This observation will
be very useful in Section 6.2.

##  3 The dimer integrable system

For further details on this section, see [GK13].

###  3.1 The dimer model in a torus.

Let $\Gamma=(B\cup W,E)$ be a bipartite graph embedded in a torus
${\mathbb{T}}$ such that $|B|=|W|$ and such that the faces of $\Gamma$, that
is, the connected components of the complement of $\Gamma$, are disks. We
denote by $F$ the set of faces of $\Gamma$. An edge weight on $\Gamma$ is a
function ${\rm wt}:E\rightarrow{\mathbb{C}}^{\times}$. Two edge weights ${\rm
wt}_{1}$ and ${\rm wt}_{2}$ are said to be gauge equivalent if there is a
function $g:B\cup W\rightarrow{\mathbb{C}}^{\times}$ such that for every edge
$e=\\{{\rm b},{\rm w}\\}$ with ${\rm b}\in B,{\rm w}\in W$, we have ${\rm
wt}_{2}(e)={\rm wt}_{1}(e)g({\rm w})g({\rm b})^{-1}$. Let
$\mathcal{L}_{\Gamma}$ denote the space of edge weights modulo gauge
equivalence.

Equivalently, we can consider the graph $\Gamma$ as a cell complex whose $0$
and $1$-cells are $B\cup W$ and $E$ respectively. Considering each edge
$e={\rm bw}$ to be oriented from ${\rm b}$ to ${\rm w}$, we have the nonzero
cellular chain groups

$C_{0}(\Gamma,{\mathbb{Z}})={\mathbb{Z}}B\oplus{\mathbb{Z}}W,\quad
C_{1}(\Gamma,{\mathbb{Z}})={\mathbb{Z}}E,$

with boundary homomorphism $\partial:C_{1}(\Gamma,{\mathbb{Z}})\rightarrow
C_{0}(\Gamma,{\mathbb{Z}})$ given by $\partial(e)={\rm w}-{\rm b}$, so that
$H_{1}(\Gamma,{\mathbb{Z}})=\text{Ker }\partial$. Dually, we have cellular
cochain groups
$C^{q}(\Gamma,{\mathbb{C}}^{\times}):=\text{Hom}_{\mathbb{Z}}(C_{q}(\Gamma,{\mathbb{Z}}),{\mathbb{C}}^{\times}),$
for $q\in\\{0,1\\}$, with coboundary homomorphism
$\delta:C^{0}(\Gamma,{\mathbb{C}}^{\times})\rightarrow
C^{1}(\Gamma,{\mathbb{C}}^{\times})$ given by $\delta(g)(e)=\frac{g({\rm
w})}{g({\rm b})}$. Since an edge weight is a $1$-cochain and two edge weights
are gauge equivalent if and only if they differ by a $1$-coboundary, we have

$\displaystyle\mathcal{L}_{\Gamma}=H^{1}(\Gamma,{\mathbb{C}}^{\times}):=C^{1}(\Gamma,{\mathbb{C}}^{\times})/\delta(C^{0}(\Gamma,{\mathbb{C}}^{\times}))$
(13)

. We denote by $[{\rm wt}]$ the cohomology class represented by the cochain
${\rm wt}$.

For $[L]\in H_{1}(\Gamma,{\mathbb{Z}})$, we denote by ${[\rm wt]}([L])$ the
result of evaluating the cohomology class $[{\rm wt}]$ on the homology class
$[L]$ yielding an alternating product of edge weights around $L$. Explicitly,
if the $L$ is the $1$-cycle ${\rm w}_{1}\xrightarrow[]{e_{1}}{\rm
b}_{1}\xrightarrow[]{e_{2}}{\rm w}_{2}\xrightarrow[]{e_{3}}{\rm
b}_{2}\xrightarrow[]{e_{4}}\cdots\xrightarrow[]{e_{2n-2}}{\rm
w}_{n}\xrightarrow[]{e_{2n-1}}{\rm b}_{n}\xrightarrow[]{e_{2n}}{\rm w}_{1}\in
H_{1}(\Gamma,{\mathbb{Z}})$, we have

$[{\rm wt}]([L])=\prod_{i=1}^{n}\frac{{\rm wt}(e_{2i})}{{\rm wt}(e_{2i-1})}.$

Since
$\mathcal{L}_{\Gamma}=\text{Hom}_{\mathbb{Z}}(H_{1}(\Gamma,{\mathbb{Z}}),{\mathbb{C}}^{\times})$
is an algebraic torus, the algebra $\mathcal{O}_{\mathcal{L}_{\Gamma}}$ of
regular functions on $\mathcal{L}_{\Gamma}$ consists of the characters
$\chi_{[L]}$ for $[L]\in H_{1}(\Gamma,{\mathbb{Z}})$ defined by
$\chi_{[L]}([{\rm wt}]):=[{\rm wt}]([L])$.

We now give a description of $\mathcal{O}_{\mathcal{L}_{\Gamma}}$ in terms of
a basis. For a face $f$ of $\Gamma$, let $\partial f$ denote the
counterclockwise oriented cycle given by the walk along the boundary of $f$.
Let $a$ and $b$ denote two cycles in $\Gamma$ generating
$H_{1}({\mathbb{T}},{\mathbb{Z}})$. Let $X_{f}:=\chi_{[\partial f]}.$ Then

$\displaystyle\mathcal{O}_{\mathcal{L}_{\Gamma}}={\mathbb{C}}[X_{f}^{\pm
1},\chi_{[a]}^{\pm 1},\chi_{[b]}^{\pm 1}]/(1-\mbox{$\prod_{f\in F}$}X_{f}),$
(14)

where the relation $\prod_{f\in F}X_{f}=1$ comes from the relation $\sum_{f\in
F}[\partial f]=0$ in $H_{1}(\Gamma,{\mathbb{Z}})$.

######  Example 3.1.

For $n$ even, consider the graph $\Gamma_{n}$ in ${\mathbb{T}}$ for which a
fundamental domain is obtained by gluing the cylinder graphs $G_{i}$ shown in
Figure 2 for $i\in\\{1,2,\dots,\frac{n}{2}\\}$ in the order $G_{1}G_{2}\cdots
G_{\frac{n}{2}}$, so that ${\rm w}_{2n+1}$ is identified with ${\rm w}_{1}$
and ${\rm w}_{2n+2}$ with ${\rm w}_{2}$. The faces of $\Gamma_{n}$ are labeled
by $X_{i},Y_{i}$ for $i\in\\{1,2,\dots,n\\}$ as shown in Figure 2.

${\rm w}_{4i-2}$${\rm w}_{4i-3}$${\rm b}_{4i-2}$${\rm
b}_{4i-3}$$Y_{2i}$$X_{2i}$${\rm w}_{4i-1}$${\rm w}_{4i-1}$${\rm
w}_{4i}$$X_{2i-1}$$Y_{2i-1}$$Y_{2i-1}$$X_{2i+1}$$Y_{2i+1}$$Y_{2i+1}$${\rm
w}_{4i+2}$${\rm w}_{4i+1}$${\rm b}_{4i}$${\rm b}_{4i-1}$ Figure 2: The
building block graph $G_{i}$, which is a graph on a cylinder, since the top
and bottom sides of the rectangle are identified. Each face is labeled by its
face weight.

#### Zig-zag paths and Newton polygon.

A zig-zag path in $\Gamma$ is a path that turns maximally left at white
vertices and maximally right at black vertices. Let $\mathcal{Z}$ denote the
set of zig-zag paths of $\Gamma$. Each zig-zag path $\beta\in\mathcal{Z}$
defines a homology class $[\beta]\in H_{1}({\mathbb{T}},{\mathbb{Z}})$. Label
the zig-zag paths $\beta_{1},\beta_{2},\dots,\beta_{|\mathcal{Z}|}$ so that
the $[\beta_{i}]$ regarded as vectors in
$H_{1}({\mathbb{T}},{\mathbb{Z}})\otimes{\mathbb{R}}\cong{\mathbb{R}}^{2}$ are
in counterclockwise order. We construct a closed convex integral polygon
$N(\Gamma)$ (or just $N$ when $\Gamma$ is clear from the context) by placing
the $[\beta_{i}]$ such that the head of $[\beta_{i}]$ is the tail of
$[\beta_{i+1}]$. Each edge of $\Gamma$ is contained in two zig-zag paths that
traverse the edge in opposite directions, so we have
$\sum_{\beta\in\mathcal{Z}}[\beta]=0$, which shows that $N$ constructed as
above is a closed polygon. $N$ is unique up to translation and is called the
Newton polygon of $\Gamma$. The name Newton polygon will be justified at the
end of this section by that fact that this polygon arises as the Newton
polygon of the characteristic polynomial. By construction, the set of
primitive edge vectors of $N$ is in bijection with $\mathcal{Z}$, but this
bijection is not canonical when there is more than one zig-zag path with the
same homology class.

A graph $\Gamma$ is said to be minimal if any lift of a zig-zag path to the
universal cover of ${\mathbb{T}}$ has no self intersections and any lifts of
two zig-zag paths to the universal cover of ${\mathbb{T}}$ do not form
parallel bigons (pairs of zig-zag paths oriented the same way intersecting
twice).

$[\xi_{1}]$$[\xi_{3}]$$[\xi_{4}]$$[\xi_{2}]$$[\zeta_{n}]$$[\zeta_{n-2}]$$[\zeta_{2}]$$[\zeta_{1}]$$[\zeta_{3}]$$[\zeta_{n-1}]$$(0,0)$$(2,0)$$\left(0,\frac{n}{2}\right)$$\left(2,\frac{n}{2}\right)$

$(0,0)$$(2,0)$$\left(0,\frac{n-1}{2}\right)$$\left(2,\frac{n+1}{2}\right)$$[\xi_{1}]$$[\xi_{2}]$$[\sigma]$$[\zeta_{2}]$$[\zeta_{n-1}]$$[\zeta_{1}]$$[\zeta_{3}]$$[\zeta_{n}]$

Figure 3: The Newton polygon of $\Gamma_{n}$ for even $n$ (left) and odd $n$
(right).

######  Example 3.2.

The graph $\Gamma_{n}$ has $n+4$ zig-zag paths:

$\displaystyle\xi_{1}$ $\displaystyle={\rm w}_{1},{\rm b}_{1},{\rm w}_{3},{\rm
b}_{3},\dots,{\rm w}_{2n-1},{\rm b}_{2n-1},{\rm w}_{1},\quad[\xi_{1}]=(1,0),$
$\displaystyle\xi_{2}$ $\displaystyle={\rm w}_{1},{\rm b}_{2n-1},{\rm
w}_{2n},{\rm b}_{2n-3},\dots,{\rm w}_{4},{\rm b}_{1},{\rm
w}_{1},\quad[\xi_{2}]=(-1,0),$ $\displaystyle\xi_{3}$ $\displaystyle={\rm
w}_{2},{\rm b}_{2},{\rm w}_{4},{\rm b}_{4},\dots,{\rm w}_{2n},{\rm
b}_{2n},{\rm w}_{2},\quad[\xi_{3}]=(1,0),$ $\displaystyle\xi_{4}$
$\displaystyle={\rm w}_{2},{\rm b}_{2n},{\rm w}_{2n-1},{\rm
b}_{2n-2},\dots,{\rm w}_{3},{\rm b}_{2},{\rm w}_{2},\quad[\xi_{4}]=(-1,0),$
$\displaystyle\zeta_{2i-1}$ $\displaystyle={\rm w}_{4i-1},{\rm b}_{4i-3},{\rm
w}_{4i},{\rm b}_{4i-2},{\rm w}_{4i-1},\quad[\zeta_{2i-1}]=(0,1),$
$\displaystyle\zeta_{2i}$ $\displaystyle={\rm w}_{4i-1},{\rm b}_{4i},{\rm
w}_{4i},{\rm b}_{4i-1},{\rm w}_{4i-1},\quad[\zeta_{2i}]=(0,-1),$

where $i\in\\{1,2,\dots,\frac{n}{2}\\}$ and we use the basis
$(\gamma_{z},\gamma_{w})$ to identify $H_{1}({\mathbb{T}},{\mathbb{Z}})$ with
${\mathbb{Z}}^{2}$ (see Figure 5 for the case $n=1$). Therefore the Newton
polygon of $\Gamma_{n}$ is (see Figure 3)

$\text{Convex-hull
}\left\\{(0,0),(2,0),\left(0,\frac{n}{2}\right),\left(2,\frac{n}{2}\right)\right\\}.$

It is easily checked that $\Gamma_{n}$ is minimal.

#### Conjugate surface and Poisson structure.

Thickening the edges of $\Gamma$, we obtain a ribbon graph. Equivalently a
ribbon graph is a graph along with the data of a cyclic order of edges around
each vertex. The ribbon graph obtained from $\Gamma$ has the cyclic order
induced from the embedding in ${\mathbb{T}}$. Let $\widehat{\Gamma}$ be the
ribbon graph obtained from $\Gamma$ by reversing the cyclic order at all white
vertices. The boundary components of $\widehat{\Gamma}$ are in bijection with
the zig-zag paths of $\Gamma$. Gluing in disks along these boundary components
of $\widehat{\Gamma}$, we obtain a surface $\widehat{S}$ called the conjugate
surface, along with an embedding of $\Gamma$ in $\widehat{S}$. Let
$\epsilon_{\widehat{S}}$ denote the intersection form on
$H_{1}(\widehat{S},{\mathbb{Z}})$: For $[L_{1}],[L_{2}]\in
H_{1}(\widehat{S},{\mathbb{Z}})$, choose cycles $L_{1}$ and $L_{2}$
representing the homology classes and intersecting transversely. Then

$\displaystyle\epsilon_{\widehat{S}}([L_{1}],[L_{2}])=\sum_{p\in L_{1}\cap
L_{2}}\epsilon_{p}(L_{1},L_{2}),$ (15)

where $\epsilon_{p}(L_{1},L_{2})$ is the local intersection index, with sign
chosen so that it is positive if $L_{2}$ crosses $L_{1}$ at $p$ from its right
side to its left side. The embedding $\iota:\Gamma\hookrightarrow\widehat{S}$
induces the homomorphism of homology groups
$\iota_{*}:H_{1}(\Gamma,{\mathbb{Z}})\rightarrow
H_{1}(\widehat{S},{\mathbb{Z}})$. We define the alternating form
$\epsilon_{\Gamma}$ on $H_{1}(\Gamma,{\mathbb{Z}})$ by
$\epsilon_{\Gamma}([L_{1}],[L_{2}]):=\epsilon_{\widehat{S}}(\iota_{*}[L_{1}],\iota_{*}[L_{2}])$.

For ${[L_{1}]},{[L_{2}]}\in H_{1}(\Gamma,{\mathbb{Z}})$, define the Poisson
bracket

$\displaystyle\\{\chi_{[L_{1}]},\chi_{[L_{2}]}\\}_{\Gamma}:=\epsilon_{\Gamma}([L_{1}],[L_{2}])\chi_{[L_{1}]}\chi_{[L_{2}]}.$
(16)

The faces of $\Gamma$ in $\widehat{S}$ become the zig-zag paths of $\Gamma$ in
${\mathbb{T}}$, so we have $\\{\chi_{[L_{1}]},\chi_{[L_{2}]}\\}=0$ for all
$[L_{2}]$ if and only if
$[L_{1}]\in\bigoplus_{\beta\in\mathcal{Z}}{\mathbb{Z}}\cdot[\beta]$. Therefore
the functions $C_{\beta}:=\chi_{[\beta]},\beta\in\mathcal{Z}$ generate the
center of the Poisson algebra $\mathcal{O}_{\mathcal{L}_{\Gamma}}$. These
functions are called Casimirs.

#### Elementary transformations and mutations.

There are two local modifications of bipartite graphs called elementary
transformations shown in Figure 7. An elementary transformation
$s:\Gamma\rightarrow\Gamma^{\prime}$ induces a unique up to isotopy
homeomorphism of conjugate surfaces
$\hat{s}:\widehat{S}_{\Gamma}\rightarrow\widehat{S}_{\Gamma^{\prime}}$
[GK13]*Lemma 4.1, which in turn induces an isomorphism
$\hat{s}_{*}:H_{1}(\Gamma,{\mathbb{Z}})\rightarrow
H_{1}(\Gamma^{\prime},{\mathbb{Z}})$. Associated to the elementary
transformation $s$ is a Poisson birational map of weights
$\mu_{s}:\mathcal{L}_{\Gamma}\rightarrow\mathcal{L}_{\Gamma^{\prime}}$:

  1. 1.

The spider move $s$ at face $f$: For $[L^{\prime}]\in
H_{1}(\Gamma^{\prime},{\mathbb{Z}})$, let
$[L]=(\hat{s}_{*})^{-1}([L^{\prime}])$. We define

$\mu_{s}^{*}(\chi_{[L^{\prime}]})=\chi_{[L]}(1+X_{f}^{-\text{sign
}\epsilon_{\Gamma}([L],\partial f)})^{-\epsilon_{\Gamma}([L],\partial f)}.$
(17)

  2. 2.

Shrinking/expanding degree two vertices: We define
$\mu_{s}^{*}(\chi_{[L^{\prime}]})=\chi_{[L]}$.

Since elementary transformations are local, they do not change homology
classes of zig-zag paths, and therefore the Newton polygon. Gluing the Poisson
affine varieties $\mathcal{L}_{\Gamma}$ for all $\Gamma$ minimal with
$N(\Gamma)=N$ using these Poisson birational maps, we obtain the Poisson space
$\mathcal{X}_{N}$ called the dimer cluster Poisson variety associated to $N$.
$\mathcal{X}_{N}$ is a cluster Poisson variety as defined by Fock and
Goncharov [FG], and will be the phase space of the dimer integrable system.

$\mathrm{b}$$\mathrm{w}$$f$

$\leftrightarrow$ $\mathrm{b}$$\mathrm{w}$$f_{b}$$f_{r}$$f_{l}$

Figure 4: Adding/removing a bigon inside a face.

#### Inserting/removing a bigon.

Figure 4 shows the insertion of a bigon between vertices ${\rm w}\in W$ and
${\rm b}\in B$ belonging to a common face $f$, with parameter $t$. This
divides $f$ into three new faces, the bigon $f_{b}$ and the face $f_{l}$
(resp. $f_{r}$) to the left (resp. right) of $f_{b}$ when traversing the bigon
from $\rm w$ to $\rm b$. Let $\Gamma_{b}$ denote the graph obtained. The
embedding $i_{b}:\Gamma\hookrightarrow\Gamma_{b}$ induces a homomorphism
$(i_{b})_{*}:H_{1}(\Gamma,{\mathbb{Z}})\rightarrow
H_{1}(\Gamma_{b},{\mathbb{Z}})$. We define the induced map of weights
$\mu_{t}:\mathcal{L}_{\Gamma}\rightarrow\mathcal{L}_{\Gamma_{b}}$ on a basis
as follows: If $[L]$ is topologically nontrivial in
$H_{1}({\mathbb{T}},{\mathbb{Z}})$ or is the boundary of a face of $\Gamma$
set $\mu_{t}^{*}(\chi_{(i_{b})_{*}[L]})=\chi_{[L]}$. Define also
$\mu_{t}^{*}(X_{f_{l}})=t$ and $\mu_{t}^{*}(X_{f_{b}})=-1$. Note that the
second equation implies that in any cocycle, the weights of the two edges of
the bigon sum to zero. On the other hand, if we have bigon with
$X_{f_{b}}=-1$, we may remove it. This induces a map of weights
$\mu_{b}^{\prime}:\\{X_{f_{b}}=-1\\}\rightarrow\mathcal{L}_{\Gamma}$, where
$\\{X_{f_{b}}=-1\\}$ denotes the subvariety in $\mathcal{L}_{\Gamma_{b}}$,
given by $\mu_{b}^{*}(\chi_{[L]})=\chi_{(i_{b})_{*}[L]}$.

###  3.2 Dimer covers and Kasteleyn theory.

$\gamma_{z}$$\gamma_{w}$${\rm w}_{2}$${\rm w}_{1}$${\rm b}_{2}$${\rm
b}_{1}$${\rm w}_{3}$${\rm w}_{3}$${\rm w}_{4}$${\rm w}_{2}$${\rm w}_{1}$${\rm
b}_{4}$${\rm b}_{3}$$z$$aw$$d$$z$$h$$e$$1$$-bw$$c$$1$$-f$$g$ Figure 5: Edge
weights, Kasteleyn sign and $M_{0}$ (thick edges) on $\Gamma_{2}$.

A dimer cover $M$ of $\Gamma$ is a subset of $E$ such that each vertex of
$\Gamma$ is incident to exactly one edge in $M$. Let $\mathcal{M}$ denote the
set of dimer covers of $\Gamma$. If we fix a reference dimer cover $M_{0}$,
then we can associate to each dimer cover a homology class

$M\mapsto[M-M_{0}]\in H_{1}({\mathbb{T}},{\mathbb{Z}}),$

where as before we orient $e=\\{{\rm b},{\rm w}\\}$ from ${\rm b}$ to ${\rm
w}$. Given $[\rm wt]\in\mathcal{L}_{\Gamma}$, each dimer cover also gets a
weight $[{\rm wt}](M-M_{0})$. If $\Gamma$ is minimal, we can describe the
Newton polygon in terms of dimer covers.

######  Proposition 3.3 ([GK13]*Theorem 3.12).

For a minimal bipartite graph $\Gamma$ in ${\mathbb{T}}$, we have

$N(\Gamma)=\text{Convex-hull }\\{[M-M_{0}]:M\text{ is a dimer cover of
}\Gamma\\},$

up to a translation.

Let $R$ be a fundamental rectangle of ${\mathbb{T}}$ and let
$\gamma_{z},\gamma_{w}$ be cycles in ${\mathbb{T}}$ such that
$[\gamma_{z}],[\gamma_{w}]$ generate $H_{1}(\Gamma,{\mathbb{Z}})$. We choose
$\gamma_{z},\gamma_{w}$ parallel to the sides of $R$ as shown in Figure 5.
Isotoping if necessary, we assume that the edges of $\Gamma$ intersect
$\gamma_{z},\gamma_{w}$ transversely. Applying
$\text{Hom}_{\mathbb{Z}}(\cdot,{\mathbb{C}}^{\times})$ to the surjection
$H_{1}(\Gamma,{\mathbb{Z}})\rightarrow H_{1}({\mathbb{T}},{\mathbb{Z}})$, we
get an inclusion $H^{1}({\mathbb{T}},{\mathbb{C}}^{\times})\hookrightarrow
H^{1}(\Gamma,{\mathbb{C}}^{\times})$. Let $[\phi^{\prime}]\in
H^{1}(\Gamma,{\mathbb{C}}^{\times})$ be in the image of
$H^{1}({\mathbb{T}},{\mathbb{C}}^{\times})$. In other words,
$X_{f}([\phi^{\prime}])=1$ for all $f\in F$. We choose a cochain $\phi$
representing $[\phi^{\prime}]$ as follows: Let
$z:=[\phi^{\prime}]([\gamma_{z}]),w=[\phi^{\prime}]([\gamma_{w}])$, and define

$\phi(e):=z^{(e,\gamma_{w})}w^{(e,-\gamma_{z})},$

where $(\cdot,\cdot)$ is the intersection index.

$[\kappa]\in H^{1}(\Gamma,{\mathbb{C}}^{\times})$ is called a Kasteleyn sign
if

  1. 1.

$X_{[L]}([\kappa])=\pm 1$ for all $[L]\in H_{1}(\Gamma,{\mathbb{Z}})$.

  2. 2.

$X_{f}([\kappa])=(-1)^{\frac{|\partial f|}{2}+1}$ for all $f\in F$, where
$|\partial f|$ is the number of edges in $\partial f$.

Let $\kappa:E\rightarrow{\mathbb{C}}^{\times}$ be a cochain representing the
Kasteleyn sign $[\kappa]$. We define the Kasteleyn matrix

$K(z,w):{\mathbb{C}}[z^{\pm 1},w^{\pm 1}]^{B}\rightarrow{\mathbb{C}}[z^{\pm
1},w^{\pm 1}]^{W}$

by

$K{(z,w)}_{{\rm w},{\rm b}}:=\sum_{e=\\{{\rm b},{\rm w}\\}\in E}{\rm
wt}(e)\kappa(e)\phi(e).$

######  Theorem 3.4 ([Kas]).

We have

$\frac{1}{{\rm wt}(M_{0})\kappa(M_{0})\phi(M_{0})}\text{det
}K(z,w)=\sum_{M\in\mathcal{M}}\text{sign}([M-M_{0}])[{\rm
wt}]([M-M_{0}])[\phi]([M-M_{0}]),$

where $\text{sign}([M-M_{0}])$ is a sign that depends on $[\kappa]$ and is
irrelevant for our purposes.

Moreover,

$P(z,w):=\frac{1}{{\rm wt}(M_{0})\kappa(M_{0})\phi(M_{0})}\text{det }K(z,w)$

is called the characteristic polynomial and
$\Sigma:=\\{(z,w)\in({\mathbb{C}}^{\times})^{2}:P(z,w)=0\\}$ is called the
spectral curve of $(\Gamma,[{\rm wt}])$. Although $K(z,w)$ depends on the
choice of cochains representing $[{\rm wt}]$ and $[\kappa]$, the spectral
curve is independent of these choices. Moreover $N(\Gamma)$ is the Newton
polygon of $P(z,w)$, that is, the convex hull of the pairs
$(i,j)\in{\mathbb{Z}}^{2}$ such that $z^{i}w^{j}$ has a nonzero coefficient in
$P(z,w)$.

######  Example 3.5.

Consider the graph $\Gamma_{2}$ with edge-weights, Kasteleyn sign, $\phi$ and
reference dimer cover $M_{0}$ as shown in Figure 5. The Kasteleyn matrix is

$K(\Gamma_{2})=\blockarray{ccccc}{\rm b}_{1}&{\rm b}_{2}{\rm b}_{3}{\rm
b}_{4}\\\ \block{[cccc]c}10z0{\rm w}_{1}\\\ 010z{\rm w}_{2}\\\ g-bwhaw{\rm
w}_{3}\\\ -fced{\rm w}_{4}\\\ ,$

and the spectral curve is

$P(z,w)=\frac{1}{bfz^{2}w}\left(-dh+(dg+ch)z-cgz^{2}+aew+(be+af)zw+bfz^{2}w\right).$
(18)

#### Hamiltonians.

Let $N^{\circ}$ denote the interior of $N$. For $[\gamma]\in N^{\circ}\cap
H_{1}(\Gamma,{\mathbb{Z}})$, let

$H_{[\gamma]}:=\sum_{M\in\mathcal{M}:[M-M_{0}]=[\gamma]}{[\rm wt]}([M-M_{0}])$

denote the coefficient of $[\phi]([\gamma])$ (up to a sign) in $P(z,w)$.

######  Proposition 3.6 ([GK13]*Theorem 1.2).

The generic level sets of the Casimirs are symplectic leaves of
$\mathcal{X}_{N}$. The Hamiltonians $H_{[\gamma]}$ for $[\gamma]\in
H_{1}({\mathbb{T}},{\mathbb{Z}})\cap N^{\circ}$ mutually commute, making these
symplectic leaves into algebraic integrable systems.

##  4 The dimer model in a cylinder

Let $\Gamma_{{\mathbb{A}}}=(B\cup W,E)$ be a bipartite graph embedded in a
cylinder ${\mathbb{A}}$ such that:

  1. 1.

The faces of $\Gamma_{\mathbb{A}}$ (including boundary faces) are disks.

  2. 2.

Every vertex on the boundary of ${\mathbb{A}}$ is white and of degree $1$.

  3. 3.

Let $S$ and $T$ denote the boundary white vertices on the two components of
the boundary of ${\mathbb{A}}$, called the source and target vertices
respectively. Let $W_{\text{int}}=W\setminus(S\cup T)$ denote the set of
internal white vertices. We assume that $|S|=|T|$ and $|B|=|W|-|S|$.

  4. 4.

$\Gamma_{\mathbb{A}}$ has a dimer cover $M_{0}$ that uses all the vertices in
$S$ and none of the vertices in $T$.

Here by a dimer cover of $\Gamma_{\mathbb{A}}$, we mean a matching that uses
all the vertices in $B$ and a in $|B|$-element subset of $W$ exactly once.
Note that assumptions 2 and 4 imply that the black vertices incident to the
white vertices in $S$ are all different. We call graphs $\Gamma_{\mathbb{A}}$
satisfying these conditions balanced cylinder graphs.

An edge-weight on $\Gamma_{\mathbb{A}}$ is a function ${\rm
wt}:E\rightarrow{\mathbb{C}}^{\times}$. Two edge weights ${\rm wt}_{1}$ and
${\rm wt}_{2}$ are gauge equivalent if there is a function $g:B\cup
W\rightarrow{\mathbb{C}}^{\times}$ satisfying $g({\rm w})=1$ for all ${\rm
w}\in S\cup T$ such that for every edge $e=\\{{\rm b},{\rm w}\\}$ with ${\rm
b}\in B,{\rm w}\in W$, we have ${\rm wt}_{2}(e)=g({\rm b})^{-1}{\rm
wt}_{1}(e)g({\rm w})$. In other words we only allow gauge transformations at
interior vertices. The space of edge-weights modulo gauge transformations is
the relative cohomology group $H^{1}(\Gamma_{\mathbb{A}},S\cup
T,{\mathbb{C}}^{\times})$. As before, we denote by $[{\rm wt}]$ the cohomology
class represented by ${\rm wt}$.

Let $\mathcal{M}$ denote the set of dimer covers of $\Gamma_{\mathbb{A}}$. For
$M\in\mathcal{M}$, we define its weight to be ${\rm wt}(M)=\prod_{e\in M}{\rm
wt}(e)$. For $M\in\mathcal{M}$, let $\partial M$ denote the set of boundary
white vertices incident to $M$. For example, $\partial M_{0}=S$.

###  4.1 Torus to cylinder

$\beta$$\gamma_{w}$$\beta$$T$$S$

$\gamma_{w}$$\beta$$T$$S$$B_{ST}$$B_{S}$

Figure 6: Cutting along a zig-zag path and labeling of vertices in
$\Gamma_{ST}$. Removing the shaded region from ${\mathbb{T}}$, we get
${\mathbb{A}}$.

In this section, we outline a general procedure to construct a graph
$\Gamma_{\mathbb{A}}$ in a cylinder satisfying the assumptions in Section 4
from a graph $\Gamma$ in a torus ${\mathbb{T}}$. Let $\beta$ be a zig-zag path
in $\Gamma$. Without loss of generality, we assume that there are no
$2$-valent black vertices in $\beta$. Changing the fundamental domain if
necessary, we can assume that $[\beta]=[\gamma_{w}]$. Split each black vertex
in $\beta$ to create a $2$-valent white vertex. $[\gamma_{w}]$ has a
representative cycle $\gamma_{w}$ in ${\mathbb{T}}$ that goes through each of
the newly created $2$-valent white vertices and does not intersect $\Gamma$
anywhere else. Cutting ${\mathbb{T}}$ along $\gamma_{w}$, we obtain a cylinder
${\mathbb{A}}$ and a graph $\Gamma_{\mathbb{A}}$ embedded in it. The
$2$-valent white vertices become $S$ and $T$, where $T$ is connected to
$\beta$ (see Figure 6). Since $|B(\Gamma)|=|W(\Gamma)|$, we have
$|B(\Gamma_{\mathbb{A}})|=|W(\Gamma_{\mathbb{A}})|-|S|$. There is a dimer
cover $M_{0}$ in $\Gamma$ that contains half the edges in $\beta$ (see for
example [GK13]*Theorem 3.12), which becomes a dimer cover in
$\Gamma_{\mathbb{A}}$ such that $\partial M_{0}=S$.

###  4.2 Kasteleyn theory in ${\mathbb{A}}$

Suppose $\Gamma_{\mathbb{A}}$ is a balanced cylinder graph. An element
$[\kappa]\in H^{1}(\Gamma_{\mathbb{A}},S\cup T,{\mathbb{C}}^{\times})$ is
called a Kasteleyn sign if

  1. 1.

$X_{[L]}([\kappa])=\pm 1$ for all $[L]\in H_{1}(\Gamma_{\mathbb{A}},S\cup
T,{\mathbb{Z}})$.

  2. 2.

$X_{f}([\kappa])=(-1)^{\frac{|\partial f|}{2}+1}$ for all $f\in F$, where
$|\partial f|$ is the number of edges in $\partial f$.

The existence of a Kasteleyn sign is shown in [CR2]. Moreover there is the
following version of Kasteleyn’s theorem:

######  Theorem 4.1 ([CR2]*Theorem 2.4).

Let $I\subset S\cup T$ such that $|I|=|S|$, and let $K_{{\mathbb{A}},I}$
denote the submatrix of the Kasteleyn matrix with rows indexed by white
vertices in $I\cup W_{\text{int}}$ and columns indexed by black vertices in
$B$. Then we have

$\frac{1}{{\rm wt}(M_{0})\kappa(M_{0})}\text{det
}K_{{\mathbb{A}},I}=\sum_{\partial M=I}\text{sign}([M-M_{0}])[{\rm
wt}]([M-M_{0}]),$

where $[M-M_{0}]$ is the relative homology class in
$H_{1}({\mathbb{A}},\partial{\mathbb{A}},{\mathbb{Z}})$ defined by the
relative cycle $M-M_{0}$.

Let $B_{S}$ denote the set of black vertices incident to $S$. $B_{S}$ is
matched to $S$ by $M_{0}$. The Kasteleyn matrix $K_{\mathbb{A}}$ of
$\Gamma_{\mathbb{A}}$ has the block matrix form

$K_{\mathbb{A}}=\blockarray{ccc}B_{S}&B\setminus B_{S}\\\ \block{[cc]c}I0S\\\
K_{1}K_{2}T\\\ K_{3}K_{4}W_{\text{int}}\\\ .$ (19)

For generic $[{\rm wt}]$, $K_{4}$ is invertible using Theorem 4.1 with $I=S$,
since $M_{0}$ is a dimer cover with $\partial M_{0}=S$ that will appear as a
summand in $\text{det }K_{4}=\text{det }K_{{\mathbb{A}},S}$. Define the Schur
complement

$\displaystyle L$ $\displaystyle:=K_{\mathbb{A}}/K_{4}$
$\displaystyle=\begin{bmatrix}I\\\ K_{1}\end{bmatrix}-\begin{bmatrix}0\\\
K_{2}\end{bmatrix}K_{4}^{-1}K_{3}$ $\displaystyle=\begin{bmatrix}I\\\
\Pi\end{bmatrix},$ (20)

where $\Pi:=K_{1}-K_{2}K_{4}^{-1}K_{3}$. To get an explicit formula for the
entries of $\Pi$, notice that for ${\rm w}_{i}\in S,{\rm w}_{j}\in T$, the
square submatrix $L_{S\setminus\\{{\rm w}_{i}\\}\cup\\{{\rm w}_{j}\\}}$ of $L$
with rows indexed by ${S\setminus\\{{\rm w}_{i}\\}\cup\\{{\rm w}_{j}\\}}$ is
the Schur complement $K_{{\mathbb{A}},S\setminus\\{{\rm w}_{i}\\}\cup\\{{\rm
w}_{j}\\}}/K_{4}$. Using $\text{det }K_{4}=\text{det }K_{{\mathbb{A}},S}$ and
Theorem A.1, we have

$\Pi_{{\rm w}_{j},{\rm b}_{i}}=(-1)^{|S|-i}\det L_{S\setminus\\{{\rm
w}_{i}\\}\cup\\{{\rm w}_{j}\\}}=(-1)^{|S|-i}\frac{\text{det
}K_{{\mathbb{A}},S\setminus\\{{\rm w}_{i}\\}\cup\\{{\rm w}_{j}\\}}}{\text{det
}K_{{\mathbb{A}},S}}.$

The $\Pi$ matrix has the following multiplicativity property that is useful
for explicit computations.

######  Proposition 4.2.

Suppose $\Gamma$ is a graph obtained by gluing graphs $\Gamma_{i}$ for
$i=1,\dots,n$ from left to right, so that $S(\Gamma_{i})$ is identified with
$T(\Gamma_{i+1})$. Then

$\Pi(\Gamma)=\Pi(\Gamma_{1})\Pi(\Gamma_{2})\cdots\Pi(\Gamma_{n}).$

Here $S(\Gamma_{i})$ and $T(\Gamma_{i+1})$ denote the source vertices of
$\Gamma_{i}$ and target vertices of $\Gamma_{i+1}$ respectively.

We omit the proof since it is very similar to that of Theorem 4.4.

#### Remark.

The matrix $\Pi$ is closely related to the boundary measurement matrix of
[GSTV]. The reference dimer $M_{0}$ makes $\Gamma_{\mathbb{A}}$ into a
directed network $\mathcal{N}$ as follows: Orient each edge $e={\rm b}{\rm w}$
contained in $M_{0}$ from ${\rm w}\rightarrow{\rm b}$, and assign it weight
$\frac{1}{{\rm wt}(e)}$, and each edge not contained in $M_{0}$ from ${\rm
w}\rightarrow{\rm b}$ and assign it weight ${\rm wt}(e)$. Each directed path
in $\mathcal{N}$ gets a weight that is the product of weights of all edges
appearing in it. Then using Theorem 4.1, we have

$\displaystyle\frac{1}{{\rm wt}(M_{0})\kappa(M_{0})}\text{det
}K_{{\mathbb{A}},S}\Pi_{{\rm w}_{j},{\rm b}_{i}}$
$\displaystyle=(-1)^{|S|-i}\frac{\text{det }K_{{\mathbb{A}},S\setminus\\{{\rm
w}_{i}\\}\cup\\{{\rm w}_{j}\\}}}{{\rm wt}(M_{0})\kappa(M_{0})}$
$\displaystyle=(-1)^{|S|-i}\sum_{\partial M=S\setminus\\{{\rm
w}_{i}\\}\cup\\{{\rm w}_{j}\\}}\text{sign}([M-M_{0}])[{\rm wt}]([M-M_{0}]).$

Notice that if $\partial M=S\setminus\\{{\rm w}_{i}\\}\cup\\{{\rm w}_{j}\\}$,
then $M-M_{0}$ is a directed path in $\mathcal{N}$ from ${\rm w}_{i}$ to ${\rm
w}_{j}$, so $\frac{1}{{\rm wt}(M_{0})\kappa(M_{0})}\text{det
}K_{{\mathbb{A}},S}\Pi$ is the boundary measurement matrix of $\mathcal{N}$.
This is the reason for calling $S$ and $T$ the source and target vertices
respectively .

######  Example 4.3.

Consider the graph $\Gamma_{2}$ from Figure 5. Let $\Gamma_{{\mathbb{A}},2}$
denote the graph in the cylinder obtained from $\Gamma_{2}$ by applying the
construction in Section 4.1 with the zig-zag path $\zeta_{2}$. In this case,
we simply cut along the vertical side of the fundamental rectangle. The
sources are the two white vertices ${\rm w}_{1}$ and ${\rm w}_{2}$ on the
right boundary, while the targets are their copies on the left boundary that
we denote by ${\rm w}_{5}$ and ${\rm w}_{6}$. The set $B_{S}$ is $\\{{\rm
b}_{3},{\rm b}_{4}\\}$. The Kasteleyn matrix is

$K_{\mathbb{A}}=\blockarray{ccccc}{\rm b}_{3}&{\rm b}_{4}{\rm b}_{1}{\rm
b}_{2}\\\ \block{[cccc]c}1000{\rm w}_{1}\\\ 0100{\rm w}_{2}\\\ 0010{\rm
w}_{5}\\\ 0001{\rm w}_{6}\\\ hag-b{\rm w}_{3}\\\ ed-fc{\rm w}_{4}\\\ ,$

and

$\Pi=\frac{1}{bf-cg}\begin{bmatrix}ch+be&ac+bd\\\ eg+fh&dg+af\end{bmatrix}.$
(21)

###  4.3 Spectral curve of $\Gamma$ from $\Gamma_{{\mathbb{A}}}$.

We now show how the spectral curve of $\Gamma$ can be constructed from
$\Gamma_{{\mathbb{A}}}$. In $\Gamma$, split the white vertices that are in the
image of $S$ and $T$ under the projection of $\Gamma_{{\mathbb{A}}}$ to
${\mathbb{T}}$, so that we now have two copies of these white vertices which
we identify with $S$ and $T$ respectively, connected by degree two black
vertices. Let $\Gamma_{ST}$ denote the graph obtained. Let $B_{ST}$ denote the
newly created degree two black vertices. Perturb $\gamma_{w}$ so that it goes
transversely through all the edges connecting $B_{ST}$ with $T$ (see Figure
6). Let $K_{\mathbb{A}}(w)$ denote the Kasteleyn matrix for
$\Gamma_{\mathbb{A}}$ with weight ${\rm wt}(e)w^{(e,-\gamma_{z})}$, and let
$\Pi(w)$ denote the matrix in (20). We extend a Kasteleyn sign $\kappa$ on
$\Gamma$ to $\Gamma_{ST}$ by defining $\kappa(e)=1$ if $e$ is an edge between
$B_{ST}$ and $T$ and $\kappa(e)=-1$ if $e$ is an edge between $B_{ST}$ and
$S$.

######  Theorem 4.4.

$\Sigma=\\{(z,w)\in({\mathbb{C}}^{\times})^{2}:\text{det }(zI+\Pi(w))=0\\}$ is
the spectral curve.

###### Proof.

The Kasteleyn matrix $K(z,w)$ of $\Gamma_{ST}$ has the block matrix form

$K(z,w)=\blockarray{ccc}B(\Gamma_{{\mathbb{A}}})&B_{ST}\\\
\block{[c|c]c}-IS\\\ K_{{\mathbb{A}}}(w)zIT\\\ 0W_{\text{int}}\\\ .$

So we have (with $B_{S},K_{4}$ as in (19))

$K(z,w)/K_{4}=\blockarray{ccc}B_{S}&B_{ST}\\\ \block{[c|c]c}I-IS\\\
\Pi(w)zIT\\\ .$

By Theorem A.1, we get $\det{K(z,w)}=\text{det }K_{4}\text{ det
}{(zI+\Pi(w))}$. ∎

######  Example 4.5.

Consider again $\Gamma_{2}$ for which we have

$\Pi(w)=\frac{1}{-cg+bfw}\begin{bmatrix}ch+bew&(ac+bd)w\\\
eg+fh&dg+afw\end{bmatrix}$

from (21). We have

$\text{det
}(zI+\Pi(w))=\frac{1}{-cg+bfw}(aew+afwz+bewz+bfwz^{2}-cgz^{2}+chz+dgz-dh),$

which agrees with (18).

##  5 TCD maps on cylinders

###  5.1 TCD maps

We now describe the cokernel of $K$ from a projective point of view in terms
of triple crossing diagram maps, which we abbreviate to TCD maps. Let $\Gamma$
be an arbitrary bipartite graph (not necessarily embedded on a torus) with
$|W|>|B|$. Assume that the black vertices of $\Gamma$ are all trivalent, which
we can always do using split and join moves. A TCD map is a collection of
points $(P_{\rm w})_{{\rm w}\in W}\in{\mathbb{C}\mathrm{P}}^{|W|-|B|-1}$ such
that for each ${\rm b}\in B$, the three points $P_{\rm w}$ for ${\rm w}$
incident to ${\rm b}$ are distinct and are all contained in a line.

Given a generic weight ${\rm wt}$ on $\Gamma$, we obtain a TCD map as follows:
consider the exact sequence

$0\rightarrow{\mathbb{C}}^{B}\xrightarrow[]{K}{\mathbb{C}}^{W}\rightarrow\text{coker
}K\rightarrow 0,$

where $\text{coker }K$ is ${(|W|-|B|)}$-dimensional. Let $v_{\rm
w}\in\text{coker }K$ be the image of $e_{\rm w}$. Then the projectivizations
$P_{\rm w}$ of the vectors $v_{\rm w}$ define a TCD map. Clearly the
definition is invariant under gauge equivalence. On the other hand, given a
TCD map, we recover the edge-weights modulo gauge transformations from the
equations of the lines associated to the black vertices.

  * Remark.

Note that the cokernel is only defined up to isomorphism. Different choices
for a representative of the isomorphism class of the cokernel give different
TCD maps related by projective transformations.

$\mathrm{w}_{1}$$\mathrm{w}_{3}$$\mathrm{w}_{2}$$\mathrm{w}_{4}$

$\leftrightarrow$
$\mathrm{w}_{4}$$\mathrm{w}_{2}$$\mathrm{w}_{1}$$\mathrm{w}_{3}$
$\mathrm{w}_{1}$$\mathrm{w}_{4}$$\mathrm{w}_{2}$$\mathrm{w}_{3}$$\mathrm{w}$
$\leftrightarrow$
$\mathrm{w}_{4}$$\mathrm{w}_{3}$$\mathrm{w}_{1}$$\mathrm{w}_{2}$$\mathrm{w}^{\prime}$

Figure 7: Elementary transformations allowed in TCD maps: spider move (left)
and resplit (right).

The advantage of working with $\Gamma$ with trivalent black vertices is that
we can keep track of both the geometric dynamics and the invariants while
performing local moves. For TCD maps there are only two allowed elementary
transformations, the spider move and the resplit, see Figure 7. In a generic
situation, the points associated to white vertices after one of these two
moves are determined by the combinatorics. Indeed, no points change when
performing the spider move and the new point appearing in the resplit is
determined as the intersection of the two lines represented by the two black
vertices. It is a straight-forward calculation to verify that the edge-weights
of $\Gamma$ are compatible with the geometric constraints. In
${\mathbb{C}\mathrm{P}}^{1}$ however, there is no incidence geometry. In this
case, we _define_ the new white vertex in the resplit such that the edge-
weights correspond to a local move as dictated by the dimer model. Another
useful property of TCD maps is that the face-variables can be recovered as
multi-ratios as stated in the following Lemma.

######  Lemma 5.1 ([AGPR]*Proposition 2.6).

For a loop $L={\rm w}_{1}\rightarrow{\rm b}_{1}\rightarrow{\rm
w}_{2}\rightarrow{\rm b}_{2}\rightarrow\cdots\rightarrow{\rm
w}_{n}\rightarrow{\rm b}_{n}\rightarrow{\rm w}_{1}$, let ${\rm
w}_{i}^{\prime}$ denote the third white vertex incident to ${\rm b}_{i}$ that
is not in $\\{{\rm w}_{i},{\rm w}_{i+1}\\}$. Then we have

$[{\rm wt}]([L])=[\kappa]([L])^{-1}\operatorname{mr}{(P_{{\rm w}_{1}},P_{{\rm
w}_{1}^{\prime}},P_{{\rm w}_{2}},P_{{\rm w}_{2}^{\prime}},\dots,P_{{\rm
w}_{n}},P_{{\rm w}_{n}^{\prime}})}.$

Moreover, we make the following observation. Assume the white vertices are
labeled as in Figure 7. Then

$\displaystyle\operatorname{mr}(P_{\mathrm{w}_{1}},P_{\mathrm{w}},P_{\mathrm{w}_{2}},P_{\mathrm{w}_{3}},P_{\mathrm{w}^{\prime}},P_{\mathrm{w}_{4}})=-1$
(22)

holds. Note that this equation has a high degree of symmetry, it also holds if
we swap $\mathrm{w}\leftrightarrow\mathrm{w}^{\prime}$ or
$\mathrm{w}_{1}\leftrightarrow\mathrm{w}_{3}$ or
$\mathrm{w}_{2}\leftrightarrow\mathrm{w}_{4}$.

###  5.2 TCD map on $\Gamma_{\widehat{\mathbb{A}}}$ from $\Gamma$

Suppose $\Gamma_{\mathbb{A}}$ is a graph in ${\mathbb{A}}$ obtained from a
graph $\Gamma$ in ${\mathbb{T}}$ as in Section 4.1. We assume further that
there is a dimer cover $M_{1}$ such that $\partial M_{1}=T$, or equivalently
that the matrix $\Pi$ in (20) is invertible.

Let
$\widehat{\mathbb{A}}:=H_{1}({\mathbb{T}},{\mathbb{R}})/{\mathbb{Z}}[\gamma_{w}]$
denote the infinite cylinder covering ${\mathbb{A}}$. Note that
$\widehat{\mathbb{A}}=\bigcup_{k\in{\mathbb{Z}}}{\mathbb{A}}+k[\gamma_{z}]$,
that is $\widehat{\mathbb{A}}$ is obtained by gluing together infinitely many
copies of ${\mathbb{A}}$. Let

$\displaystyle\Gamma_{\widehat{\mathbb{A}}}=(B(\Gamma_{\widehat{\mathbb{A}}})\cup
W(\Gamma_{\widehat{\mathbb{A}}}),E(\Gamma_{\widehat{\mathbb{A}}}))$ (23)

denote the preimage of $\Gamma$ under the covering map
$\widehat{\mathbb{A}}\rightarrow{\mathbb{T}}$. Fix a white vertex ${\rm w}\in
W(\Gamma_{\widehat{\mathbb{A}}})$ and choose a large enough $m$ so that ${\rm
w}$ is in
${\mathbb{A}}_{m}:=\bigcup_{k\in[-m,m]\cap{\mathbb{Z}}}({\mathbb{A}}+k\gamma_{z})\subset\widehat{\mathbb{A}}$.
Let
$\Gamma_{{\mathbb{A}}_{m}}:=\Gamma_{\widehat{\mathbb{A}}}\cap{\mathbb{A}}_{m}$.

######  Lemma 5.2.

We have $\text{coker }K_{{{{\mathbb{A}}_{m}}}}\cong\text{coker
}K_{\mathbb{A}}\cong{\mathbb{C}}^{|W|-|B|}$ for all $m\geq 0$. Moreover with
these identifications, the image of $e_{{\rm w}}$ in ${\mathbb{C}}^{|W|-|B|}$
under the cokernel map of $K_{{\mathbb{A}}_{m}}$ is independent of $m$, where
$e_{\rm w}$ is the unit basis vector corresponding to ${\rm w}$ in
${\mathbb{C}}^{W(\Gamma_{{\mathbb{A}}_{m}})}$.

###### Proof.

Suppose $m>{m^{\prime}}$. $K_{{{{\mathbb{A}}_{m}}}}$ has the block form

$K_{{{{\mathbb{A}}_{m}}}}=\blockarray{ccc}B(\Gamma_{{{\mathbb{A}}_{m^{\prime}}}})&B(\Gamma_{{{\mathbb{A}}_{m}}})\setminus
B(\Gamma_{{{\mathbb{A}}_{m^{\prime}}}})\\\
\block{[cc]c}K_{{{\mathbb{A}}_{m^{\prime}}}}*W(\Gamma_{{{\mathbb{A}}_{m^{\prime}}}})\\\
0K^{\prime}W(\Gamma_{{{\mathbb{A}}_{m}}})\setminus
W(\Gamma_{{{\mathbb{A}}_{m^{\prime}}}})\\\ ,$

where $K^{\prime}$ is invertible by existence of $M_{0}$ and $M_{1}$, and
Theorem 4.1. Therefore
$K_{{{{\mathbb{A}}_{m}}}}/K^{\prime}=K_{{{\mathbb{A}}_{m^{\prime}}}}$ and so
the second statement follows from Theorem A.2. By Theorem A.2 with
${m^{\prime}}=0$ we get $\text{coker }K_{{{\mathbb{A}}_{m}}}\cong\text{coker
}K_{\mathbb{A}}\cong{\mathbb{C}}^{|W|-|B|}$ for all $m\geq 0$. ∎

We define a TCD map
$P:W(\Gamma_{\widehat{\mathbb{A}}})\rightarrow{\mathbb{C}\mathrm{P}}^{|W(\Gamma_{{\mathbb{A}}})|-|B(\Gamma_{{\mathbb{A}}})|-1}$
as follows: For ${\rm w}\in W(\Gamma_{\widehat{\mathbb{A}}})$, we choose $m$
sufficiently large so that ${\rm w}\in W(\Gamma_{{\mathbb{A}}_{m}})$. Let
$v_{\rm w}$ denote the image of $e_{\rm w}$ in
${\mathbb{C}}^{|W(\Gamma_{{\mathbb{A}}})|-|B(\Gamma_{{\mathbb{A}}})|}$ as in
Lemma 5.2, and define $P_{\rm
w}\in{\mathbb{C}\mathrm{P}}^{|W(\Gamma_{{\mathbb{A}}})|-|B(\Gamma_{{\mathbb{A}}})|-1}$
to be the projectivization of $v_{\rm w}$. Lemma 5.2 guarantees that the
definition is independent of the choice of $m$.

###  5.3 Monodromy of a TCD map on $\Gamma_{\widehat{\mathbb{A}}}$

A pair $(P,M)$ where
$P:W(\Gamma_{\widehat{\mathbb{A}}})\rightarrow{\mathbb{C}\mathrm{P}}^{|W(\Gamma_{{\mathbb{A}}})|-|B(\Gamma_{{\mathbb{A}}})|-1}$
is a TCD map and
$M\in\operatorname{PGL}_{|W(\Gamma_{{\mathbb{A}}})|-|B(\Gamma_{{\mathbb{A}}})|}$
is called a twisted TCD map if $M(P_{{\rm w}})=P_{{\rm w}+\gamma_{z}}$ for all
${\rm w}\in W(\Gamma_{\widehat{\mathbb{A}}}).$ $M$ is called the monodromy of
$P$.

Now suppose
$P:W(\Gamma_{\widehat{\mathbb{A}}})\rightarrow{\mathbb{C}\mathrm{P}}^{|W(\Gamma_{{\mathbb{A}}})|-|B(\Gamma_{{\mathbb{A}}})|-1}$
is a TCD map on $\Gamma_{\widehat{\mathbb{A}}}$ as in Section 5.2. The
existence of $M_{1}$ means that the map
${\mathbb{C}}^{T}\rightarrow\text{coker }K_{\mathbb{A}}$ given by $e_{\rm
w}\mapsto v_{\rm w}$ is an isomorphism. In other words, this map defines a
change of basis for $\text{coker }K_{\mathbb{A}}$ making $v_{\rm w},{\rm w}\in
T$ the standard basis vectors. Let $[\Pi]$ denote the class of $\Pi$ in
$\operatorname{PGL}_{|W(\Gamma_{{\mathbb{A}}})|-|B(\Gamma_{{\mathbb{A}}})|}$,
where $\Pi$ is the matrix in (20).

######  Proposition 5.3.

$P$ is a twisted TCD map. The
$\operatorname{PGL}_{|W(\Gamma_{{\mathbb{A}}})|-|B(\Gamma_{{\mathbb{A}}})|}$-matrix
$[\Pi]$ is the monodromy of $P$ in the basis $\\{v_{\rm w}\\}_{{\rm w}\in T}$.

###### Proof.

Let $T=\\{{\rm w}_{1},\dots,{\rm w}_{n}\\}$ be the set of target vertices in
$W(\Gamma_{{\mathbb{A}}})$. Since $\begin{bmatrix}I\\\ \Pi{}\end{bmatrix}$ is
the Schur complement of $K_{\mathbb{A}}$, by Theorem A.2 we have

$\begin{bmatrix}v_{{\rm w}_{1}+\gamma_{z}}&\dots&v_{{\rm
w}_{n}+\gamma_{z}}&v_{{\rm w}_{1}}&\dots&v_{{\rm
w}_{n}}\end{bmatrix}\begin{bmatrix}I\\\ \Pi{}\end{bmatrix}=0,$

where $v_{{\rm w}_{i}}$ is the standard basis vector $e_{i}$. From this, we
get $v_{{\rm w}_{i}+\gamma_{z}}=-\Pi{}v_{{\rm w}_{i}}$. The same argument
applied to the translated graph $\Gamma_{{\mathbb{A}}+k\gamma_{z}}$ gives
$v_{{\rm w}_{i}+(k+1)\gamma_{z}}=-\Pi{}v_{{\rm w}_{i}+k\gamma_{z}}$. This
implies $v_{{\rm w}_{i}+k\gamma_{z}}=(-\Pi{})^{k}v_{{\rm w}_{i}}$ for all
$k\in{\mathbb{Z}}$. For ${\rm w}\in W(\Gamma_{{\mathbb{A}}})$, since
$\\{v_{{\rm w}_{i}}\\}$ is a basis, there exists $a_{i}$ such that $v_{\rm
w}=\sum_{i=1}^{n}a_{i}v_{{\rm w}_{i}}$. Then we have

$\displaystyle(-\Pi{})^{k}v_{{\rm w}}$
$\displaystyle=\sum_{i=1}^{n}a_{i}(-\Pi{})^{k}v_{{\rm w}_{i}}$
$\displaystyle=\sum_{i=1}^{n}a_{i}v_{{\rm w}_{i}+k\gamma_{z}}$
$\displaystyle=v_{{\rm w}+k\gamma_{z}},$

where the last equality follows from
$K_{{\mathbb{A}}}{}=K_{{{\mathbb{A}}+k\gamma_{z}}}{}$. Any white ${\rm
w}^{\prime}\in W(\Gamma_{\widehat{\mathbb{A}}})$ is of the form ${\rm
w}+m\gamma_{z}$ for some ${\rm w}\in W(\Gamma_{{\mathbb{A}}})$. Then ${\rm
w}^{\prime}+\gamma_{z}={\rm w}+(m+1)\gamma_{z}$, so that we have

$v_{{\rm w}^{\prime}}=(-\Pi{})^{m}v_{{\rm w}},\quad v_{{\rm
w}^{\prime}+\gamma_{z}}=(-\Pi{})^{m+1}v_{\rm w},$

and therefore $v_{{\rm w}^{\prime}+\gamma_{z}}=-\Pi{}v_{{\rm w}^{\prime}}$.
Projectivizing, we get the statement of the proposition. ∎

##  6 Twisted TCD maps from $\mathcal{U}_{n}$ for $n$ even

For $n$ even, recall the graph $\Gamma_{n}$ from Example 3.1 and let $N$
denote its Newton polygon computed in Example 3.2. Suppose the faces and zig-
zag paths of $\Gamma_{n}$ are labeled as in Figure 2 and Example 3.2
respectively. The set
$\\{X_{1},\dots,X_{n},Y_{1},\dots,Y_{n},\chi_{[\zeta_{1}]},\chi_{[\xi_{1}]}\\}$
is a set of generators for $\mathcal{O}_{{\mathcal{L}}_{\Gamma_{n}}}$ and the
only relation among them is $\prod_{i=1}^{n}X_{i}Y_{i}=1$.

Let $\mathcal{X}_{N,\vec{\alpha}}$ (resp.
$\mathcal{L}_{\Gamma_{n},\vec{\alpha}}$) denote the subvariety of
$\mathcal{X}_{N}$ (resp. $\mathcal{L}_{\Gamma_{n}}$) where

$\displaystyle\chi_{[\zeta_{2i-1}]}$ $\displaystyle=1-\alpha_{2i-1},$ (24)
$\displaystyle\chi_{[\zeta_{2i}]}$ $\displaystyle=\frac{1}{1-\alpha_{2i}},$
(25)

for all $i\in\\{1,2,\dots,n\\}$. Recall from Section 2 that
$\mathcal{U}_{n}\cong({\mathbb{C}}\setminus\\{0,-1\\})^{n}$.

######  Definition 6.1.

We define a rational map

$\displaystyle\pi_{\vec{\alpha}}:\mathcal{X}_{N,\vec{\alpha}}\supset\mathcal{L}_{\Gamma_{n},\vec{\alpha}}\rightarrow({\mathbb{C}}\setminus\\{0,-1\\})^{n}$
(26)

such that

$\displaystyle\quad\pi_{\vec{\alpha}}^{*}u_{i}=Y_{i},$ (27)

for all $i\in\\{1,2,\dots,n\\}$.

######  Lemma 6.2.

We have

$\displaystyle\pi_{\vec{\alpha}}^{*}v_{i}=X_{i}$

for all $i\in\\{1,2,\dots,n\\}$.

###### Proof.

It follows from

$X_{i}Y_{i}=\begin{cases}{\chi_{[\zeta_{i}]}}{\chi_{[\zeta_{i-1}]}}&\text{ if
$i$ is odd};\\\ \frac{1}{{\chi_{[\zeta_{i}]}}{\chi_{[\zeta_{i-1}]}}}&\text{ if
$i$ is even},\end{cases}$

and the equations (24), (25) and (10)). ∎

Next, we check that:

######  Lemma 6.3.

$\pi_{\vec{\alpha}}$ is Poisson.

###### Proof.

$Y_{1},Y_{2},\dots,Y_{n}$ and $\chi_{[\xi_{1}]}$ are coordinates on
$\mathcal{X}_{N,\vec{\alpha}}$. The only nonzero Poisson brackets on
$\mathcal{X}_{N,\vec{\alpha}}$ in these coordinates are

$\displaystyle\\{Y_{i},Y_{i+1}\\}=Y_{i}Y_{i+1},\quad i=1,2,\dots,n.$ (28)

We compute that

$\displaystyle\pi_{\vec{\alpha}}^{*}\\{Y_{i},Y_{i+1}\\}=\pi_{\vec{\alpha}}^{*}(Y_{i}Y_{i+1})=u_{i}u_{i+1}=\\{\pi_{\vec{\alpha}}^{*}Y_{i},\pi_{\vec{\alpha}}^{*}Y_{i+1}\\}.$
(29)

∎

###  6.1 TCD map and the pair of curves

${\rm w}_{4i-2}$${\rm w}_{4i-3}$${\rm b}_{4i-2}$${\rm b}_{4i-3}$${\rm
w}_{4i-1}$${\rm w}_{4i-1}$${\rm
w}_{4i}$$1$$-u_{2i-1}$$1$$1$$-\frac{1-\alpha_{2i-1}}{u_{2i-1}}w$$1$${\rm
w}_{4i+2}$${\rm w}_{4i+1}$${\rm b}_{4i}$${\rm
b}_{4i-1}$$1$${\alpha_{2i-1}u_{2i-1}u_{2i}}$$\alpha_{2i-1}$$1$$\alpha_{2i-1}u_{2i-1}u_{2i}$$\alpha_{2i-1}(1-\alpha_{2i})w$

${\rm w}_{2n}$${\rm w}_{2n-1}$${\rm b}_{2n-1}$${\rm b}_{2n-1}$${\rm
b}_{2n}$${\rm w}_{2n+2}$${\rm
w}_{2n+1}$$1$$\frac{1}{\alpha_{n}}$$1$$\frac{1}{\alpha_{n}u_{n}}$$-\frac{1-\alpha_{n}}{\alpha_{n}u_{n}}w$$-\frac{1}{\alpha_{n}}$

Figure 8: Edge weights and Kasteleyn signs for the building block graphs
$G_{i}$ (left) as well as $G_{n}^{\text{odd}}$ (right).

Let $[{\rm wt}]\in\mathcal{X}_{N,\vec{\alpha}}$ such that
$u^{-1}\circ\pi_{\vec{\alpha}}([{\rm wt}])=(p,q,M)$. We choose edge weights
representing $[{\rm wt}]$ and Kasteleyn signs as in Figure 8.

#### Remark.

We have additionally the freedom to choose $\chi_{\xi_{1}}([{\rm wt}]).$
However this is a Casimir and for our purposes may be absorbed into $z$ since
it changes the Hamiltonians by a multiplicative factor, and does not affect
the TCD map.

Note that we have
$|W(\Gamma_{n,{\mathbb{A}}})|-|B(\Gamma_{n,{\mathbb{A}}})|=2$. Let
$P:\Gamma_{n,\widehat{\mathbb{A}}}\rightarrow{\mathbb{C}\mathrm{P}}^{1}$
denote the TCD map associated to $[{\rm wt}]$.
$\Gamma_{n,\widehat{\mathbb{A}}}$ is a union of infinitely many copies of the
building block graph $G_{i},i\in{\mathbb{Z}}$. We label the vertices of
$\Gamma_{n,\widehat{\mathbb{A}}}$ as in Figure 2.

######  Lemma 6.4.

We have (see Figure 1)

$P_{{\rm w}_{k}}=\begin{cases}q_{2i-1}&\text{ if $k=4i-3$};\\\ q_{2i}&\text{
if $k=4i-1$};\\\ p_{2i-1}&\text{ if $k=4i-2$};\\\ p_{2i}&\text{ if
$k=4i$},\end{cases}$

up to a projective transformation, for all $i\in{\mathbb{Z}}$.

###### Proof.

The discrete curves $p$ and $q$ are determined up to projective
transformations by the cross-ratios (7) and (8). Using Lemma 5.1, we see from
(31) and Lemma 6.2 that the $P_{\rm w}$ have the same cross-ratios. ∎

######  Corollary 6.5.

The monodromy matrix of the TCD map $P$ coincides with the monodromy matrix
$M$ of the pair of discrete curves $p,q$.

We now compute the monodromy matrix. The Kasteleyn matrix of $G_{i}$ is

$K_{G_{i}}(w)=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
{\alpha_{2i-1}u_{2i-1}u_{2i}}&{\alpha_{2i-1}u_{2i-1}u_{2i}}&1&-u_{2i-1}\\\
\alpha_{2i-1}{1-\alpha_{2i}}{w}&\alpha_{2i-1}&-\frac{1-\alpha_{2i-1}}{u_{2i-1}}w&1\end{bmatrix}$

so that

$\displaystyle\Pi_{G_{i}}(w)=\frac{\alpha_{2i-1}}{({1-\alpha_{2i-1})w-1}}\begin{bmatrix}u_{2i-1}(u_{2i}+{(1-\alpha_{2i})w)}&u_{2i-1}(1+u_{2i})\\\
{{({1-\alpha_{2i}})w}+u_{2i}{{(1-\alpha_{2i-1})}{w}}}&1+u_{2i}({{1-\alpha_{2i-1}}})w\end{bmatrix}.$

We have $\text{det
}\Pi_{G_{i}}(w)=\left(\frac{\alpha_{2i-1}}{{(1-\alpha_{2i-1})}{w}-1}\right)\left({{(1-\alpha_{2i})}{w}-1}\right){\alpha_{2i-1}u_{2i-1}u_{2i}}$.
By Proposition 4.2, the monodromy matrix is

$\Pi(w)=\Pi_{G_{1}}(w)\Pi_{G_{2}}(w)\cdots\Pi_{G_{\frac{n}{2}}}(w)$

and so we get

$\text{det }\Pi(1)={\alpha_{[n]}u_{[n]}}.$ (30)

###  6.2 Hamiltonians

The following lemma is elementary.

######  Lemma 6.6.

Suppose $M$ is a $2\times 2$ matrix and $P(z)=\text{det }(zI+M)$ is its
characteristic polynomial. Then $P(z)=z^{2}+\text{tr }Mz+\text{det }M$.

Let $P(z,w)$ denote the characteristic polynomial of $\Gamma_{n}$, normalized
so that it has Newton polygon as in Figure 3 with the vertex in top right
corner corresponding to $z^{2}w^{\frac{n}{2}}$, and such that the coefficient
of $z^{2}w^{\frac{n}{2}}$ is
$\frac{1}{\alpha_{\text{odd}}}\prod_{i=1}^{\frac{n}{2}}{(1-\alpha_{2i-1})}$.
To find the normalization explicitly, we know from Theorem 4.4 and Lemma 6.6
that $P(z,w)$ is up to normalization equal to $\text{det
}(zI+\Pi(w))=z^{2}+\text{tr }\Pi(w)z+\text{det }\Pi(w)$, which is equal to

$\displaystyle z^{2}$
$\displaystyle+z\frac{\alpha_{\text{odd}}}{\prod_{i=1}^{\frac{n}{2}}((1-\alpha_{2i-1})w-1)}\text{
tr
}\prod_{i=1}^{\frac{n}{2}}\begin{bmatrix}u_{2i-1}(u_{2i}+{(1-\alpha_{2i})w)}&u_{2i-1}(1+u_{2i})\\\
{{({1-\alpha_{2i}})w}+u_{2i}{{(1-\alpha_{2i-1})}{w}}}&1+u_{2i}({{1-\alpha_{2i-1}}})w\end{bmatrix}+$
$\displaystyle\frac{\alpha_{\text{odd}}}{\prod_{i=1}^{\frac{n}{2}}((1-\alpha_{2i-1})w-1)}{\alpha_{\text{odd}}}u_{[n]}{\prod_{i=1}^{\frac{n}{2}}((1-\alpha_{2i})w-1)},$

which is

$\frac{\alpha_{\text{odd}}}{\prod_{i=1}^{\frac{n}{2}}((1-\alpha_{2i-1})w-1)}\left(\left(\frac{1}{\alpha_{\text{odd}}}\prod_{i=1}^{\frac{n}{2}}{(1-\alpha_{2i-1})}\right)z^{2}w^{\frac{n}{2}}+\text{
lower degree terms}\right).$

Let $H_{(1,k)}$ denote the coefficient of $zw^{k}$ in $P(z,w)$ for
$k=0,\dots,\frac{n}{2}$, so that when $k\in\\{1,2,\dots,\frac{n}{2}-1\\}$,
they are the Hamiltonians of the dimer integrable system. Then we have

$\displaystyle\sum_{k=0}^{\frac{n}{2}}H_{(1,k)}w^{k}$
$\displaystyle=\frac{\prod_{i=1}^{\frac{n}{2}}((1-\alpha_{2i-1})w-1)}{\alpha_{\text{odd}}}\text{tr
}\Pi(w)$ $\displaystyle=\text{tr
}\prod_{i=1}^{\frac{n}{2}}\begin{bmatrix}u_{2i-1}(u_{2i}+{(1-\alpha_{2i})w)}&u_{2i-1}(1+u_{2i})\\\
{{({1-\alpha_{2i}})w}+u_{2i}{{(1-\alpha_{2i-1})}{w}}}&1+u_{2i}({{1-\alpha_{2i-1}}})w\end{bmatrix}.$

Make the substitution $\beta_{i}=1-\alpha_{i}$ and consider the above
expression as a polynomial in $\beta_{1},\beta_{2},\dots,\beta_{n}$. Since
each $(1-\alpha_{i})$ inside the matrices appears with a $w$, the homogeneous
component of degree $k$ in $\beta_{1},\beta_{2},\dots,\beta_{n}$ is
$H_{(1,k)}w^{k}$.

The main result of this section is the following simple procedure for
converting between the AFIT Hamiltonians [AFIT] and the dimer Hamiltonians.

######  Theorem 6.7.

The homogeneous degree $k$ component of $\sum_{k=0}^{\frac{n}{2}}H_{(1,k)}$ as
a polynomial in the variables $\alpha_{1},\alpha_{2},\dots,\alpha_{n}$ is, up
to a sign, equal to
$\sqrt{Y_{[n]}\alpha_{[n]}}\pi_{\vec{\alpha}}^{*}\circ\Lambda_{\overrightarrow{\alpha}}^{*}\left(\frac{F_{k}(c)}{\sqrt{c_{[n]}}}\right)$.
For $k\in\\{1,2,\dots,\frac{n}{2}-1\\}$, these are AFIT Hamiltonians since
$\sqrt{Y_{[n]}\alpha_{[n]}}$ is a Casimir. The homogeneous degree $k$
component of $\sum_{k=0}^{\frac{n}{2}}H_{(1,k)}$ as a polynomial in
$1-\alpha_{1},1-\alpha_{2},\dots,1-\alpha_{n}$ is the dimer Hamiltonian
$H_{(1,k)}$.

###### Proof.

By Corollary 6.5, $\Pi(1)$ is conjugated to
$\pi_{\vec{\alpha}}^{*}\circ\Lambda_{\overrightarrow{\alpha}}^{*}M$ in
$\operatorname{PGL}_{2}$. Since $\frac{1}{\text{det }M}\text{tr}^{2}M$ is a
$\operatorname{PGL}_{2}$-conjugacy invariant, we have

$\pi_{\vec{\alpha}}^{*}\circ\Lambda_{\overrightarrow{\alpha}}^{*}\left(\frac{1}{\sqrt{\text{det
}M}}\text{tr}M\right)=\pm\frac{1}{\sqrt{\text{det
}(\Pi(1)})}\text{tr}(\Pi(1)).$

Using Theorem 2.3 and (30), we get

$\sqrt{Y_{[n]}\alpha_{[n]}}\sum_{k=0}^{\frac{n}{2}}(-1)^{k}\pi_{\vec{\alpha}}^{*}\circ\Lambda_{\overrightarrow{\alpha}}^{*}\left(\frac{F_{k}(c)}{\sqrt{c_{[n]}}}\right)=\pm\sum_{k=0}^{\frac{n}{2}}H_{(1,k)}.$

Note that the left side is a polynomial in $\alpha_{i}$ since
$\pi_{\vec{\alpha}}^{*}\circ\Lambda_{\overrightarrow{\alpha}}^{*}\sqrt{c_{[n]}}$
has a factor $\sqrt{\alpha_{[n]}}$ which cancels the same factor in the
numerator. Then the homogeneous component of degree $k$ in the variables
$\alpha_{1},\alpha_{2},\dots,\alpha_{n}$ on the left is
$(-1)^{k}\sqrt{Y_{[n]}\alpha_{[n]}}\pi_{\vec{\alpha}}^{*}\circ\Lambda_{\overrightarrow{\alpha}}^{*}\left(\frac{F_{k}(c)}{\sqrt{c_{[n]}}}\right)$.
∎

##  7 Odd $n$

For $n$ odd, let $\Gamma_{n}$ be obtained by gluing the graphs
$G_{1}G_{2}\cdots G_{\frac{n-3}{2}}G_{\frac{n-1}{2}}G_{n}^{\text{odd}}$ from
left to right and identifying ${\rm w}_{2n+1}$ with ${\rm w}_{1}$ and ${\rm
w}_{2n+2}$ with ${\rm w}_{2}$. The Newton polygon of $\Gamma_{n}$ is (see
Figure 3)

$\text{Convex-hull
}\left\\{(0,0),(2,0),\left(0,\frac{n-1}{2}\right),\left(2,\frac{n+1}{2}\right)\right\\}.$

We label the zig-zag paths as follows:

$\displaystyle\xi_{1}$ $\displaystyle={\rm w}_{1},{\rm b}_{1},{\rm w}_{3},{\rm
b}_{3},\dots,{\rm w}_{2n-1},{\rm b}_{2n},{\rm w}_{1},\quad[\xi_{1}]=(1,0),$
$\displaystyle\xi_{2}$ $\displaystyle={\rm w}_{2},{\rm b}_{2},{\rm w}_{4},{\rm
b}_{4},\dots,{\rm w}_{2n},{\rm b}_{2n-1},{\rm w}_{2},\quad[\xi_{2}]=(1,0),$
$\displaystyle\sigma$ $\displaystyle={\rm w}_{1},{\rm b}_{2n-1},{\rm
w}_{2n},{\rm b}_{2n-2},\dots,{\rm w}_{4},{\rm b}_{1},{\rm
w}_{1},\quad[\sigma]=(-2,-1),$ $\displaystyle\zeta_{2i-1}$ $\displaystyle={\rm
w}_{4i-1},{\rm b}_{4i-3},{\rm w}_{4i},{\rm b}_{4i-2},{\rm
w}_{4i-1},\quad[\zeta_{2i-1}]=(0,1),$ $\displaystyle\zeta_{2i}$
$\displaystyle={\rm w}_{4i-1},{\rm b}_{4i},{\rm w}_{4i},{\rm b}_{4i-1},{\rm
w}_{4i-1},\quad[\zeta_{2i}]=(0,-1),$ $\displaystyle\zeta_{n}$
$\displaystyle={\rm w}_{2n+1},{\rm b}_{2n},{\rm w}_{2n+2},{\rm
b}_{2n-1},\quad[\zeta_{2i-1}]=(0,1),$

where $i\in\\{1,2,\dots,\frac{n-1}{2}\\}$.

Let $N$ denote the Newton polygon of $\Gamma_{n}$. We define a rational map
$\pi_{\vec{\alpha}}:\mathcal{X}_{N,\vec{\alpha}}\supset\mathcal{L}_{\Gamma_{n},\vec{\alpha}}\rightarrow({\mathbb{C}}\setminus\\{0,-1\\})^{n}:$

$\displaystyle\quad\pi_{\vec{\alpha}}^{*}u_{i}=Y_{i},$ (31)

for all $i\in\\{1,2,\dots,n\\}$, which is checked to be Poisson as in the even
case. We have

$\displaystyle\Pi_{G_{n}^{\text{odd}}}(w)$
$\displaystyle=\frac{\alpha_{n}}{({1-\alpha_{n}}){w}-1}\begin{bmatrix}1&(1-\alpha_{n})w\\\
u_{n}&u_{n}\end{bmatrix},$ (32) $\displaystyle\text{det
}\Pi_{G_{n}^{\text{odd}}}(w)$
$\displaystyle=\frac{\alpha_{n}}{({1-\alpha_{n}}){w}-1}(-{\alpha_{n}u_{n}}),$
$\displaystyle\text{det }\Pi_{G_{n}^{\text{odd}}}(1)$
$\displaystyle=\alpha_{n}u_{n}.$

Let $\Pi(w)$ denote the monodromy matrix of $\Gamma_{n}$. As in the even case,
we have $\text{det }\Pi(1)={\alpha_{[n]}u_{[n]}}.$ Let $P(z,w)$ denote the
characteristic polynomial of $\Gamma_{n}$ normalized as follows

$\text{det }(zI+\Pi(w))=\frac{\alpha_{\text{odd}}}{\prod_{i\text{
odd}}((1-\alpha_{2i})w-1)}P(z,w).$

Let $H_{(1,k)}$ denote the coefficient of $zw^{k}$ in $P(z,w)$ for
$k=1,\dots,\frac{n+1}{2}$. Note that $(1-\alpha_{n})$ appears with a $w$ in
(32), so as before $H_{(1,k)}w^{k}$ is the homogeneous component of degree $k$
in $1-\alpha_{1},1-\alpha_{2},\dots,1-\alpha_{n}$. Theorem 6.7 holds verbatim,
after making the necessary adjustment of the indices.

######  Theorem 7.1.

The homogeneous degree $k$ component of $\sum_{k=0}^{\frac{n}{2}}H_{(1,k)}$ as
a polynomial in the variables $\alpha_{1},\alpha_{2},\dots,\alpha_{n}$ is, up
to a sign, equal to
$\sqrt{Y_{[n]}\alpha_{[n]}}\pi_{\vec{\alpha}}^{*}\circ\Lambda_{\overrightarrow{\alpha}}^{*}\left(\frac{F_{k}(c)}{\sqrt{c_{[n]}}}\right)$.
For $k\in\\{1,2,\dots,\frac{n-1}{2}\\}$, these are AFIT Hamiltonians since
$\sqrt{Y_{[n]}\alpha_{[n]}}$ is a Casimir. The homogeneous degree $k$
component of $\sum_{k=0}^{\frac{n}{2}}H_{(1,k)}$ as a polynomial in
$1-\alpha_{1},1-\alpha_{2},\dots,1-\alpha_{n}$ is the dimer Hamiltonian
$H_{(1,k)}$.

##  8 Cross-ratio dynamics via local moves

$q_{2}$$q_{2}$$p_{2}$$q_{1}$$p_{1}$$q_{n}$$q_{n}$$p_{n}$

$\rightarrow$ $q_{2}$$q_{2}$$p_{2}$$q_{1}$$p_{1}$$p_{1}$$q_{n}$$q_{n}$$p_{n}$
$\rightarrow$ $q_{2}$$q_{2}$$p_{2}$$q_{1}$$p_{1}$$p_{1}$$q_{n}$$q_{n}$$p_{n}$
$\rightarrow$
$q_{2}$$q_{2}$$p_{2}$$q_{1}$$p_{1}$$p_{1}$$r_{1}$$q_{n}$$q_{n}$$p_{n}$

Figure 9: Insertion of $r_{1}$ in the fundamental domain.

$q_{3}$$p_{3}$$q_{2}$$q_{2}$$p_{2}$$q_{1}$$p_{1}$$r_{1}$

$\rightarrow$
$q_{3}$$p_{3}$$q_{2}$$q_{2}$$p_{2}$$q_{1}$$r_{2}$$r_{1}$$\rightarrow$
$q_{3}$$p_{3}$$q_{2}$$q_{2}$$p_{2}$$q_{1}$$r_{2}$$r_{1}$$\rightarrow$
$q_{3}$$p_{3}$$q_{2}$$q_{2}$$p_{2}$$q_{1}$$r_{2}$$r_{1}$$\rightarrow$
$q_{3}$$p_{3}$$q_{2}$$q_{2}$$r_{3}$$q_{1}$$r_{2}$$r_{1}$

Figure 10: Beginning of the sequence of local moves that propagates $r$
through $(p,q)$.

In this section we show that cross-ratio dynamics, that is the map
$\nu_{\overrightarrow{\alpha}}:(p,q,M)\mapsto(q,r,M)$ is identified with a
certain sequence of local moves on the dimer side. We prepare with an
observation on multi-ratios.

######  Lemma 8.1.

Assume $p,q,r$ are nondegenerate twisted curves such that $p\neq r$ and both
$p$ and $r$ are $\overrightarrow{\alpha}$-related to $q$. Then

$\displaystyle\operatorname{mr}(p_{i},q_{i},r_{i},r_{i+1},q_{i+1},p_{i+1})=-1$
(33)

holds for all $i\in{\mathbb{Z}}$.

###### Proof.

The Lemma follows directly from the fact that both

$\displaystyle\operatorname{cr}(p_{i},q_{i},q_{i+1},p_{i+1})$
$\displaystyle=\operatorname{cr}(q_{i},r_{i},r_{i+1},q_{i+1}),$ (34)
$\displaystyle\operatorname{mr}(p_{i},q_{i},r_{i},r_{i+1},q_{i+1},p_{i+1})$
$\displaystyle=-\frac{\operatorname{cr}(p_{i},q_{i},q_{i+1},p_{i+1})}{\operatorname{cr}(q_{i},r_{i},r_{i+1},q_{i+1})}$
(35)

hold for all $i\in{\mathbb{Z}}$. ∎

Now we construct the sequence of moves.

######  Theorem 8.2.

Suppose $[{\rm wt}]\in\mathcal{X}_{N,\vec{\alpha}}$ is such that
$u^{-1}\circ\pi_{\vec{\alpha}}([{\rm wt}])=(p,q,M)$. Consider the sequence of
moves shown in Figures 9, 10, and let $\mu$ denote the induced birational map
of weights. Then the pair of curves $(q,r,M)$ is
$u^{-1}\circ\pi_{\vec{\alpha}}\circ\mu([{\rm wt}])$. In other words, the
following diagram commutes:

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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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###### Proof.

By Lemma 6.4, it suffices to trace what happens to the TCD map associated to
$[{\rm wt}]$ through the sequence of moves. Let us initially assume that we
deal with a periodic instead of a twisted curve. Assume $n$ is even. We
proceed in four steps:

  1. 1.

First, we need to insert $r_{1}$ into the graph, see Figure 9 for an
illustration. To do this, we begin by splitting $p_{1}$ into two copies of
$p_{1}$ and a new black vertex ${\rm b}$. Then we add a bigon with vertices
${\rm b},q_{1}$, such that the bigon is inside the face also bounded by
$p_{2}$ and $p_{n}$. Finally we split ${\rm b}$ into two new black vertices of
degree three while also generating the new white vertex corresponding to
$r_{1}$.

  2. 2.

We apply a sequence of spider moves and resplits to propagate $r_{1}$ through
the graph, see Figure 10. To replace both $p_{2i-1}$ and $p_{2i}$ with
$r_{2i}$ and $r_{2i+1}$ we need to apply a resplit, then two spider moves and
then a resplit again. Each time we apply a resplit, we check that the points
before and after are related as in Lemma 8.1. This ensures that we are not
changing the face-weights of $\Gamma$, as discussed in Section 5.1. We apply
these moves until we have replaced $p_{n}$ with a new copy of $r_{1}$.

  3. 3.

After the moves of step 2, there is still one copy of $p_{1}$ left in the
graph. The situation at $p_{1}$ looks like in the rightmost graph in Figure 9,
except that the roles of $p$ and $r$ are interchanged. We can therefore absorb
$p_{1}$ again by doing step 1 backwards. Here, it is essential that the two
copies of $r_{1}$ coincide with each other in ${\mathbb{C}\mathrm{P}}^{1}$.
After a rescaling at $r_{1}$, we can assume the lifts in ${\mathbb{C}}^{2}$
coincide as well. This ensures that the corresponding edge-weights in the
contraction and the removal of the bigon agree up to a common factor. This
common factor can be set to ${-1}$ by a rescaling at one of the involved black
vertices.

  4. 4.

Finally, we translate $\Gamma_{n}$ by $\frac{1}{2}\gamma_{w}$ to interchange
$q$ and $r$.

If $n$ is not even, the procedure is almost identical. We also split $p_{1}$,
but we insert the bigon in the face also bounded by $p_{2}$ and $q_{n}$.
Propagation and absorbtion of $p_{1}$ works as in the even case, see Figure 11
for an illustration. This concludes the proof in the periodic case. In the
twisted case, we apply the same sequence of moves in the fundamental domain.
Instead of inserting $r_{1}$, we simultaneously insert $r_{1+m\gamma_{z}}$ for
all $m\in{\mathbb{Z}}$. We proceed analogously in Step 3, where we contract
pairs of $p_{1+m\gamma_{z}}$ for all $m\in{\mathbb{Z}}$. ∎

Since the weights of the two edges of a bigon sum to zero, the weight of any
dimer cover that uses one of the edges of the bigon is canceled by the weight
of the dimer cover that uses the other edge. Therefore the dimer Hamiltonians
are unchanged on inserting a bigon. Since the dimer Hamiltonians are also
preserved by the elementary transformations ([GK13]*Theorem 4.7), we obtain:

######  Corollary 8.3.

The dimer Hamiltonians are invariant under cross-ratio dynamics.

$q_{2}$$q_{2}$$p_{2}$$q_{1}$$p_{1}$$p_{1}$$r_{1}$$p_{n}$$p_{n}$$q_{n}$

$\dashrightarrow$
$q_{2}$$q_{2}$$p_{2}$$q_{1}$$p_{1}$$r_{1}$$p_{n}$$p_{n}$$q_{n}$$r_{n}$$r_{n}$$\rightarrow$
$q_{2}$$q_{2}$$p_{2}$$q_{1}$$r_{1}$$r_{1}$$p_{1}$$r_{n}$$r_{n}$$q_{n}$$\rightarrow$
$q_{2}$$q_{2}$$p_{2}$$q_{1}$$r_{1}$$r_{1}$$p_{1}$$p_{n}$$p_{n}$$q_{n}$

Figure 11: Key steps in cross-ratio dynamics via local moves for $n$ odd.

## Acknowledgements

TG thanks Nick Ovenhouse for discussions about networks in a cylinder. SR
thanks Ivan Izmestiev for discussions on cross-ratio dynamics during a visit
at TU Wien. NA was supported by the Deutsche Forschungsgemeinschaft (DFG)
Collaborative Research Center TRR 109 “Discretization in Geometry and
Dynamics”. SR is partially supported by the Agence Nationale de la Recherche,
Grant Number ANR-18-CE40-0033 (ANR DIMERS) and by the CNRS grant Tremplin@INP.

##  Appendix A Schur complement

Suppose $M=\begin{bmatrix}A&B\\\ C&D\end{bmatrix}$ is a $(p+q)\times(r+q)$
block matrix with $D$ an invertible $q\times q$ matrix. Let $V:=\text{coker
}M$. Let $e_{i}$ denote the $i$th basis column vector, and let $v_{i}\in V$
denote the image of $e_{i}$ under the cokernel map. Let $M/D:=A-BD^{-1}C$
denote the Schur complement.

######  Theorem A.1 (Schur determinant formula [Schur]).

If $M$ is a square matrix ($p=r$), then we have

$\text{det}(M)=\text{det}(D)\text{det}(M/D).$

######  Theorem A.2.

$\text{coker }M/D\cong\text{coker }M$, and under this identification the
cokernel map of $M/D$ is $e_{i}\mapsto v_{i}$ for $i=1,\dots,p$.

###### Proof.

After a change of basis, $M$ takes the block diagonal form
$\begin{bmatrix}M/D&0\\\ 0&D\end{bmatrix}$:

$M=\begin{bmatrix}I&BD^{-1}\\\ 0&I\end{bmatrix}\begin{bmatrix}M/D&0\\\
0&D\end{bmatrix}\begin{bmatrix}I&0\\\ D^{-1}C&I\end{bmatrix}.$

Since $D$ is invertible, we have $\text{coker }D=0$. Therefore we have
$\text{coker }M\cong\text{coker }M/D\oplus\text{coker }D\cong\text{coker
}M/D$. Under the change of basis $\begin{bmatrix}I&BD^{-1}\\\
0&I\end{bmatrix}$ of ${\mathbb{C}}^{p+q}$, we have

$e_{i}\mapsto\begin{bmatrix}I&BD^{-1}\\\
0&I\end{bmatrix}^{-1}e_{i}=e_{i}\text{ for }i=1,2,\dots,p,$

from which we see that the cokernel map of $M/D$, when we identify
$\text{coker }M/D$ with $\text{coker }M=V$, is given by $e_{i}\mapsto v_{i}$.
∎

## References

Technische Universität Berlin, Institute of Mathematics, Strasse des 17. Juni
136, 10623 Berlin, Germany

E-mail address: affolter at math.tu-berlin.de

University of Michigan, Department of Mathematics, 2844 East Hall, 530 Church
Street, Ann Arbor, MI 48109-1043, USA

E-mail address: georgete at umich.edu

Université Paris-Saclay, CNRS, CEA, Institut de physique théorique, 91191 Gif-
sur-Yvette, France

E-mail address: sanjay.ramassamy at ipht.fr