Functional Relations on Anisotropic Potts Models

Functional Relations on Anisotropic Potts Models:
from Biggs Formula to the Tetrahedron Equation

Boris BYCHKOV ab, Anton KAZAKOV abc and Dmitry TALALAEV abc

B. Bychkov, A. Kazakov and D. Talalaev

a) Faculty of Mathematics, National Research University Higher School of
Economics,
a) Usacheva 6, 119048, Moscow, Russia<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
b) Centre of Integrable Systems, P.G. Demidov Yaroslavl State University,
b) Sovetskaya 14, 150003, Yaroslavl, Russia

c) Faculty of Mechanics and Mathematics, Moscow State University, 119991
Moscow, Russia

Received July 06, 2020, in final form March 26, 2021; Published online April
07, 2021

We explore several types of functional relations on the family of multivariate
Tutte polynomials: the Biggs formula and the star-triangle ($Y-\Delta$)
transformation at the critical point $n=2$. We deduce the theorem of
Matiyasevich and its inverse from the Biggs formula, and we apply this
relation to construct the recursion on the parameter $n$. We provide two
different proofs of the Zamolodchikov tetrahedron equation satisfied by the
star-triangle transformation in the case of $n=2$ multivariate Tutte
polynomial, we extend the latter to the case of valency 2 points and show that
the Biggs formula and the star-triangle transformation commute.

tetrahedron equation; local Yang–Baxter equation; Biggs formula; Potts model;
Ising model

82B20; 16T25; 05C31

##  1 Introduction

The theory of polynomial invariants of graphs in its current state uses many
methods and tools of integrable statistical mechanics. This phenomenon
demonstrates the inherent intrusion of mathematical physics methods into
topology and combinatorics. In this paper, the main subject of research is
functional relations in the family of polynomial invariants for framed graphs,
in particular for multivariate Tutte polynomials [22], their specializations
for Potts models, multivariate chromatic and flow polynomials.

The flow generating function is closely related to the problems of electrical
networks on a graph over a finite field. Each flow defines a discrete harmonic
function, and non-zero flows can be interpreted as harmonic functions with a
completely non-zero gradient. Specifically, we discuss the full flow
polynomial which is a linearization of the flow polynomial and, in particular,
corresponds to the point of the compactification of the parameter space for
the Biggs model.

One of the central tools of the paper is the Biggs formula (Lemma 2.13), which
connects $n$-Potts models for different parameter values as a convolution with
some weight over all edge subgraphs. In particular, we offer a new proof of
the theorem of Matiyasevich 2.18 about the connection of a flow and chromatic
polynomial, as a special case of the Biggs formula. This interpretation allows
us to construct an inverse statement of the theorem of Matiyasevich. Moreover,
using the connection between the flow and the complete flow polynomial, we
obtain a shift of parameters in the Potts models (Theorem 2.25).

The fundamental type of correspondences on the space of the aforementioned
invariants is the star-triangle type relations (also known as “wye-delta”
relations) and the associated deletion-contraction relations. In a sense, the
kinship between these relations is analogous to the role of the tetrahedron
equation in the local Yang–Baxter equation. Despite the fact that the
invariance of the Ising model with respect to the star-triangle transformation
is very well known [2], we have not found in the literature a full proof of
the fact that the action of this transformation on the weights of an
anisotropic system is a solution of the tetrahedron equation that corresponds
to the orthogonal solution of the local Yang–Baxter equation: Theorem 4.1
(parts of this statement were mentioned in [15, 17, 21]). We offer here two
new proofs of this fact. We find them instructive due to their anticipated
relation to the theory of positive orthogonal grassmannians [14].

The identification of the Potts model and the multivariate Tutte polynomial
allows us to assert the existence of a critical point for the parameter $n$ in
the family of Tutte polynomials. Namely, for $n=2$, this model has a groupoid
symmetry generated by a family of transformations defined by the trigonometric
solution of the Zamolodchikov tetrahedron equation. We extend the star-
triangle transformation for the graphs of lower valency in Section 5. In this
way, we obtain a $14$-term correspondence. This extension commute with the
Biggs formula. We should mention the relation of this subject with the theory
of cluster algebras. We suppose that the multivariate Tutte polynomial on
standard graphs at the critical point $n=2$ corresponds to the orthogonal
version of the Lusztig variety [4] in the case of the unipotent group and the
electrical variety [12] for the symplectic group.

###  1.1 Organization of the paper

In Section 2, we concentrate our attention on the Biggs formalism in the Ising
and Potts type models. We define the main recurrence relations and also
identify the Tutte polynomial with the Potts model. Then, we apply the Biggs
formula to the proof of theorem of Matiyasevich and propose its inverse
version. We examine in details the recursion of the Potts model with respect
to the parameter $n$.

In Section 3, we show that, if $n=2$, then the Potts model is invariant with
respect to the star-triangle transformation given by the orthogonal solution
for the local Yang–Baxter equation and the corresponding solution for the
Zamolodchikov tetrahedron equation. In Section 4 we provide two different
proofs for this fact. Both of them are interesting in the context of cluster
variables on the space of Ising models. The first proof operates with the
space of boundary measurement matrices and the second with the matrix of
boundary partition function.

In Section 5, we show that the Biggs formula considered as a correspondence on
the set of multivariate Tutte polynomials commutes with the star-triangle
transformation.

##  2 Biggs interaction models

###  2.1 $\boldsymbol{n}$-Potts models and Tutte polynomial

We define the anisotropic Biggs model (interaction model) on an undirected
graph $G$ with the set of edges $E$ and the set of vertices $V$ (a graph can
have multiple edges and loops) as follows:

  * •

a state $\sigma$ is a map $\sigma\colon V\rightarrow R$, where $R$ is a
commutative ring with the unit,

  * •

the weight of the state $\sigma$ is defined by the formula

$\displaystyle W_{G}(\sigma):=\prod\limits_{e\in E}i_{e}(\delta(e)),$

where $\delta(e)=\sigma(v)-\sigma(w)$; the edge $e$ connects the vertices $v$
and $w$, the functions $i_{e}\colon R\rightarrow\mathbb{C}$ are even: $\forall
b\in R\colon i_{e}(b)=i_{e}(-b)$,

  * •

the partition function $Z(G)$ of a model is the following sum

$\displaystyle Z(G)=\sum_{\sigma}W_{G}(\sigma),$

where the summation is taken over all possible states $\sigma$.

Let us consider the most simple Biggs interaction models:

######  Definition 2.1.

If $R\cong\mathbb{Z}_{n}$ and functions $i_{e}$ given as

$\displaystyle\begin{cases}i_{e}(0)=\alpha_{e},\\\ i_{e}(a)=\beta_{e},&\forall
a\neq 0\in R,\end{cases}$

we call such model the anisotropic $n$-Potts model with the set of parameters
$\alpha_{e}$ and $\beta_{e}$ and we denote it by $M(G;i_{e})$.

In addition, if the maps $i_{e}=i$ do not depend on edges, then we call such
model the isotropic $n$-Potts model (or just $n$-Potts model) with parameters
$\alpha$ and $\beta$. We denote it by $M(G;i)$, also we use the notation
$M(G;\alpha,\beta)$.

######  Remark 2.2.

In the case $R\cong\mathbb{Z}_{2}$, $i(0)=\exp\big{(}\frac{J}{kT}\big{)}$ and
$i(1)=\exp\big{(}{-}\frac{J}{kT}\big{)}$ this model can be identified with the
classic isotropic Ising model [2]. Therefore we will call any anisotropic or
isotropic $2$-Potts model just Ising model.

######  Definition 2.3.

Consider an anisotropic $n$-Potts model $M(G;i_{e})$, we denote its partition
function as $Z_{n}(G)$. In addition, if $n=2$, we omit index $2$ and write
just $Z(G)$.

######  Remark 2.4.

For the empty graph, we define the partition function of any $n$-Potts model
to be equal to $1$, and for a disjoint set of $m$ points to be equal to
$n^{m}$.

Now we will consider the combinatorial properties of the anisotropic $n$-Potts
models (compare with [3, Theorem 3.2]):

######  Theorem 2.5.

Consider an anisotropic $n$-Potts model $M(G;i)$ and its partition function
$Z_{n}(G)$.

  * •

Let graph $G$ be the disjoint union of graphs $G_{1}$ and $G_{2}$, then

$\displaystyle Z_{n}(G)=Z_{n}(G_{1})Z_{n}(G_{2}).$

Figure 1: The joining of two graphs by the vertex $v$.

  * •

Let graph $G$ be the joining of graphs $G_{1}$ and $G_{2}$ by the vertex $v$,
then

$\displaystyle nZ_{n}(G)=Z_{n}(G_{1})Z_{n}(G_{2}).$

  * •

Consider a graph $G$ and its edge $e$, where $e$ is neither a bridge nor a
loop. Consider the graph $G/e$ obtained by contraction of $e$, and the graph
$G\backslash e$ obtained by deletion of $e$. Then the following formula holds

$\displaystyle
Z_{n}(G)=(\alpha_{e}-\beta_{e})Z_{n}(G/e)+\beta_{e}Z_{n}(G\backslash e).$

###### Proof.

1\. The statement directly follows from Definition 2.1.

2\. Let us rewrite the partition function $Z_{n}(G)$:

$\displaystyle
Z_{n}(G)=\sum\limits_{k\in\\{0,\ldots,n-1\\}}\sum\limits_{\sigma\colon\sigma(v)=k}W_{G}(\sigma).$

Notice that $i(\sigma(v)-\sigma(w))=i(\sigma(v)+1-\sigma(w)-1)$, therefore for
any $i\neq j$ we have the following identity

$\displaystyle\sum\limits_{\sigma\colon\sigma(v)=i}W_{G}(\sigma)=\sum\limits_{\sigma\colon\sigma(v)=j}W_{G}(\sigma).$

Hence we obtain

$\displaystyle
Z_{n}(G)=n\sum\limits_{\sigma\colon\sigma(v)=i}W_{G}(\sigma),\qquad\forall\,i\in\\{0,\ldots,n-1\\}.$

Let us introduce the partial partition functions
$X_{k}:=\\!\\!\sum\limits_{\sigma\colon\sigma(v)=k}\\!\\!W_{G_{1}}(\sigma)$
and
$Y_{k}:=\\!\\!\sum\limits_{\sigma\colon\sigma(v)=k}\\!\\!W_{G_{2}}(\sigma)$,
then we could rewrite

$\displaystyle
Z_{n}(G_{1})Z_{n}(G_{2})=\bigg{(}\sum_{k}\sum_{\sigma\colon\sigma(v)=k}W_{G_{1}}(\sigma)\bigg{)}\bigg{(}\sum_{k}\sum_{\sigma\colon\sigma(v)=k}W_{G_{2}}(\sigma)\bigg{)}$
$\displaystyle\hphantom{Z_{n}(G_{1})Z_{n}(G_{2})}{}=(X_{0}+X_{1}+\dots+X_{n-1})(Y_{0}+Y_{1}+\dots
Y_{n-1})=n^{2}X_{0}Y_{0}$
$\displaystyle\hphantom{Z_{n}(G_{1})Z_{n}(G_{2})}{}=n(X_{0}Y_{0}+X_{1}Y_{1}+\dots+X_{n-1}Y_{n-1})=n\sum_{k}\sum_{\sigma\colon\sigma(v)=k}W_{G}(\sigma)=nZ_{n}(G).$

3\. Let the edge $e$ is neither a bridge nor a loop and denote by $X$ the
income in the partition function of all states such that the values of the
ends of $e$ coincide and by $Y$ another part of the partition function (that
of the distinct values of the ends of $e$), then

$\displaystyle Z_{n}(G)=\alpha_{e}X+\beta_{e}Y,\qquad Z_{n}(G\backslash
e)=X+Y,\qquad Z_{n}(G/e)=X$

and we obtain the statement. ∎

Now let us recall the definition of the Tutte polynomial of a graph $G$.

######  Definition 2.6.

Let us define the Tutte polynomial $T_{G}(x,y)$ by the deletion-contraction
recurrence relation:

  1. 1.

If an edge $e$ is neither a bridge nor a loop, then $T_{G}(x,y)=T_{G\backslash
e}(x,y)+T_{G/e}(x,y)$.

  2. 2.

If the graph $G$ consists of $i$ bridges and $j$ loops, then
$T_{G}(x,y)=x^{i}y^{j}$.

######  Theorem 2.7 ([9, 23]).

Let $F(G)$ be a function of a graph $G$ satisfuing the following conditions:

  * •

$F(G)=1$, if $G$ consists of only one vertex.

  * •

$F(G)=aF(G\backslash e)+bF(G/e)$, if an edge $e$ is not a bridge neither a
loop.

  * •

$F(G)=F(G_{1})F(G_{2})$, if either $G=G_{1}\sqcup G_{2}$ or the intersection
$G_{1}\cap G_{2}$ consists of only one vertex.

Then

$\displaystyle
F(G)=a^{c(G)}b^{r(G)}T_{G}\bigg{(}\frac{F(K_{2})}{b},\frac{F(L)}{a}\bigg{)},$

where $K_{2}$ is a complete graph on two vertices, $L$ is a loop,
$r(G)=v(G)-k(G)$ is a rank of $G$ and $c(G)=e(G)-r(G)$ is a corank. Here and
below $e(G)$ is the number of edges in the graph $G$.

Now we are ready to connect the partition function $Z_{n}(G)$ of the isotropic
$n$-Potts model $M(G;\alpha,\beta)$ with the Tutte polynomial $T_{G}(x,y)$ of
the same graph $G$ using a well-known trick (for instance see [3]). Let us
consider the weighted partition function

$\displaystyle\frac{Z_{n}(G)}{n^{k(G)}},$

where $k(G)$ is the number of connected components in the graph $G$. It is
easy to verify that the weighted partition function
$\frac{Z_{n}(G)}{n^{k(G)}}$ satisfies Theorem 2.7, therefore the following
theorem holds:

######  Theorem 2.8 (Theorem 3.2 [3]).

The partition function $Z_{n}(G)$ of the $n$-Potts model $M(G;\alpha,\beta)$
coincides with the Tutte polynomial of a graph $G$ up to a multiplicative
factor

$\displaystyle
Z_{n}(G)=n^{k(G)}\beta^{c(G)}(\alpha-\beta)^{r(G)}T_{G}\bigg{(}\frac{\alpha+(n-1)\beta}{\alpha-\beta},\frac{\alpha}{\beta}\bigg{)}.$

######  Example 2.9 (the bad coloring polynomial [9]).

Consider a graph $G$ and all possible colorings of $V(G)$ in $n$ colors.
Define the bad coloring polynomial as

$\displaystyle B_{G}(n,t)=\sum\limits_{j}b_{j}(G,n)t^{j},$

here $b_{j}(G,n)$ is the number of colorings such that each of them has
exactly $j$ bad edges (we call an edge “bad” if its ends have the same
colors). So, easy to see that $B_{G}(n,t)=Z_{n}(G)$, here $Z_{n}(G)$ is the
partition function of the $n$-Potts model $M(G;t,1)$. Hence, using the Theorem
2.8 we immediately obtain

$\displaystyle
B_{G}(n,t+1)=n^{k(G)}t^{r(G)}T_{G}\bigg{(}\frac{t+n}{t},t+1\bigg{)}.$

###  2.2 $\boldsymbol{n}$-Potts models and the theorem of Matiyasevich

The connection between the $n$-Potts models and Tutte polynomials allows us to
give a simple proof of the theorem of Matiyasevich about the chromatic and
flow polynomials, but at first we introduce a few definitions.

######  Definition 2.10.

A graph $A$ is called a spanning subgraph of a graph $G$, if graphs $G$ and
$A$ share the same set of vertices: $V(G)=V(A)$, and the set of edges $E(A)$
is the subset of the set of edges $E(G)$.

######  Definition 2.11.

A graph $A$ is called an edge induced subgraph (Figure 2) of a graph $G$, if
$A$ is induced by a subset of the set $E(G)$. Every edge induced subgraph $A$
of a graph $G$ could be completed to the spanning subgraph $A^{\prime}$ by
adding all the vertices of $G$ which is not contained in the subgraph $A$.

######  Definition 2.12.

For a $n$-Potts model $M(G;i)$ we introduce the normalized partition function
as follows

$\displaystyle\widetilde{Z}_{n}(G)=\frac{Z_{n}(G)}{n^{v(G)}}.$

Figure 2: Edge induced and spanning subgraphs.

We start with the following lemma, which is a generalization of the high
temperature formula for the Ising model:

######  Lemma 2.13 (Biggs formula [5]).

Let us consider two $n$-Potts models $M_{1}(G;i_{1})$ with parameters
$\alpha_{1}$, $\beta_{1}$ and $M_{2}(G;i_{2})$ with parameters $\alpha_{2}$,
$\beta_{2}$. Then the normalized partition function $Z^{1}_{n}(G)$ of the
first model could be expressed in terms of the normalized partition functions
of the models of all edge induced subgraphs of the second model:

$\displaystyle\widetilde{Z}^{1}_{n}(G)=q^{e(G)}\sum\limits_{A\subseteq
G}\left(\frac{p}{q}\right)^{e(A)}\widetilde{Z}^{2}_{n}(A),$

where $p=\frac{\alpha_{1}-\beta_{1}}{\alpha_{2}-\beta_{2}}$, and
$q=\frac{\alpha_{2}\beta_{1}-\alpha_{1}\beta_{2}}{\alpha_{2}-\beta_{2}}$
$\big{(}$we assume that $\widetilde{Z}^{i}_{n}(\varnothing)=1\big{)}$.

###### Proof.

Let us notice that $i_{1}=p\cdot i_{2}+q$, therefore

$\displaystyle\widetilde{Z}^{1}_{n}(G)=\sum\limits_{\sigma\colon
V(G)\rightarrow\mathbb{Z}_{n}}\prod_{e}i_{1}(\delta(e))=\sum\limits_{\sigma\colon
V(G)\rightarrow\mathbb{Z}_{n}}\prod_{e}(pi_{2}(\delta(e))+q)$
$\displaystyle\hphantom{\widetilde{Z}^{1}_{n}(G)}{}=\sum\limits_{\sigma\colon
V(G)\rightarrow\mathbb{Z}_{n}}\sum\limits_{A\subseteq
G}p^{e(A)}q^{e(G)-e(A)}\prod_{e\in E(A)}i_{2}(\delta(e)).$

In order to complete the proof we consider the following term for a fixed $A$:

$\displaystyle\sum\limits_{\sigma\colon
V(G)\rightarrow\mathbb{Z}_{n}}\\!\\!\\!p^{e(A)}q^{e(G)-e(A)}\\!\\!\\!\\!\prod_{e\in
E(A)}\\!\\!i_{2}(\delta(e))=q^{e(G)}\bigg{(}\frac{p}{q}\bigg{)}^{e(A)}\\!\\!\sum\limits_{\sigma\colon
V(G)\rightarrow\mathbb{Z}_{n}}\prod_{e\in E(A)}i_{2}(\delta(e))$
$\displaystyle\hphantom{\sum\limits_{\sigma\colon
V(G)\rightarrow\mathbb{Z}_{n}}\\!\\!\\!p^{e(A)}q^{e(G)-e(A)}\\!\\!\\!\\!\prod_{e\in
E(A)}\\!\\!i_{2}(\delta(e))}{}=q^{e(G)}\bigg{(}\frac{p}{q}\bigg{)}^{e(A)}n^{v(G)-v(A)}\sum\limits_{\sigma\colon
V(A)\rightarrow\mathbb{Z}_{n}}\prod_{e\in E(A)}i_{2}(\delta(e))$
$\displaystyle\hphantom{\sum\limits_{\sigma\colon
V(G)\rightarrow\mathbb{Z}_{n}}\\!\\!\\!p^{e(A)}q^{e(G)-e(A)}\\!\\!\\!\\!\prod_{e\in
E(A)}\\!\\!i_{2}(\delta(e))}{}=n^{v(G)}q^{e(G)}\sum\limits_{A\subseteq
G}\bigg{(}\frac{p}{q}\bigg{)}^{e(A)}\widetilde{Z}^{2}_{n}(A).$ ∎

######  Proposition 2.14.

Consider two anisotropic $n$-Potts models $M_{1}\big{(}G;i^{1}_{e}\big{)}$ and
$M_{2}\big{(}G;i^{2}_{e}\big{)}$. In the same fashion we can obtain

$\displaystyle\widetilde{Z}^{1}_{n}(G)=\prod_{e\in
G}q_{e}\sum\limits_{A\subseteq G}\prod_{e\in
A}\frac{p_{e}}{q_{e}}\widetilde{Z}^{2}_{n}(A),$ (2.1)

here
$p_{e}=\frac{\alpha^{1}_{e}-\beta^{1}_{e}}{\alpha^{2}_{e}-\beta^{2}_{e}}$, and
$q_{e}=\frac{\alpha^{2}_{e}\beta^{1}_{e}-\alpha^{1}_{e}\beta^{2}_{e}}{\alpha^{2}_{e}-\beta^{2}_{e}}$.

We consider further the chromatic and flow polynomials, first of all remain
some well-known definitions.

######  Definition 2.15.

A coloring of the set of vertices $V(G)$ is said to be proper if the ends of
each edge have different colors.

######  Definition 2.16.

Let $G$ be a graph with the edge set $E(G)$ and the vertex set $V(G)$, let us
choose a fixed edge orientation on $G$. Then, a function $f\colon
E\rightarrow\mathbb{Z}_{n}$ is called a nowhere-zero $n$-flow if the following
conditions hold:

  * •

$\forall e\in E(G)\colon f(e)\neq 0$,

  * •

$\forall v\in V(G)\colon\sum\limits_{e\in M^{+}(v)}f(e)=\sum\limits_{e\in
M^{-}(v)}f(e)$, where $M^{+}(v)$ (respectively $M^{-}(v)$) is the set of edges
each of them is directed to (respectively from) $v$.

Next, we formulate one of the classic results of graph theory which can be
found for instance in [9]:

######  Theorem 2.17.

The number of proper colorings of a graph $G$ in $n$ colors is the following
polynomial $($called chromatic polynomial$)$ in the variable $n$:

$\displaystyle\chi_{G}(n)=(-1)^{v(G)-k(G)}n^{k(G)}T_{G}(1-n,0).$

The number of nowhere-zero $n$-flows of a graph $G$ is independent on the
choice of orientation and is obtained by the following polynomial $($called
flow polynomial$)$ in the variable $n$:

$\displaystyle C_{G}(n)=(-1)^{e(G)+v(G)+k(G)}T_{G}(0,1-n).$

Now we are ready to formulate and prove the theorem of Matiyasevich:

######  Theorem 2.18 (Matiyasevich [20]).

Let us consider a graph $G$, its chromatic polynomial $\chi_{G}$ and its flow
polynomial $C_{G}$, then

$\displaystyle\chi_{G}(n)=\frac{(n-1)^{e(G)}}{n^{e(G)-v(G)}}\sum\limits_{A\subseteq
G}\frac{C_{A}(n)}{(1-n)^{e(A)}},$

where the summation goes through all spanning subgraphs $A$.

###### Proof.

Let us consider two $n$-Potts models with the special parameters: the model
$M_{1}(G;i_{1})$ with the parameters $\alpha_{1}=0$, $\beta_{1}=1$ and the
model $M_{2}(G;i_{2})$ with the parameters $\alpha_{2}=1-n$, $\beta_{2}=1$. By
Theorem 2.8 we could express the partition function of the first model in
terms of the chromatic polynomial

$\displaystyle\chi_{G}(n)=(-1)^{v(G)-k(G)}n^{k(G)}T_{G}(1-n,0)=\frac{(-1)^{v(G)-k(G)-r(G)}n^{k(G)}Z^{1}_{n}(G)}{n^{k(G)}}$
$\displaystyle\hphantom{\chi_{G}(n)}=(-1)^{v(G)-k(G)-r(G)}n^{v(G)}\widetilde{Z}^{1}_{n}(G)=n^{v(G)}\widetilde{Z}^{1}_{n}(G).$

So we have

$\displaystyle\widetilde{Z}^{1}_{n}(G)=\frac{\chi_{G}(n)}{n^{v(G)}}.$

Analogously, we express the partition function of the second model in terms of
the flow polynomial

$\displaystyle
C_{G}(n)=(-1)^{e(G)+v(G)+k(G)}T_{G}(0,1-n)=\frac{(-1)^{e(G)+v(G)+k(G)-r(G)}Z^{2}_{n}(G)}{n^{k(G)}n^{v(G)-k(G)}}$
$\displaystyle\hphantom{C_{G}(n)}{}=(-1)^{e(G)}\widetilde{Z}^{2}_{n}(G).$
(2.2)

So we have

$\displaystyle\widetilde{Z}^{2}_{n}(G)=(-1)^{e(G)}C_{G}(n).$ (2.3)

Then by Lemma 2.13 after the substitutions (2.2) and (2.3) we obtain

$\displaystyle\frac{\chi_{G}(n)}{n^{v(G)}}=\frac{(n-1)^{e(G)}}{n^{e(G)}}\sum_{A^{\prime}\subseteq
G}\frac{C_{A^{\prime}}(n)}{(1-n)^{e(A^{\prime})}},$

where the summation goes through all edge induced subgraphs $A^{\prime}$.

We finish the proof by noticing that the edge induced subgraph differs from
the spanning subgraph by the set of isolated vertices. Therefore we can
complete each edge induced subgraph to its corresponding spanning subgraph and
then replace the summation over all edge induced subgraph by the summation
over all spanning subgraph, because the value of the each flow polynomial
$C_{A^{\prime}}$ remains the same and finally we obtain

$\displaystyle\chi_{G}(n)=\frac{(n-1)^{e(G)}}{n^{e(G)-v(G)}}\sum_{A\subseteq
G}\frac{C_{A}(n)}{(1-n)^{e(A)}}.$ ∎

Note that we could produce series of statements that look like Theorem 2.18:

######  Theorem 2.19.

Let us consider a graph $G$, then we can obtain the following formulas

$\displaystyle
n^{k(G)}\beta_{1}^{c(G)}(\alpha_{1}-\beta_{1})^{r(G)}T_{G}\bigg{(}\frac{\alpha_{1}+(n-1)\beta_{1}}{\alpha_{1}-\beta_{1}},\frac{\alpha_{1}}{\beta_{1}}\bigg{)}=q^{e(G)}\sum_{A\subseteq
G}\bigg{(}\frac{p}{q}\bigg{)}^{e(A)}\chi_{A}(n),$ (2.4)

where $p=-\alpha_{1}+\beta_{1}$, $q=\alpha_{1}$, and the summation (here and
below) goes through all spanning subgraphs $A$,

$\displaystyle C_{G}(n)=(n-1)^{e(G)}\sum_{A\subseteq
G}\frac{n^{e(A)-v(G)}}{(1-n)^{e(A)}}\chi_{A}(n),$ (2.5) $\displaystyle
n^{k(G)-v(G)}\beta_{1}^{c(G)}(\alpha_{1}-\beta_{1})^{r(G)}T_{G}\bigg{(}\frac{\alpha_{1}+(n-1)\beta_{1}}{\alpha_{1}-\beta_{1}},\frac{\alpha_{1}}{\beta_{1}}\bigg{)}$
$\displaystyle\qquad{}=q_{1}^{e(G)}\sum_{A}\bigg{(}\frac{p_{1}}{q_{1}}\bigg{)}^{e(A)}(-1)^{e(A)}C_{A}(n),$
(2.6)

where $p_{1}=\frac{\beta_{1}-\alpha_{1}}{n}$,
$q_{1}=\frac{\alpha_{1}-(1-n)\beta_{1}}{n}$,

$\displaystyle(-1)^{e(G)}C_{G}(n)$
$\displaystyle\qquad=q_{2}^{e(G)}\sum_{A}\bigg{(}\frac{p_{2}}{q_{2}}\bigg{)}^{e(A)}n^{k(A)-v(G)}\beta_{1}^{c(A)}(\alpha_{1}-\beta_{1})^{r(A)}T_{A}\bigg{(}\frac{\alpha_{1}+(n-1)\beta_{1}}{\alpha_{1}-\beta_{1}},\frac{\alpha_{1}}{\beta_{1}}\bigg{)},$
(2.7)

where $p_{2}=\frac{n}{\beta_{1}-\alpha_{1}}$ and
$q_{2}=\frac{\alpha_{1}-(1-n)\beta_{1}}{\alpha_{1}-\beta_{1}}$.

###### Proof.

Let us consider two $n$-Potts models:

  * •

Models $M_{1}(G;\alpha_{1},\beta_{1})$ and $M_{2}(G;0,1)$ for the proof of the
formula (2.4).

  * •

The specification of the first case: $M_{1}(G;1-n,1)$ and the same
$M_{2}(G;0,1)$ for the proof of the formula (2.5).

  * •

Models $M_{1}(G;1-n,1)$ and $M_{2}(G;\alpha_{1},\beta_{1})$ with the
parameters $\alpha_{1}$, $\beta_{1}$ for the proof of the formula (2.6).

  * •

And finally, models $M_{1}(G;\alpha_{1},\beta_{1})$ and $M_{2}(G;1-n,1)$ for
the proof of formula (2.7).

Now it is left to repeat step by step the proof of Theorem 2.18 for these two
models. ∎

######  Remark 2.20.

We notice that the formula (2.5) naturally can be considered as “inversion” of
Theorem 2.18.

###  2.3 Shifting the order in the Potts models

Biggs Lemma 2.13 allows us to relate the values of the partition functions of
the $n$-Potts models with fixed $n$, but different values of parameters
$\alpha$ and $\beta$. The goal of the current subsection is to present a
method for connecting partition functions of the $n$-Potts models for
different $n$. We will call it shifting order formulas.

The first method is based on the multiplicativity property of the complete
flow polynomial.

######  Definition 2.21.

Let $G$ be a graph with the edge set $E(G)$ and the vertex set $V(G)$, let us
chose a fixed edge orientation on $G$. Then, a function $f\colon
E\rightarrow\mathbb{Z}_{n}$ is called an $n$-flow if the following condition
holds

$\displaystyle\forall v\in V(G)\colon\ \sum\limits_{e\in
M^{+}(v)}f(e)=\sum\limits_{e\in M^{-}(v)}f(e),$

here again $M^{+}(v)$ (respectively $M^{-}(v)$) is the set of edges each of
them is directed to (respectively from) $v$.

Let us formulate a few well known results concerning a flow polynomial and a
number of all $n$-flows. The proofs could be found for example in [22].

######  Proposition 2.22.

Denote the number of all $n$-flows on a graph $G$ by ${FC}_{G}(n)$, then
${FC}_{G}(n)$ is independent of the choice of an orientation and the following
identity holds

$\displaystyle{FC}_{G}(n)=\sum\limits_{A\subseteq G}C_{A}(n),$

where the summation goes through all spanning subgraphs $A$ of the graph $G$.

######  Proposition 2.23.

The number of all $n$-flows on a graph is the following polynomial $($called
complete flow polynomial$)$

$\displaystyle{FC}_{G}(n)=n^{e(G)-v(G)+k(G)},$

where $e(G)$, $v(G)$, $k(G)$ are numbers of edges, vertices and connected
components in the graph $G$ correspondingly.

######  Proposition 2.24.

The flow polynomial $C_{G}(n)$ of a graph $G$ could be expressed in terms of
the complete flow polynomials of its spanning subgraphs by the following
identity:

$\displaystyle C_{G}(n)=\sum_{A\subseteq G}(-1)^{e(G)-e(A)}{FC}_{A}(n).$

The complete flow polynomial ${FC}_{G}(n)$ is a multiplicative invariant:
${FC}_{G}(n_{1}n_{2})={FC}_{G}(n_{1})\allowbreak\times{FC}_{G}(n_{2})$,
therefore we are ready to formulate the following theorem:

######  Theorem 2.25.

The partition function $Z_{n_{1}n_{2}}(G)$ of the $n_{1}n_{2}$-Potts model
$M(G;\alpha_{1},\beta_{1})$ could be expressed in terms of the partition
functions $Z_{n_{1}}(A)$ and $Z_{n_{2}}(A)$ of the $n_{1}$-Potts model
$M_{1}(A;\alpha_{1},\beta_{1})$ and $n_{2}$-Potts model
$M_{2}(A;\alpha_{1},\beta_{1})$ of all spanning subgraphs $A$ of the graph $G$
correspondingly.

###### Proof.

Indeed, by Theorem 2.8 and the formula (2.6) we have

$\displaystyle
Z_{n_{1}n_{2}}(G)=\gamma_{G}T_{G}\bigg{(}\frac{\alpha_{1}+(n_{1}n_{2}-1)\beta_{1}}{\alpha_{1}-\beta_{1}},\frac{\alpha_{1}}{\beta_{1}}\bigg{)}=\sum_{A\subseteq
G}\lambda_{A}C_{A}(n_{1}n_{2}).$

From Proposition 2.24 we obtain

$\displaystyle\sum_{A\subseteq G}\lambda_{A}C_{A}(n_{1}n_{2})=\sum_{A\subseteq
G}\lambda_{A}\bigg{(}\sum_{A^{\prime}\subseteq
A}(-1)^{e(A)-e(A^{\prime})}{FC}_{A^{\prime}}(n_{1}n_{2})\bigg{)}$
$\displaystyle\hphantom{\sum_{A\subseteq
G}\lambda_{A}C_{A}(n_{1}n_{2})}{}=\sum_{A\subseteq
G}\omega_{A}{FC}_{A}(n_{1}n_{2})=\sum_{A\subseteq
G}\omega_{A}{FC}_{A}(n_{1}){FC}_{A}(n_{2}),$

notice that we used for the second resummations the following simple
observation: if $X$ is a spanning subgraph of $Y$, which is a spanning
subgraph of graph $Z$, so $X$ is a spanning subgraph of a graph $Z$. We omit
this remark below.

The Proposition 2.22 implies

$\displaystyle\sum_{A\subseteq
G}\omega_{A}{FC}_{A}(n_{1}){FC}_{A}(n_{2})=\sum_{A\subseteq
G}\omega_{A}\bigg{(}\sum_{A^{\prime}\subseteq
A}C_{A^{\prime}}(n_{1})\bigg{)}\bigg{(}\sum_{A^{\prime\prime}\subseteq
A}C_{A^{\prime\prime}}(n_{2})\bigg{)}$
$\displaystyle\hphantom{\sum_{A\subseteq
G}\omega_{A}{FC}_{A}(n_{1}){FC}_{A}(n_{2})}{}=\sum_{A^{\prime}\subseteq
G}\sum_{A^{\prime\prime}\subseteq
G}\mu_{A^{\prime}A^{\prime\prime}}C_{A^{\prime}}(n_{1})C_{A^{\prime\prime}}(n_{2}).$

Finally, with the help of formula (2.7) and Theorem 2.8 we obtain

$\displaystyle\sum_{A^{\prime}\subseteq G}\sum_{A^{\prime\prime}\subseteq
G}\mu_{A^{\prime}A^{\prime\prime}}\bigg{(}\sum_{B\subseteq
A^{\prime}}\delta_{B}Z_{n_{1}}(B)\bigg{)}\bigg{(}\sum_{C\subseteq
A^{\prime\prime}}\delta_{C}Z_{n_{2}}(C)\bigg{)}=\sum_{A^{\prime}\subseteq
G}\sum_{A^{\prime\prime}\subseteq
G}\eta_{A^{\prime}A^{\prime\prime}}Z_{n_{1}}(A^{\prime})Z_{n_{2}}(A^{\prime\prime}),$

where $\eta_{A^{\prime}A^{\prime\prime}}$ are some constants, appeared after
the resummations. ∎

######  Remark 2.26 (convolution formula [16]).

It seems extremely interesting and fruitful to compare Lemma 2.13 and Theorem
2.25 with the convolution formula

$\displaystyle T_{G}(x,y)=\sum_{A\subseteq E(G)}T_{G|A}(0,y)T_{G/A}(x,0),$

here the summation is over all possible subsets of $E(G)$, here $G|A$ is a
graph obtained by the restriction of $G$ on the edge subset $A$ and $G/A$ is a
graph obtained from $G$ by the contraction of all edges from $A$ (see [9] for
more details).

Our second method is based on the Tutte identity for the chromatic polynomial:

######  Theorem 2.27 ([9]).

Consider a graph $G$ with the set of edge $V(G)$ then the following formula
holds

$\displaystyle\chi_{G}(n_{1}+n_{2})=\sum_{B\subseteq
V(G)}\chi_{G|B}(n_{1})\chi_{G|B^{c}}(n_{2}),$

where $G|B$ $(G|B^{c})$ is the restriction of $G$ on the vertex subset $B\in
V(G)$ $(B^{c}\in V(G)$, where $B^{c}=V(G)\setminus B)$.

Using this fact we can formulate the following theorem:

######  Theorem 2.28.

The partition function $Z_{n_{1}+n_{2}}(G)$ of the $n_{1}+n_{2}$-Potts model
$M(G;\alpha_{1},\beta_{1})$ could be expressed in terms of the partition
functions $Z_{n_{1}}(A)$ and $Z_{n_{2}}(A)$ of the $n_{1}$-Potts model
$M_{1}(A;\alpha_{1},\beta_{1})$ and $n_{2}$-Potts model
$M_{2}(A;\alpha_{1},\beta_{1})$ of all spanning subgraphs $A$ of the graph $G$
correspondingly.

###### Proof.

The proof is very similar to the proof of Theorem 2.25. Again, from Theorem
2.8 and the formula (2.4) we have

$\displaystyle
Z_{n_{1}+n_{2}}(G)=\gamma_{G}T_{G}\bigg{(}\frac{\alpha_{1}+(n_{1}+n_{2}-1)\beta_{1}}{\alpha_{1}-\beta_{1}},\frac{\alpha_{1}}{\beta_{1}}\bigg{)}=\sum_{A\subseteq
G}\lambda_{A}\chi_{A}(n_{1}+n_{2}).$

From Theorem 2.27 we obtain

$\displaystyle\sum_{A\subseteq
G}\lambda_{A}\chi_{A}(n_{1}+n_{2})=\sum_{A\subseteq
G}\lambda_{A}\bigg{(}\sum_{B\subseteq
V(A)}\chi_{A|B}(n_{1})\chi_{A|B^{c}}(n_{2})\bigg{)}$
$\displaystyle\hphantom{\sum_{A\subseteq
G}\lambda_{A}\chi_{A}(n_{1}+n_{2})}{}=\sum_{A\subseteq
G}\lambda_{A}\Bigg{(}\sum_{B\subseteq V(A)}\\!\\!\bigg{(}\sum_{A_{1}\subseteq
A|B}\\!\\!\omega_{A_{1}}Z_{n_{1}}(A_{1})\bigg{)}\bigg{(}\sum_{A_{2}\subseteq
G|B^{c}}\\!\\!\omega_{A_{2}}Z_{n_{2}}(A_{2})\bigg{)}\Bigg{)}=$
$\displaystyle\hphantom{\sum_{A\subseteq
G}\lambda_{A}\chi_{A}(n_{1}+n_{2})}{}=\sum_{A\subseteq G}\sum_{B\subseteq
V(A)}\sum_{A_{1}\subseteq A|B}\sum_{A_{2}\subseteq
A|B^{c}}\mu_{A_{1}A_{2}}Z_{n_{1}}(A_{1})Z_{n_{2}}(A_{2}).$

Let us complete each subgraph $A_{1}$ (each $A_{2}$) to the corresponding
spanning subgraph of $G$ by adding isolating vertices

$\displaystyle\sum_{A\subseteq G}\sum_{B\subseteq V(A)}\sum_{A_{1}\subseteq
A|B}\sum_{A_{2}\subseteq
A|B^{c}}\mu_{A_{1}A_{2}}Z_{n_{1}}(A_{1})Z_{n_{2}}(A_{2})=\sum_{A^{\prime}\subseteq
G}\sum_{A^{\prime\prime}\subseteq
G}\eta_{A^{\prime}A^{\prime\prime}}Z_{n_{1}}(A^{\prime})Z_{n_{2}}(A^{\prime\prime}),$

where $\eta_{A^{\prime}A^{\prime\prime}}$ are again some constants, appeared
after the resummations. ∎

##  3 Star-triangle equation for Ising and Potts models

###  3.1 General properties

Let us rewrite the partition function of the anisotropic $n$-Potts models in
the so called Fortuin–Kasteleyn representation:

######  Proposition 3.1 (compare with the formula $(2.7)$ from [22]).

Consider the anisotropic $n$-Potts model $M(G;i_{e})$, then its partition
function could be expressed as follows

$\displaystyle Z_{n}(G)=\sum_{\sigma}\prod_{e\in
E}(\beta_{e}+(\alpha_{e}-\beta_{e})\delta(\sigma_{e}))=\prod_{e\in
E}\beta_{e}\sum_{\sigma}\prod_{e\in E}(1+(t_{e}-1)\delta(\sigma_{e})),$ (3.1)

where $\delta(\sigma_{e})$ is a value of standard Kronecker delta function of
the values of $\sigma$ on the boundary vertices of the edge $e$ and
$t_{e}=\frac{\alpha_{e}}{\beta_{e}}$ is a reduced weight of the edge $e$.

###### Proof.

Indeed, it is easy to see that if $\sigma(v)=\sigma(w)$:

$\displaystyle
i_{e}(\delta(e))=i_{e}(\sigma(v)-\sigma(w))=\alpha_{e}=\beta_{e}+(\alpha_{e}-\beta_{e})\delta(\sigma_{e})=\beta_{e}+(\alpha_{e}-\beta_{e})\delta(\sigma(v),\sigma(w)),$

and if $\sigma(v)\neq\sigma(w)$:

$\displaystyle
i_{e}(\delta(e))=i_{e}(\sigma(v)-\sigma(w))=\beta_{e}=\beta_{e}+(\alpha_{e}-\beta_{e})\delta(\sigma_{e})=\beta_{e}+(\alpha_{e}-\beta_{e})\delta(\sigma(v),\sigma(w)).\\!\\!\\!\\!$
∎

Also, we introduce the boundary partition function of the $n$-Potts models:

######  Definition 3.2.

Let $G$ be a graph (with possible loops and multiple edges) with the set of
vertices $V$, the set of edges $E$ and the boundary subset $S\subseteq V$ of
enumerated vertices: $S=\\{v_{1},v_{2},\dots,v_{k}\\}$. The boundary partition
function on $G$ is defined by the following expression

$\displaystyle Z_{n;S(\textbf{A})}(G)=\sum_{\sigma_{\textbf{A}}}\prod_{e\in
E}(\beta_{e}+(\alpha_{e}-\beta_{e})\delta(\sigma_{e})),$

where $\textbf{A}=\\{a_{1},a_{2},\dots,a_{k}\\}$, $\forall i\colon
a_{i}\in\mathbb{Z}_{n}$ is the set of fixed values, and the summation is over
such states $\sigma_{\textbf{A}}$ that $\sigma_{\textbf{A}}(v_{i})=a_{i}$.

######  Remark 3.3.

If $n=2$, we will omit the index $2$ and will write just
$Z_{S(\textbf{A})}(G)$.

The next Lemma connects boundary and ordinary partition functions:

######  Lemma 3.4.

Consider two graphs $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$ with the
only common vertices in the boundary subset $S=\\{v_{1},v_{2},\dots,v_{n}\\}$
in $V_{1}$ and $V_{2}$. We can glue these graphs and obtain the third graph
$G=(V,E)$, where $E=E_{1}\sqcup E_{2}$, $V=V_{1}\cup_{S}V_{2}$. Then, the
following identity holds

$\displaystyle
Z_{n}(G)=\sum_{\textbf{A}}Z_{n;S(\textbf{A})}(G_{1})Z_{n;S(\textbf{A})}(G_{2}),$

where the summation is over all possible sets A.

Figure 3: $G$ is obtained by merging of $S=\\{v_{1},v_{2},v_{3},v_{4}\\}$.

###### Proof.

The formula is obtained directly from the Proposition 3.1 and Definitions 3.2.
Indeed, by the Definition 3.2 we can write down
$Z_{n}(G)=\sum\limits_{\textbf{A}}Z_{n;S(\textbf{A})}(G),$ but also
$Z_{n;S(\textbf{A})}(G)=Z_{n;S_{1}(\textbf{A})}(G_{1})Z_{n;S_{2}(\textbf{A})}(G_{2})$.
∎

######  Remark 3.5.

The latter property of a partition function (Lemma 3.4) allow us to consider
$n$-Potts model partition function as a discrete version of the topological
quantum field theory in the Atiyah formalism [1], where

$\displaystyle TQFT\colon\ {\rm Cob}\to{\rm Vect}$

is a functor from the category of cobordisms to the category of vector spaces.

###  3.2 The case $\boldsymbol{n=2}$

In this subsection we consider the case $n=2$. Our first goal is to find such
conditions that the partition function (3.1) is invariant under the star-
triangle transformation which changes the subgraph $\Omega$ to the subgraph
$\Omega^{\prime}$. We derive these conditions with the use of the boundary
partition functions: consider a graph $G$ with the subgraph $\Omega$, then
using Lemma 3.4 for graphs $\Omega$ and $G-\Omega$ we obtain the following
identity

$\displaystyle
Z(G)=\sum_{\textbf{A}}Z_{S(\textbf{A})}(\Omega)Z_{S(\textbf{A})}(G-\Omega),$

where $S=\\{v_{1},v_{2},v_{3}\\}$ (Figure 4). After the star-triangle
transformation, we obtain a graph $G^{\prime}$ with the following partition
function

$\displaystyle
Z(G^{\prime})=\sum_{\textbf{A}}Z_{S(\textbf{A})}(\Omega^{\prime})Z_{S(\textbf{A})}(G^{\prime}-\Omega^{\prime}).$

Figure 4: Star-triangle transformation.

Due to the fact that the star-triangle transformation does not change edges of
the graph $G-\Omega$, we deduce that $\forall\textbf{A}\colon
Z_{S(\textbf{A})}(G-\Omega)=Z_{S(\textbf{A})}(G^{\prime}-\Omega^{\prime})$.
Therefore, the sufficient and necessary conditions for the invariance of the
partition function are the following

$\displaystyle\forall\textbf{A}\colon\
Z_{S(\textbf{A})}(\Omega)=Z_{S(\textbf{A})}(\Omega^{\prime}).$ (3.2)

We write them down in detail. Let us note that these conditions do not depend
on the states of the vertices (see Figure 5), but depend on the number and the
positions of the vertices with equal states. Therefore, we have the following
possibilities:

Figure 5: Different possibilities.

  * •

two states in the triangle are the same, then the central vertex either has
the same state or has the different state, then
$\alpha_{1}\beta_{2}\beta_{3}+\beta_{1}\alpha_{2}\alpha_{3}\mapsto\alpha^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3}$
and two more maps after permuting indexes,

  * •

all states are the same, then
$\alpha_{1}\alpha_{2}\alpha_{3}+\beta_{1}\beta_{2}\beta_{3}\mapsto\alpha^{\prime}_{1}\alpha^{\prime}_{2}\alpha^{\prime}_{3}$.

In this way we obtain the following equations

$\displaystyle\begin{cases}\alpha_{1}\beta_{2}\beta_{3}+\beta_{1}\alpha_{2}\alpha_{3}=\alpha^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3},\\\
\alpha_{2}\beta_{1}\beta_{3}+\beta_{2}\alpha_{1}\alpha_{3}=\alpha^{\prime}_{2}\beta^{\prime}_{1}\beta^{\prime}_{3},\\\
\alpha_{3}\beta_{1}\beta_{2}+\beta_{3}\alpha_{1}\alpha_{2}=\alpha^{\prime}_{3}\beta^{\prime}_{1}\beta^{\prime}_{2},\\\
\alpha_{1}\alpha_{2}\alpha_{3}+\beta_{1}\beta_{2}\beta_{3}=\alpha^{\prime}_{1}\alpha^{\prime}_{2}\alpha^{\prime}_{3}.\end{cases}$
(3.3)

After the substitution

$\displaystyle t_{i}=\frac{\alpha_{i}}{\beta_{i}}$

we rewrite (3.3) as

$\displaystyle\begin{cases}\beta_{1}\beta_{2}\beta_{3}(t_{1}+t_{2}t_{3})=\beta^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3}t^{\prime}_{1},\\\
\beta_{1}\beta_{2}\beta_{3}(t_{2}+t_{1}t_{3})=\beta^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3}t^{\prime}_{2},\\\
\beta_{1}\beta_{2}\beta_{3}(t_{3}+t_{1}t_{2})=\beta^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3}t^{\prime}_{3},\\\
\beta_{1}\beta_{2}\beta_{3}(t_{1}t_{2}t_{3}+1)=\beta^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3}t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3}.\end{cases}$

This set of equations defines a correspondence which preserves the Ising model
partition function if we mutate the graph $G$ to $G^{\prime}$. Let us denote
the product $\beta_{1}\beta_{2}\beta_{3}$ by $\beta$ and the product
$\beta^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3}$ by $\beta^{\prime}$.
Then, we obtain the following map from the $(t,\beta)$-variables to the
$(t^{\prime},\beta^{\prime})$ variables, we will call it $\widetilde{F}$,

$\displaystyle\widetilde{F}(t_{1},t_{2},t_{3},\beta)=(t^{\prime}_{1},t^{\prime}_{2},t^{\prime}_{3},\beta^{\prime})\colon$
$\displaystyle
t^{\prime}_{1}=\sqrt{\dfrac{(t_{1}+t_{2}t_{3})(t_{1}t_{2}t_{3}+1)}{(t_{2}+t_{1}t_{3})(t_{3}+t_{1}t_{2})}},$
$\displaystyle
t^{\prime}_{2}=\sqrt{\dfrac{(t_{2}+t_{1}t_{3})(t_{1}t_{2}t_{3}+1)}{(t_{1}+t_{2}t_{3})(t_{3}+t_{1}t_{2})}},$
$\displaystyle
t^{\prime}_{3}=\sqrt{\dfrac{(t_{3}+t_{1}t_{2})(t_{1}t_{2}t_{3}+1)}{(t_{1}+t_{2}t_{3})(t_{2}+t_{1}t_{3})}},$
$\displaystyle\beta^{\prime}=\beta\sqrt{\frac{(t_{1}+t_{2}t_{3})(t_{3}+t_{1}t_{2})(t_{2}+t_{1}t_{3})}{(t_{1}t_{2}t_{3}+1)}}.$
(3.4)

######  Remark 3.6.

Formally speaking to define a map on the space of edge weight adopted to the
star-triangle transformation we have to resolve the map $\widetilde{F}$
somehow for the parameters $\beta_{i}$. For example one can take the following
one

$\beta^{\prime}_{i}=\beta_{i}(\beta^{\prime}/\beta)^{1/3}.$

Actually, the choice of a resolution is not important in what follows.

######  Remark 3.7.

We choose the positive branch of the root function for real positive values of
variables $t_{i}$ for purposes emphasized further. This is relevant to the
almost positive version of the orthogonal grassmanian. See [11] for more
details about the connection of the Ising model and positive orthogonal
grassmanian.

###  3.3 The case $\boldsymbol{n\neq 2}$

Let us demonstrate how the method, described above, works for the star-
triangle transformation in the case $n\geq 3$. Using the same ideas as in the
previous subsection we could obtain the following conditions

$\begin{cases}\beta_{1}\beta_{2}\beta_{3}(t_{1}+t_{2}t_{3}+n-2)=\beta^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3}t^{\prime}_{1},\\\\[2.15277pt]
\beta_{1}\beta_{2}\beta_{3}(t_{2}+t_{1}t_{3}+n-2)=\beta^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3}t^{\prime}_{2},\\\\[2.15277pt]
\beta_{1}\beta_{2}\beta_{3}(t_{3}+t_{1}t_{2}+n-2)=\beta^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3}t^{\prime}_{3},\\\\[2.15277pt]
\beta_{1}\beta_{2}\beta_{3}(t_{1}t_{2}t_{3}+n-1)=\beta^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3}t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3},\\\\[2.15277pt]
\beta_{1}\beta_{2}\beta_{3}(t_{1}+t_{2}+t_{3}+n-3)=\beta^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3}.\end{cases}$
(3.5)

Here the last equation follows from the extra case in which all states are
different.

Figure 6: The extra case.

In general, the system (3.5) does not have a solution and the star-triangle
transformation is not possible. But, if $t_{i}$ satisfy the special condition,
partition function of the $n$-Potts model is still invariant under the star-
triangle transformation.

######  Proposition 3.8.

The system (3.5) together with equation

$\displaystyle
t_{1}t_{2}t_{3}=t_{1}t_{2}+t_{2}t_{3}+t_{3}t_{1}+(n-1)(t_{1}+t_{2}+t_{3})+n^{2}-3n+1$
(3.6)

has a solution in terms of prime variables.

###### Proof.

Using the first three and the last equations of (3.5) we immediately obtain
the expressions for $t^{\prime}_{i}$ and
$\frac{\beta^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3}}{\beta_{1}\beta_{2}\beta_{3}}$:

$\displaystyle\begin{cases}\dfrac{\beta^{\prime}_{1}\beta^{\prime}_{2}\beta^{\prime}_{3}}{\beta_{1}\beta_{2}\beta_{3}}=t_{1}+t_{2}+t_{3}+n-3,\\\\[8.61108pt]
t^{\prime}_{1}=\dfrac{t_{1}+t_{2}t_{3}+n-2}{t_{1}+t_{2}+t_{3}+n-3},\\\\[8.61108pt]
t^{\prime}_{2}=\dfrac{t_{2}+t_{1}t_{3}+n-2}{t_{1}+t_{2}+t_{3}+n-3},\\\\[8.61108pt]
t^{\prime}_{3}=\dfrac{t_{3}+t_{1}t_{2}+n-2}{t_{1}+t_{2}+t_{3}+n-3}.\end{cases}$

Substitute these expressions into the fourth equation of (3.5) and obtain the
equation

$\displaystyle
t_{1}t_{2}t_{3}+n-1=\frac{(t_{1}+t_{2}t_{3}+n-2)(t_{2}+t_{1}t_{3}+n-2)(t_{3}+t_{1}t_{2}+n-2)}{(t_{1}+t_{2}+t_{3}+n-3)^{2}}.$
(3.7)

By the straightforward computation, we retrieve that the identity (3.6) is the
consequence from the equation (3.7). ∎

######  Corollary 3.9.

Partition function of the $n$-Potts model $(n\geq 3)$ is invariant under the
star-triangle transformation if and only if the system (3.5) with the equation
(3.6) hold.

Below we present two nontrivial specialization of the partition function of
$n$-Potts model which are agreed with the system (3.5), (3.6).

######  Example 3.10.

Consider a graph $G$ and equip each $e\in E(G)$ with sign $+$ or $-$. Let us
consider the $n$-Potts model $M_{k}(G,\alpha_{e},\beta_{e})$ with following
parameters:

  * •

for all $e\in E$ equipped with $+$ the parameters $\alpha_{e}$, $\beta_{e}$
equal $\alpha_{e}=A_{+}=-t^{-\frac{3}{4}}$, $\beta_{e}=B_{+}=t^{\frac{1}{4}}$,

  * •

for all $e\in E$ equipped with $-$ the parameters $\alpha_{e}$, $\beta_{e}$
equal $\alpha_{e}=A_{-}=-t^{\frac{3}{4}}$, $\beta_{e}=B_{-}=t^{-\frac{1}{4}}$,

  * •

and $n=t+\frac{1}{t}+2$ (we suppose that parameter $t$ is chosen such that
$n\in\mathbb{N}$).

Let the graph $G$ has a triangle subgraph, the edges of which have signs $+$,
$-$, $-$. The reduced weights of edges are
$t_{1}=\frac{A_{+}}{B_{+}}=-\frac{1}{t}$, $t_{2}=\frac{A_{-}}{B_{-}}=-t$,
$t_{3}=\frac{A_{-}}{B_{-}}=-t$. It is easy to see that these $t_{i}$ satisfy
the equation (3.6).

We notice that the signed graph $G$ could be considered as the signed Tait
graph for a diagram $D(K)$ of a knot $K$ ([24], the chapter “Knot invariants
from edge-interaction models”). Moreover, the value of the Jones polynomial of
the knot $K$ at the point $n$ is closely related with the partition function
of the $n$-Potts model $M_{k}(G,\alpha_{e},\beta_{e})$ (see [24, equation
(7.17)]). Thus, the identification of the third Reidemeister move of the
diagram $D(K)$ with the star-triangle transformation of the signed graph $G$
is agreed with the star-triangle transformation defined by the system (3.5),
(3.6) for the $n$-Potts model $M_{k}(G,\alpha_{e},\beta_{e})$.

Our second example is about the models of bond percolation. Firstly, we
briefly give their definitions:

######  Definition 3.11 (bond percolation [13]).

Consider a graph $G$. An edge $e\in E(G)$ is considered to be open with
probability $p_{e}$ or closed with probability $1-p_{e}$. We suppose that all
edges might be closed or open independently. One is interested in
probabilistic properties of cluster formation (i.e. maximal connected sets of
closed edges of the graph $G$).

######  Example 3.12.

The bond percolation models could be considered as a limit $n\to 1$ of the
$n$-Potts models at the level of the boundary partition functions [7]. This
identification corresponds to the specialization of the system (3.5), (3.6) by
$n\to 1$.

Substitute $t_{i}=\frac{1}{p_{i}}$, $t^{\prime}_{i}=\frac{1}{p^{\prime}_{i}}$
and $n=1$ in (3.5) and (3.6), then

$\displaystyle\begin{cases}(p_{1}+p_{2}p_{3}-p_{1}p_{2}p_{3})\alpha_{1}\alpha_{2}\alpha_{3}=p^{\prime}_{2}p^{\prime}_{3}\alpha^{\prime}_{1}\alpha^{\prime}_{2}\alpha^{\prime}_{3},\\\
(p_{2}+p_{1}p_{3}-p_{1}p_{2}p_{3})\alpha_{1}\alpha_{2}\alpha_{3}=p^{\prime}_{1}p^{\prime}_{3}\alpha^{\prime}_{1}\alpha^{\prime}_{2}\alpha^{\prime}_{3},\\\
(p_{3}+p_{1}p_{2}-p_{1}p_{2}p_{3})\alpha_{1}\alpha_{2}\alpha_{3}=p^{\prime}_{1}p^{\prime}_{2}\alpha^{\prime}_{1}\alpha^{\prime}_{2}\alpha^{\prime}_{3},\\\
\alpha_{1}\alpha_{2}\alpha_{3}=\alpha^{\prime}_{1}\alpha^{\prime}_{2}\alpha^{\prime}_{3},\\\\[4.30554pt]
\dfrac{1}{p_{1}p_{2}}+\dfrac{1}{p_{2}p_{3}}+\dfrac{1}{p_{1}p_{3}}-1=\dfrac{1}{p_{1}p_{2}p_{3}}.\end{cases}$

After simplifications we obtain the condition for the star-triangle
transformation of the bond percolation models (for instance, see [13])

$\displaystyle\begin{cases}p_{1}+p_{2}p_{3}-p_{1}p_{2}p_{3}=p^{\prime}_{2}p^{\prime}_{3},\\\
p_{2}+p_{1}p_{3}-p_{1}p_{2}p_{3}=p^{\prime}_{1}p^{\prime}_{3},\\\
p_{3}+p_{1}p_{2}-p_{1}p_{2}p_{3}=p^{\prime}_{1}p^{\prime}_{2},\\\
p_{1}+p_{2}+p_{3}-1=p_{1}p_{2}p_{3}.\end{cases}$

##  4 Tetrahedron equation

The tetrahedron equation firstly was considered by A. Zamolodchikov [25] who
has constructed its solution in $S$-form. We consider the following form of
the equation

$\displaystyle T_{123}T_{145}T_{246}T_{356}=T_{356}T_{246}T_{145}T_{123},$
(4.1)

where $T_{ijk}$ is an operator acting nontrivially in the tensor product of
three vector spaces $V_{i}$, $V_{j}$, $V_{k}$, indexed by $i$, $j$ and $k$.
Tetrahedron equation is the higher order analog of the Yang–Baxter equation.
Both equations are examples of $n$-simplex equations [18] and play an
important role in hypercube combinatorics and higher Bruhat orders. For the
complete introduction to the topic see for example [21]. In this section we
present two proofs of the main theorem of the paper:

######  Theorem 4.1.

The change of variables (3.4) defines the solution of the tetrahedron equation
(4.1).

These two proofs have a lot of common points and ideas, but have the crucial
differences in the last stages. It is interesting to compare proofs for the
purpose of combining arguments of boundary partition functions and the
technique of correlation functions in the Ising–Potts models.

At first in the next subsection we prove that the change of variables (3.4)
corresponds to the variables transform in the trigonometric solution of a
local Yang–Baxter equation.

###  4.1 Local Yang–Baxter equation

Let us recall that the following change of variables
$(t_{1},t_{2},t_{3})\mapsto(t^{\prime}_{1},t^{\prime}_{2},t^{\prime}_{3})$
provides an invariance of the Ising model (3.1) under the star-triangle
transformation (3.4):

$\displaystyle\begin{cases}t^{\prime}_{1}=&\sqrt{\dfrac{(t_{1}+t_{2}t_{3})(t_{1}t_{2}t_{3}+1)}{(t_{2}+t_{1}t_{3})(t_{3}+t_{1}t_{2})}},\\\\[8.61108pt]
t^{\prime}_{2}=&\sqrt{\dfrac{(t_{2}+t_{1}t_{3})(t_{1}t_{2}t_{3}+1)}{(t_{1}+t_{2}t_{3})(t_{3}+t_{1}t_{2})}},\\\\[8.61108pt]
t^{\prime}_{3}=&\sqrt{\dfrac{(t_{3}+t_{1}t_{2})(t_{1}t_{2}t_{3}+1)}{(t_{1}+t_{2}t_{3})(t_{2}+t_{1}t_{3})}}\end{cases}$
$\displaystyle\qquad\qquad\Updownarrow$
$\displaystyle\begin{cases}t^{\prime}_{1}t^{\prime}_{2}=&\dfrac{t_{1}t_{2}t_{3}+1}{t_{3}+t_{1}t_{2}},\\\\[8.61108pt]
t^{\prime}_{2}t^{\prime}_{3}=&\dfrac{t_{1}t_{2}t_{3}+1}{t_{1}+t_{2}t_{3}},\\\\[8.61108pt]
t^{\prime}_{1}t^{\prime}_{3}=&\dfrac{t_{1}t_{2}t_{3}+1}{t_{2}+t_{1}t_{3}}.\end{cases}$
(4.2)

Following [17], we construct orthogonal hyperbolic $3\times 3$ matrices
$R_{ij}$ which solve the local Yang–Baxter equation

$\displaystyle
R_{12}(t_{3})R_{13}(S(t_{2}))R_{23}(t_{1})=R_{23}(S(t^{\prime}_{1}))R_{13}(t^{\prime}_{2})R_{12}(S(t^{\prime}_{3})),$
(4.3)

where $S(t)$ is the following involution

$\displaystyle S(t)=\frac{t-1}{t+1}.$ (4.4)

On the left hand side of (4.3) we have

$\displaystyle
R_{12}(t_{3})=\begin{pmatrix}\mathrm{i}\sinh(\log(t_{3}))&\cosh(\log(t_{3}))&0\\\
\cosh(\log(t_{3}))&-\mathrm{i}\sinh(\log(t_{3}))&0\\\ 0&0&1\end{pmatrix}\\!,$
(4.5) $\displaystyle
R_{13}(S(t_{2}))=\begin{pmatrix}\mathrm{i}\sinh(\log(S(t_{2})))&0&\cosh(\log(S(t_{2})))\\\
0&1&0\\\
\cosh(\log(S(t_{2})))&0&-\mathrm{i}\sinh(\log(S(t_{2})))\end{pmatrix}\\!,$
(4.6) $\displaystyle R_{23}(t_{1})=\begin{pmatrix}1&0&0\\\
0&\mathrm{i}\sinh(\log(t_{1}))&\cosh(\log(t_{1}))\\\
0&\cosh(\log(t_{1}))&-\mathrm{i}\sinh(\log(t_{1}))\end{pmatrix}\\!.$ (4.7)

######  Theorem 4.2.

Matrices (4.5), (4.6), (4.7) together with the rules (3.4), (4.4) give a
solution of (4.3).

###### Proof.

It can be proved by a straightforward computation. For example let us write
down the result of the product on the left hand side

$\displaystyle\begin{pmatrix}\dfrac{t_{2}\big{(}t_{3}^{2}-1\big{)}}{t_{3}(t_{2}^{2}-1)}&\dfrac{{\rm
i}\big{(}t_{1}^{2}t_{2}^{2}t_{3}^{2}-t_{1}^{2}-t_{2}^{2}+t_{3}^{2}\big{)}}{2t_{1}t_{3}\big{(}t_{2}^{2}-1\big{)}}&\dfrac{t_{1}^{2}t_{2}^{2}t_{3}^{2}-t_{1}^{2}+t_{2}^{2}-t_{3}^{2}}{2t_{1}t_{3}\big{(}t_{2}^{2}-1\big{)}}\\\\[12.91663pt]
-\dfrac{{\rm
i}t_{2}\big{(}t_{3}^{2}+1\big{)}}{t_{3}(t_{2}^{2}-1)}&\dfrac{t_{1}^{2}t_{2}^{2}t_{3}^{2}+t_{1}^{2}+t_{2}^{2}+t_{3}^{2}}{2t_{1}t_{3}\big{(}t_{2}^{2}-1\big{)}}&-\dfrac{{\rm
i}\big{(}t_{1}^{2}t_{2}^{2}t_{3}^{2}+t_{1}^{2}-t_{2}^{2}-t_{3}^{2}\big{)}}{2t_{1}t_{3}\big{(}t_{2}^{2}-1\big{)}}\\\\[12.91663pt]
\dfrac{t_{2}^{2}+1}{t_{2}^{2}-1}&\dfrac{{\rm
i}t_{2}\big{(}t_{1}^{2}+1\big{)}}{t_{1}\big{(}t_{2}^{2}-1\big{)}}&\dfrac{t_{2}\big{(}t_{1}^{2}-1\big{)}}{t_{1}\big{(}t_{2}^{2}-1\big{)}}\end{pmatrix}\\!.$
(4.8)

At the first glance the product on the right hand side looks much more
cumbersome, but occasionally all terms are simplified and the matrix on the
right hand side coincides with (4.8). ∎

###  4.2 Tetrahedron equation, first proof

Let us encode the tetrahedron equation by the Figure 7.

Figure 7: Encoding the tetrahedron equation by the standard graph.

The standard graph encodes $R$-matrices in the following way: in each inner
vertex numbered by $k$, which is the intersection of strands $i$ and $j$ we
put the matrix $R_{ij}(t_{k})$ which is the $2\times 2$ matrix in the
$4$-dimensional space with basis vectors indexed by $a$, $b$, $c$, $d$. For
instance,

$\displaystyle
R_{ac}(t_{5})=\begin{pmatrix}\mathrm{i}\sinh(\log(t_{5}))&0&\cosh(\log(t_{5}))&0\\\
0&1&0&0\\\ \cosh(\log(t_{5}))&0&-\mathrm{i}\sinh(\log(t_{5}))&0\\\
0&0&0&1\end{pmatrix}\\!.$

Let us orient each strand from the left to the right and multiply $R$-matrices
in order of the orientation, for instance for the Figure 7 we have the
following product of $R$-matrices

$\displaystyle
R_{cd}(t_{1})R_{bd}(S(t_{2}))R_{bc}(t_{3})R_{ad}(t_{4})R_{ac}(S(t_{5}))R_{ab}(t_{6}).$
(4.9)

We note that the orientation defines the product (4.9) uniquely.

Then let us apply four local Yang–Baxter equations consequently to the inner
triangles with vertices numbered $(1,2,3)$, $(1,4,5)$, $(2,4,6)$ and $(3,5,6)$
as on the Figure 8. As a result, we have one and the same standard graph as on
the Figure 7 rotated by $\pi$.

Figure 8: Local Yang–Baxter equations applied to the standard graph.

At the same time we could apply local Yang–Baxter equations in the opposite
direction: firstly to the triangle $(3,5,6)$, then $(2,4,6)$, $(1,4,5)$ and
$(1,2,3)$. Eventually in this case we will have again the same standard graph.

As the reader may have already guessed, every local Yang–Baxter equation
applied to the triangle $A$, $B$, $C$ defines the factor $T_{ABC}$ in the
tetrahedron equation (4.1). For example we obtain

$\displaystyle T_{1,2,3}\colon\
(t_{1},S(t_{2}),t_{3},t_{4},t_{5},t_{6})\mapsto(S(t^{\prime}_{1}),t^{\prime}_{2},S(t^{\prime}_{3}),t_{4},t_{5},t_{6}).$

By Theorem 4.2 the product (4.9) preserves by each local Yang–Baxter equation
encoded on the Figure 8. As a result of two sequences of four Local
Yang–Baxter equations we obtain an equality of two products of six $4\times 4$
$R$-matrices

$\displaystyle
R_{cd}(u_{1})R_{bd}(u_{2})R_{bc}(u_{3})R_{ad}(u_{4})R_{ac}(u_{5})R_{ab}(u_{6})$
$\displaystyle\qquad{}=R_{cd}(v_{1})R_{bd}(v_{2})R_{bc}(v_{3})R_{ad}(v_{4})R_{ac}(v_{5})R_{ab}(v_{6}),$
(4.10)

where the parameters $u_{i}$, $v_{j}$, $i,j=1,\ldots,6$ depend on the initial
variables $t_{i}$, on the mapping (3.4) and on the involution (4.4).

Let us consider this equation element-wise, and note that we could uniquely
express parameters in the right hand side in terms of the parameters on the
left hand side

$\displaystyle U_{1,4}=b(t_{4}),\qquad U_{2,4}=-a(t_{4})b(t_{2}),\qquad
U_{1,3}=a(t_{4})b(t_{5}),$ $\displaystyle
U_{1,2}=a(t_{4})a(t_{5})b(t_{6}),\quad U_{3,4}=a(t_{2})a(t_{4})b(t_{1}),\quad
U_{2,3}=b(t_{2})b(t_{4})b(t_{5})-a(t_{2})a(t_{5})b(t_{3}).$

Here $U$ is a matrix on the left hand side and $a,b$ are some invertible
functions, come from (4.5), (4.6), (4.7). So we could uniquely determine
$t_{4}$ from the first equation, $t_{2}$ and $t_{5}$ from the second and the
third, then $t_{1}$ and $t_{6}$, and finally $t_{3}$ from the element
$U_{2,3}$.

Let us note that this algebraic proof could be formulated in terms of the
paths on the standard graph (Figure 7) with orientation. So the equation
(4.10) provides coincidence of the parameters in the vertices given by the two
sides of the tetrahedron equation. This finishes the proof.

###  4.3 Tetrahedron equation, second proof

####  4.3.1 Involution lemma

Let us consider the map
$F(t_{1},t_{2},t_{3})=(t^{\prime}_{1},t^{\prime}_{2},t^{\prime}_{3})$, where
prime variables defined by (3.4). First of all, we formulate one technical
lemma:

######  Lemma 4.3.

The following identity holds for all $t_{1}$, $t_{2}$, $t_{3}$:

$\displaystyle S\times S\times S\circ F\circ S\times S\times S=F^{-1},$

where $S(t)=\frac{t-1}{t+1}$.

We present the proof in Appendix A.

####  4.3.2 Towards the tetrahedron equation

Let us consider any graph $G$ with a subgraph $\Gamma_{1}$ which coincides
with the leftmost graph on the Figure 9. We can transform the graph $G$ to the
graph $G^{\prime}$ with the subgraph $\Gamma_{2}$ which coincides with the
rightmost graph on the Figure 9. We could make this mutation by two different
chains of star-triangle transformations:
$F_{356}^{-1}F_{246}F_{145}^{-1}F_{123}$ and
$F_{123}^{-1}F_{145}F_{246}^{-1}F_{356}$. Both are figured out on the Figure
8. This observation turns us to the following hypothesis

$\displaystyle
F_{356}^{-1}F_{246}F_{145}^{-1}F_{123}=F_{123}^{-1}F_{145}F_{246}^{-1}F_{356}.$
(4.11)

Figure 9: The graphical representation of the left and right parts of (4.12).

This equality is equivalent to the Zamolodchikov equation

$\displaystyle\Phi_{356}\Phi_{246}\Phi_{145}\Phi_{123}=\Phi_{123}\Phi_{145}\Phi_{246}\Phi_{356},$
(4.12)

where $\Phi_{ijk}=S_{i}S_{k}F_{ijk}S_{j}$. Indeed, using Lemma 4.3 and the
simple observation that $S_{l}F_{ijk}=F_{ijk}S_{l}$, $l\neq\\{i,j,k\\}$ we can
write down

$\displaystyle
F_{356}^{-1}F_{246}F_{145}^{-1}F_{123}=S_{3}S_{5}S_{6}F_{356}S_{3}S_{5}S_{6}F_{246}S_{1}S_{4}S_{5}F_{145}S_{1}S_{4}S_{5}F_{123}$
$\displaystyle\hphantom{F_{356}^{-1}F_{246}F_{145}^{-1}F_{123}}{}=S_{2}S_{5}(S_{3}S_{6}F_{356}S_{5})(S_{2}S_{6}F_{246}S_{4})(S_{1}S_{5}F_{145}S_{4})(S_{1}S_{3}F_{123}S_{2})S_{2}S_{5},$

and

$\displaystyle
F_{123}^{-1}F_{145}F_{246}^{-1}F_{356}=S_{1}S_{2}S_{3}F_{123}S_{1}S_{2}S_{3}F_{145}S_{2}S_{4}S_{6}F_{246}S_{2}S_{4}S_{6}F_{356}$
$\displaystyle\hphantom{F_{123}^{-1}F_{145}F_{246}^{-1}F_{356}}{}=S_{2}S_{5}(S_{1}S_{3}F_{123}S_{2})(S_{1}S_{5}F_{145}S_{4})(S_{2}S_{6}F_{246}S_{4})(S_{1}S_{3}F_{356}S_{2})S_{2}S_{5}.$

Conjugating both sides of (4.11) by $S_{2}S_{5}$ we obtain the Zamolodchikov
tetrahedron equation.

####  4.3.3 Solution for the tetrahedron equation

######  Proposition 4.4.

The functions

  * •

$\dfrac{\partial\ln(Z(G))}{\partial t_{e}}$, where $e$ is any edge belonging
to $G-\Omega$, and

  * •

$\dfrac{Z_{S(\textbf{A})}(G)}{Z(G)}$, where $Z_{S(\textbf{A})}(G)$ is the
boundary partition function and $S$ is any vertex subset of $G-\Omega$ or
$G-\Omega^{\prime}$ $($see the Figure $\ref{fig startr}),$

are invariant under the star-triangle transformation. Moreover, these
functions do not depend on variables $\beta_{e}$.

######  Remark 4.5.

The function

$\displaystyle\dfrac{Z_{S(\textbf{A})}(G)}{Z(G)}$

can be interpreted as a probability of the fixed values A of spins in $S$,
related to the boundary partition function $Z_{S(\textbf{A})}(G)$.

###### Proof.

The crucial point in the demonstration of the first part of the statement is
the fact that the derivative $\frac{\partial\ln(Z(G))}{\partial t_{e_{i}}}$
does not depend on parameters $\beta$. Indeed, this follows from the explicit
form of the partition function

$\displaystyle Z(G)=\prod_{e\in E}\beta_{e}\sum_{\sigma}\prod_{e\in
E}(1+(t_{e}-1)\delta(\sigma_{e})).$

The proof of the second part of the statement is straightforward. It follows
from the Definition 3.2 and the condition (3.2). ∎

We will prove the Zamolodchikov equation in its equivalent form

$\displaystyle
F_{356}^{-1}F_{246}F_{145}^{-1}F_{123}=F_{123}^{-1}F_{145}F_{246}^{-1}F_{356}.$
(4.13)

Let us notice that due to the local nature of the star-triangle transformation
and the convolution property of the boundary partition function we have a
choice to take some suitable graph to prove equation (4.13). So let us take
the graph $\Gamma_{1}$ from the Figure 10 with the following choice of
boundary set $S_{0}$:

$\displaystyle S_{0}:=\\{v_{1},v_{2},v_{3},v_{4}\\}.$

Figure 10: The graphs $\Gamma_{1}$ and $\Gamma_{2}$.

We will prove that the values of the second-type invariant functions which are
preserved by both sides of the equation (4.13) allows us to uniquely
reconstruct weights of all edges. Explain this idea in detail, let us consider
the left hand side of the equation (4.13) and the map $F_{123}$, then for any
$\textbf{A}=\\{a_{1},\dots,a_{4}\\}$ the following identity holds

$\displaystyle\frac{Z_{S_{0}(\textbf{A})}(\Gamma_{1})}{Z(\Gamma_{1})}=\frac{Z_{S_{1}(\textbf{A}_{1})}(\Gamma_{1})}{Z(\Gamma_{1})}+\frac{Z_{S_{1}(\textbf{A}_{2})}(\Gamma_{1})}{Z(\Gamma_{1})},$

here $S_{1}=\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}$ (see Figure 11),
$\textbf{A}_{1}=\\{a_{1},\dots,a_{4},0\\}$,
$\textbf{A}_{2}=\\{a_{1},\dots,a_{4},1\\}$. The Proposition 4.4 provides

$\displaystyle\frac{Z_{S_{1}(\textbf{A}_{1})}(\Gamma_{1})}{Z(\Gamma_{1})}=\frac{Z_{S_{1}(\textbf{A}_{1})}(\Gamma^{\prime})}{Z(\Gamma^{\prime})},\qquad\frac{Z_{S_{1}(\textbf{A}_{2})}(\Gamma_{1})}{Z(\Gamma_{1})}=\frac{Z_{S_{1}(\textbf{A}_{2})}(\Gamma^{\prime})}{Z(\Gamma^{\prime})},$

where $\Gamma^{\prime}$ is obtained from $\Gamma_{1}$ by the star-triangle
transformation (see Figure 11). And therefore we deduce that

$\displaystyle\frac{Z_{S_{0}(\textbf{A})}(\Gamma_{1})}{Z(\Gamma_{1})}=\frac{Z_{S_{0}(\textbf{A})}(\Gamma^{\prime})}{Z(\Gamma^{\prime})}.$

Repeating these arguments for the remaining maps $F_{ijk}$ from the left hand
side of (4.13) we obtain

$\displaystyle\frac{Z_{S_{0}(\textbf{A})}(\Gamma_{1})}{Z(\Gamma_{1})}=\frac{Z_{S_{0}(\textbf{A})}(\Gamma_{2})}{Z(\Gamma_{2})}.$
(4.14)

In the same fashion, if we consider the right hand side of the equation
(4.13), we similarly obtain that

$\displaystyle\frac{Z_{S_{0}(\textbf{A})}(\Gamma_{1})}{Z(\Gamma_{1})}=\frac{Z_{S_{0}(\textbf{A})}(\Gamma_{2})}{Z(\Gamma_{2})}.$

Hence, in order to prove the equation (4.13) it is sufficient to prove that we
can reconstruct the parameters $t_{i}$, $i=1,\ldots,6$ from the values
${Z_{S_{0}(\textbf{A})}(\Gamma_{2})}/{Z(\Gamma_{2})}$ for different values of
A in a unique way.

Figure 11: The left hand side of (4.13). Figure 12: The graph $\Gamma_{2}$.

We understand the identity (4.14) as a system of $2^{4}$ linear equations with
unknowns $Z_{S_{0}(\textbf{A})}(\Gamma_{2})$ of the following type

$\displaystyle
Z_{S(\textbf{A})}(\Gamma_{2})\colon\quad\frac{Z_{S(\textbf{A})}(\Gamma_{2})}{Z(\Gamma_{2})}=\alpha(\textbf{A}),\qquad\forall\textbf{A}=(a_{1},\ldots,a_{4})$

which is equivalent to

$\displaystyle\sum_{\textbf{A}^{\prime}}Z_{S(\textbf{A}^{\prime})}(\Gamma_{2})=Z_{S(\textbf{A})}(\Gamma_{2})/\alpha(\textbf{A}).$

The rank of the system is equal to 15. Indeed, the rank is $\geq 15$ and we
know that there is a nontrivial solution coming from the boundary partition
functions for the graph $\Gamma_{2}$.

Hence any solution has the form

$\displaystyle
Z_{S_{0}(\textbf{A})}(\Gamma_{2})=C\cdot\alpha_{0}(a_{1},a_{2},a_{3},a_{4}),$
(4.15)

where $C$ is some constant and $a_{i}$ are the states. Now we will prove that
the parameters $t_{1},\ldots,t_{6}$ are reconstructed uniquely from the
equation (4.15).

Let us introduce some auxiliary variables and rewrite the partition function
in the following way: we have 16 states of boundary vertices
$S_{0}=\\{v_{1},v_{2},v_{3},v_{4}\\}$. Each expression
$Z_{S_{0}(\textbf{A})}(\Gamma_{2})$ is a sum of four terms corresponding to
the states of internal vertices $v_{5}$ and $v_{6}$. We consider in details
the case $S_{0}=\\{0,0,0,0\\}$. Let us denote the weights of the states of the
square $\\{v_{1},v_{6},v_{5},v_{4}\\}$ by $v$, $z$, $y$ and $x$ (Figure 13).
Then, we obtain the following equations

$\displaystyle v=t_{3}t_{5}B,\qquad$ $\displaystyle v_{1}=t_{5}t_{6}B,$
$\displaystyle z=t_{3}t_{4}t_{5}t_{6}B,\qquad$ $\displaystyle
z_{1}=t_{4}t_{5}B,$ $\displaystyle y=t_{6}t_{3}B,\qquad$ $\displaystyle
y_{1}=B,$ $\displaystyle x=t_{3}t_{4}B,\qquad$ $\displaystyle
x_{1}=t_{4}t_{6}B,$ (4.16)

where $B=\beta_{3}\beta_{4}\beta_{5}\beta_{6}$ and

$\displaystyle
v+t_{1}t_{2}z+t_{2}y+t_{1}x=\frac{C}{B_{1}}\alpha_{0}(0,0,0,0),B_{1}=\beta_{1}\beta_{2}.$

Figure 13: The auxiliary variables.

Similarly we obtain seven more equations (we omit eighth equation with
$v_{1}=1$ due to the symmetry of the model with respect to the total
involution of spins)

$\displaystyle
v_{1}+t_{1}t_{2}z_{1}+t_{2}y_{1}+t_{1}x_{1}=\frac{C}{B_{1}}b_{1}=\frac{C}{B_{1}}\alpha^{\prime}(0,0,0,1),$
$\displaystyle
t_{1}v_{1}+t_{2}z_{1}+y_{1}t_{1}t_{2}+x_{1}=\frac{C}{B_{1}}b_{2}=\frac{C}{B_{1}}\alpha^{\prime}(0,1,0,1),$
$\displaystyle
t_{1}t_{2}v_{1}+z_{1}+t_{1}y_{1}+t_{2}x_{1}=\frac{C}{B_{1}}b_{3}=\frac{C}{B_{1}}\alpha^{\prime}(0,1,1,1),$
$\displaystyle
t_{2}v_{1}+t_{1}z_{1}+y_{1}+t_{1}t_{2}x_{1}=\frac{C}{B_{1}}b_{4}=\frac{C}{B_{1}}\alpha^{\prime}(0,0,1,1),$
$\displaystyle
v+t_{1}t_{2}z+t_{2}y+t_{1}x=\frac{C}{B_{1}}a_{1}=\frac{C}{B_{1}}\alpha^{\prime}(0,0,0,0),$
$\displaystyle
t_{1}v+t_{2}z+yt_{1}t_{2}+x=\frac{C}{B_{1}}a_{2}=\frac{C}{B_{1}}\alpha^{\prime}(0,1,0,0),$
$\displaystyle
t_{1}t_{2}v+z+t_{1}y+t_{2}x=\frac{C}{B_{1}}a_{3}=\frac{C}{B_{1}}\alpha^{\prime}(0,1,1,0),$
$\displaystyle
t_{2}v+t_{1}z+y+t_{1}t_{2}x=\frac{C}{B_{1}}a_{4}=\frac{C}{B_{1}}\alpha^{\prime}(0,0,1,0).$

By straightforward calculation we retrieve

$\displaystyle
y_{1}=\frac{C}{B_{1}}\frac{-t_{2}b_{1}+t_{1}t_{2}b_{2}+b_{4}-t_{1}b_{3}}{-t_{2}^{2}+1+t_{1}^{2}t_{2}^{2}-t_{1}^{2}},\qquad
z_{1}=\frac{C}{B_{1}}\frac{t_{1}t_{2}b_{1}-t_{2}b_{2}+b_{3}-t_{1}b_{4}}{-t_{2}^{2}+1+t_{1}^{2}t_{2}^{2}-t_{1}^{2}},$
$\displaystyle
x_{1}=\frac{C}{B_{1}}\frac{t_{2}t_{1}b_{4}-t_{1}b_{1}-t_{2}b_{3}+b_{2}}{-t_{2}^{2}+1+t_{1}^{2}t_{2}^{2}-t_{1}^{2}},\qquad
v_{1}=\frac{C}{B_{1}}\frac{b_{1}+t_{1}t_{2}b_{3}-t_{2}b_{4}-t_{1}b_{2}}{-t_{2}^{2}+1+t_{1}^{2}t_{2}^{2}-t_{1}^{2}},$
(4.17) $\displaystyle
y=\frac{C}{B_{1}}\frac{-t_{2}a_{1}+t_{1}t_{2}a_{2}+a_{4}-t_{1}a_{3}}{-t_{2}^{2}+1+t_{1}^{2}t_{2}^{2}-t_{1}^{2}},\qquad
z=\frac{C}{B_{1}}\frac{t_{1}t_{2}a_{1}-t_{2}a_{2}+a_{3}-t_{1}a_{4}}{-t_{2}^{2}+1+t_{1}^{2}t_{2}^{2}-t_{1}^{2}},$
$\displaystyle
x=\frac{C}{B_{1}}\frac{t_{2}t_{1}a_{4}-t_{1}a_{1}-t_{2}a_{3}+a_{2}}{-t_{2}^{2}+1+t_{1}^{2}t_{2}^{2}-t_{1}^{2}},\qquad
v=\frac{C}{B_{1}}\frac{a_{1}+t_{1}t_{2}a_{3}-t_{2}a_{4}-t_{1}a_{2}}{-t_{2}^{2}+1+t_{1}^{2}t_{2}^{2}-t_{1}^{2}}.$

Using the auxiliary variables it is easy to see that

$\displaystyle\frac{z_{1}}{x_{1}}=\frac{v}{y}=\frac{-t_{2}a_{4}+t_{1}t_{2}a_{3}-t_{1}a_{2}+a_{1}}{t_{2}t_{1}a_{2}-t_{2}a_{1}-t_{1}a_{3}+a_{4}}=\frac{t_{1}t_{2}b_{1}-t_{2}b_{2}+b_{3}-t_{1}b_{4}}{t_{2}t_{1}b_{4}-t_{1}b_{1}-t_{2}b_{3}+b_{2}}.$

In the same fashion we obtain

$\displaystyle\frac{v_{1}}{x_{1}}=\frac{v}{x}.$

Then we can straightforwardly deduce expressions for the variables $t_{1}$ and
$t_{2}$ (equations (B.1), (B.2) in Appendix B), then obtain expressions for
the auxiliary variables from equations (4.17) and finally obtain variables
$t_{3}$, $t_{4}$, $t_{5}$, $t_{6}$ from the equations (4.16):

$\displaystyle\begin{cases}t_{3}=\sqrt{\dfrac{vyx}{zy^{2}_{1}}},\\\
t_{5}=\dfrac{v}{t_{3}y_{1}},\\\ t_{6}=\dfrac{y}{t_{3}y_{1}},\\\
t_{4}=\dfrac{x}{t_{3}y_{1}}.\end{cases}$

This completes the proof of the Zamolodchikov equation due to the fact that
there is a unique way to choose positive weights for the edges of the model to
provide the expected values of boundary partitions function for the graph
$\Gamma_{2}$.

##  5 Star-triangle transformation, Biggs formula and conclusion

The main results of the paper represent the functional relations on the space
of multivariate Tutte polynomials. This problem is a step of the program of
investigation of the framed graph structures and the related statistical
models. We examined in details the Biggs formula and applied it to the
multivariate case. We also provided a new proof of the theorem of Matiyasevich
as a partial case of such formula. The second principal result is the reveal
of the tetrahedral symmetry of the multivariate Tutte polynomial at the point
$n=2$. Therefore, we have a connection between the multivariate Tutte
polynomial, functions on Lustig cluster manifolds [4] and its electrical
analogues [12, 19]. We would like to interpret this property as the critical
point of the model described by the multivariate Tutte polynomial, and the
tetrahedral symmetry as a longstanding analog of the conformal symmetry of the
Ising model at the critical point [8].

Both correspondences are related by the following observation. Let $G$ and
$G^{\prime}$ be two graphs related by the star-triangle transformation. Let us
consider the case when the partition function is invariant with respect to
this transformation ($n=2$ or $n>2$ and the system (3.5), (3.6) holds)

$Z_{n}(G^{\prime})=Z_{n}(G).$

On the other hand the star-triangle transformation provides a groupoid
symmetry on a wide class of objects, in our case on the space of Ising models.
The Biggs formula allows us to extend this action to the points of valency $1$
and $2$. And we can obtain the 14-term relation (Theorem 5.1) by comparing the
right-hand sides of Biggs formulas for $G$ and $G^{\prime}$.

Let us explain this idea in details, consider two pairs of $n$-Potts models:
$M_{1}(G,i^{1}_{e})$ and $M_{1}(G^{\prime},i^{1}_{e})$, $M_{2}(G,i^{2}_{e})$
and $M_{2}(G^{\prime},i^{2}_{e})$ (for simplicity we denote these models
$M_{1}(G)$, $M_{2}(G)$, $M_{1}(G^{\prime})$, $M_{2}(G^{\prime})$
correspondingly). After multiplying both parts of the formula (2.1) for
$M_{1}(G)$ and $M_{2}(G)$ by $n^{v(G)}$ (by $n^{v(G^{\prime})}$ for
$M_{1}(G^{\prime})$ and $M_{2}(G^{\prime})$) we obtain

$\displaystyle Z^{1}_{n}(G)=\prod\limits_{e\in G}q_{e}\sum_{A\subseteq
G}\prod\limits_{e\in
A}\frac{p_{e}}{q_{e}}Z^{2}_{n}(A)=Z^{1}_{n}(G^{\prime})=\prod_{e\in
G^{\prime}}q^{\prime}_{e}\sum_{A^{\prime}\subseteq G^{\prime}}\prod_{e\in
A^{\prime}}\frac{p^{\prime}_{e}}{q^{\prime}_{e}}Z^{2}_{n}(A^{\prime}),$ (5.1)

here in both cases we take the sum over the set of all spanning subgraphs.

Let us rewrite the first part of the formula (5.1) by separating two kinds of
terms

$\displaystyle Z^{1}_{n}(G)=\prod_{e\in G}q_{e}\sum\limits_{A_{1}\subseteq
G}\prod_{e\in A_{1}}\frac{p_{e}}{q_{e}}Z^{2}_{n}(A_{1})+\prod_{e\in
G}q_{e}\sum\limits_{A_{2}\subseteq G}\prod_{e\in
A_{2}}\frac{p_{e}}{q_{e}}Z^{2}_{n}(A_{2}),$ (5.2)

where each subgraph $A_{1}$ contains the full triangle and each $A_{2}$
contains only a part of the triangle.

After the star-triangle transformation of the $M_{1}(G)$ and $M_{2}(G)$ we
obtain the following formula for the models $M_{1}(G^{\prime})$ and
$M_{2}(G^{\prime})$:

$\displaystyle Z^{1}_{n}(G^{\prime})=\prod_{e\in
G^{\prime}}q^{\prime}_{e}\sum\limits_{A_{1}^{\prime}\subseteq
G^{\prime}}\prod_{e\in
A^{\prime}_{1}}\frac{p^{\prime}_{e}}{q^{\prime}_{e}}Z^{2}_{n}(A_{1}^{\prime})+\prod_{e\in
G^{\prime}}q^{\prime}_{e}\sum\limits_{A^{\prime}_{2}\subseteq
G^{\prime}}\prod_{e\in
A^{\prime}_{2}}\frac{p^{\prime}_{e}}{q^{\prime}_{e}}Z^{2}_{n}(A_{2}^{\prime}),$
(5.3)

where each subgraph $A^{\prime}_{1}$ contains the full star and each
$A^{\prime}_{2}$ contains only a part of the star. Then, we compare the terms
of these formulas:

  * •

We notice that due to the star-triangle transformation
$Z_{n}^{i}(G)=Z_{n}^{i}(G^{\prime})$ and
$Z_{n}^{i}(A_{1})=Z_{n}^{i}(A_{1}^{\prime})$ (here and below $A_{1}$ is
different from $A_{1}^{\prime}$ only by the star-triangle transformation).

  * •

Also it is easy to see that $\prod_{e\in G}q_{e}\frac{1}{\prod_{e\in
A_{1}}q_{e}}=\prod_{e\in G^{\prime}}q^{\prime}_{e}\frac{1}{\prod_{e\in
A_{1}^{\prime}}q^{\prime}_{e}}$.

  * •

If the model $M_{1}(G)$ is chosen such that
$p_{1}p_{2}p_{3}=p_{1}^{\prime}p_{2}^{\prime}p_{3}^{\prime}$, we conclude that
$\prod_{e\in A_{1}}p_{e}=\prod_{e\in A^{\prime}_{1}}p^{\prime}_{e}$.

Now, we are ready to formulate the following theorem:

######  Theorem 5.1.

Consider two $n$-Potts models $M_{2}(G)$ and $M_{2}(G^{\prime})$, which are
different from each other by the star-triangle transformation. Then, the
following formula holds

$\displaystyle
q_{1}q_{2}q_{3}\bigg{(}Z^{2}_{n}(G_{0})+\frac{p_{1}}{q_{1}}Z^{2}_{n}(G_{1})+\frac{p_{2}}{q_{2}}Z^{2}_{n}(G_{2})+\frac{p_{3}}{q_{3}}Z^{2}_{n}(G_{3})+\frac{p_{1}p_{2}}{q_{1}q_{2}}Z^{2}_{n}(G_{12})+\frac{p_{1}p_{3}}{q_{1}q_{3}}Z^{2}_{n}(G_{13})$
$\displaystyle\hphantom{q_{1}q_{2}q_{3}\bigg{(}}{}+\frac{p_{2}p_{3}}{q_{2}q_{3}}Z^{2}_{n}(G_{23})\bigg{)}=q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime}\bigg{(}Z^{2}_{n}(G^{\prime}_{0})+\frac{p^{\prime}_{1}}{q^{\prime}_{1}}Z^{2}_{n}(G^{\prime}_{1})+\frac{p^{\prime}_{2}}{q^{\prime}_{2}}Z^{2}_{n}(G_{2})+\frac{p^{\prime}_{3}}{q^{\prime}_{3}}Z^{2}_{n}(G^{\prime}_{3})$
$\displaystyle\hphantom{q_{1}q_{2}q_{3}\bigg{(}}+\frac{p^{\prime}_{1}p^{\prime}_{2}}{q^{\prime}_{1}q^{\prime}_{2}}Z^{2}_{n}(G^{\prime}_{12})+\frac{p^{\prime}_{1}p^{\prime}_{3}}{q^{\prime}_{1}q^{\prime}_{3}}Z^{2}_{n}(G^{\prime}_{13})+\frac{p^{\prime}_{2}p^{\prime}_{3}}{q^{\prime}_{2}q^{\prime}_{3}}Z^{2}_{n}(G^{\prime}_{23})\bigg{)},$
(5.4)

where

$\displaystyle
p_{i}=\frac{\alpha^{1}_{i}-\beta^{1}_{i}}{\alpha^{2}_{i}-\beta^{2}_{i}},\qquad
q_{i}=\frac{\alpha^{2}_{i}\beta^{1}_{i}-\alpha^{1}_{i}\beta^{2}_{i}}{\alpha^{2}_{i}-\beta^{2}_{i}},\qquad
p^{\prime}_{i}=\frac{\alpha^{\prime 1}_{i}-\beta^{\prime
1}_{i}}{\alpha^{\prime 2}_{i}-\beta^{\prime 2}_{i}},\qquad
q_{i}^{\prime}=\frac{\alpha^{\prime 2}_{i}\beta^{\prime 1}_{i}-\alpha^{\prime
1}_{i}\beta^{\prime 2}_{i}}{\alpha^{\prime 2}_{i}-\beta^{\prime 2}_{i}},$

variables $\alpha^{k}_{i}$, $\beta^{k}_{i}$ and $\alpha^{\prime k}_{i}$,
$\beta^{\prime k}_{i}$ are related by the star-triangle transformation with
the condition $p_{1}p_{2}p_{3}=p_{1}^{\prime}p_{2}^{\prime}p_{3}^{\prime}$ and
graphs $G_{i}$ and $G_{ij}$, $i,j=0,1,2,3$ are depicted on the Figure 14.

Figure 14: The 14-term relation.

###### Proof.

We will prove this theorem using induction on $ex(G):=e(G)-3$.

We show that the base of the induction $k=0$ is trivial. Hence, let us
consider the $n$-Potts models $M_{2}(G)$ and $M_{2}(G^{\prime})$ and the
special models $M_{1}(G)$ and $M_{1}(G^{\prime})$ such that
$p_{1}p_{2}p_{3}=p_{1}^{\prime}p_{2}^{\prime}p_{3}^{\prime}$. Then, we write
the formulas (5.2) and (5.3), after the comparison for each terms using the
reasoning above we immediately obtain the result in the case $ex(G)=0$.

Then, make the step of induction. Again, let us write down the formulas (5.2)
and (5.3):

$\displaystyle Z^{1}_{n}(G)=\prod_{e\in G}q_{e}\sum\limits_{A_{1}\subseteq
G}\prod_{e\in A_{1}}\frac{p_{e}}{q_{e}}Z^{2}_{n}(A_{1})+\prod_{e\in
G}q_{e}\sum\limits_{A_{2}\subseteq G}\prod_{e\in
A_{2}}\frac{p_{e}}{q_{e}}Z^{2}_{n}(A_{2})+\frac{\prod_{e\in
G}q_{e}}{q_{1}q_{2}q_{3}}S_{1},$

where each $A_{1}$ contains the full triangle, each $A_{2}$ contains only a
part of the triangle and such that $ex(A_{2})\neq ex(G)$, and by $S_{1}$ we
denoted the left hand side of (5.4),

$\displaystyle Z^{1}_{n}(G^{\prime})=\prod_{e\in
G^{\prime}}q_{e}^{\prime}\sum\limits_{A_{1}^{\prime}\subseteq
G^{\prime}}\prod_{e\in
A^{\prime}_{1}}\frac{p^{\prime}_{e}}{q^{\prime}_{e}}Z^{2}_{n}(A_{1}^{\prime})+\prod_{e\in
G^{\prime}}q_{e}^{\prime}\sum\limits_{A^{\prime}_{2}\subseteq
G^{\prime}}\prod_{e\in
A^{\prime}_{2}}\frac{p^{\prime}_{e}}{q^{\prime}_{e}}Z^{2}_{n}(A_{2}^{\prime})+\frac{\prod_{e\in
G^{\prime}}q^{\prime}_{e}}{q^{\prime}_{1}q^{\prime}_{2}q^{\prime}_{3}}S_{2},$

where each $A^{\prime}_{1}$ contains the full star, each $A^{\prime}_{2}$
contains only a part of the star and such that $ex(A^{\prime}_{2})\neq
ex(G^{\prime})$, and by $S_{2}$ we denoted the right hand side of (5.4).

Then the induction assumption ends the proof. ∎

We consider these results in the context of numerous generalizations, both for
other models of statistical physics, and in a purely mathematical direction.
In particular, we are interested in applying this technique to the Potts model
in the presence of an external magnetic field [10], including an inhomogeneous
one. In addition, we are going to develop these methods in a more general
algebraic sense, in particular in a non-commutative situation. Partial results
of this activity have already been obtained in [6].

##  Appendix A The proof of Lemma 4.3

###### Proof.

We start by reformulating this statement in terms of equivalent rational
identities. Let us introduce the $x$-variables by the following formula

$\displaystyle F\circ S^{3}(t_{1},t_{2},t_{3})=(x_{1},x_{2},x_{3}),$
$\displaystyle
S^{3}(x_{1},x_{2},x_{3})=(t_{1}^{\prime},t_{2}^{\prime},t_{3}^{\prime}).$
(A.1)

Here $S^{3}(t_{1},t_{2},t_{3})=(S\times S\times
S)(t_{1},t_{2},t_{3})=(S(t_{1}),S(t_{2}),S(t_{3}))$. Then the statement of the
lemma is equivalent to

$\displaystyle(t_{1}^{\prime},t_{2}^{\prime},t_{3}^{\prime})=S^{3}(x_{1},x_{2},x_{3})=F^{-1}(t_{1},t_{2},t_{3}).$

This identity is equivalent to three algebraic relations (we will write down
only one of them, because the others differ just by replacing the indices)

$\displaystyle
t_{1}t_{2}=\frac{t_{1}^{\prime}t_{2}^{\prime}t_{3}^{\prime}+1}{t_{3}^{\prime}+t_{1}^{\prime}t_{2}^{\prime}}=\bigg{(}\frac{(x_{1}-1)(x_{2}-1)(x_{3}-1)}{(x_{1}+1)(x_{2}+1)(x_{3}+1)}+1\bigg{)}\bigg{/}\bigg{(}\frac{x_{3}-1}{x_{3}+1}+\frac{(x_{1}-1)(x_{2}-1)}{(x_{1}+1)(x_{2}+1)}\bigg{)}$
$\displaystyle\hphantom{t_{1}t_{2}}{}=\frac{x_{1}+x_{2}+x_{3}+x_{1}x_{2}x_{3}}{x_{3}-x_{2}-x_{1}+x_{1}x_{2}x_{3}}=\frac{(x_{1}+x_{2}+x_{3}+x_{1}x_{2}x_{3})x_{1}x_{2}x_{3}}{(x_{3}-x_{2}-x_{1}+x_{1}x_{2}x_{3})x_{1}x_{2}x_{3}}.$
(A.2)

Now let us introduce some additional variables

$\displaystyle t_{12}=x_{1}x_{2},\qquad t_{23}=x_{2}x_{3},\qquad
t_{13}=x_{1}x_{3},\qquad a_{1}=x_{1}^{2},\qquad a_{2}=x_{2}^{2},\qquad
a_{3}=x_{3}^{2}.$

We could rewrite (A.2) in the following way

$\displaystyle
t_{1}t_{2}=\frac{a_{1}t_{23}+a_{2}t_{13}+a_{3}t_{12}+t_{12}t_{23}t_{13}}{-a_{2}t_{13}-a_{1}t_{23}+a_{3}t_{12}+t_{12}t_{23}t_{13}}.$

The equations (4.2) and (A.1) with the identification $y_{i}:=S(t_{i})$
provide the following system

$\displaystyle
t_{12}=\frac{y_{1}y_{2}y_{3}+1}{y_{3}+y_{1}y_{2}}=\frac{t_{3}+t_{2}+t_{1}+t_{1}t_{2}t_{3}}{t_{3}-t_{2}-t_{1}+t_{1}t_{2}t_{3}},$
$\displaystyle
t_{13}=\frac{y_{1}y_{2}y_{3}+1}{y_{2}+y_{3}y_{1}}=\frac{t_{3}+t_{2}+t_{1}+t_{1}t_{2}t_{3}}{t_{2}-t_{1}-t_{3}+t_{1}t_{2}t_{3}},$
$\displaystyle
t_{23}=\frac{y_{1}y_{2}y_{3}+1}{y_{1}+y_{3}y_{2}}=\frac{t_{3}+t_{2}+t_{1}+t_{1}t_{2}t_{3}}{t_{1}-t_{2}-t_{3}+t_{1}t_{2}t_{3}},$
$\displaystyle
a_{1}=\frac{(y_{1}y_{2}y_{3}+1)(y_{1}+y_{2}y_{3})}{(y_{2}+y_{1}y_{3})(y_{3}+y_{1}y_{2})}=\frac{(t_{1}-t_{2}-t_{3}+t_{1}t_{2}t_{3})(t_{3}+t_{2}+t_{1}+t_{1}t_{2}t_{3})}{(t_{3}-t_{2}-t_{1}+t_{1}t_{2}t_{3})(t_{2}-t_{1}-t_{3}+t_{1}t_{2}t_{3})},$
$\displaystyle
a_{2}=\frac{(y_{1}y_{2}y_{3}+1)(y_{2}+y_{1}y_{3})}{(y_{1}+y_{2}y_{3})(y_{3}+y_{1}y_{2})}=\frac{(t_{2}-t_{1}-t_{3}+t_{1}t_{2}t_{3})(t_{3}+t_{2}+t_{1}+t_{1}t_{2}t_{3})}{(t_{3}-t_{2}-t_{1}+t_{1}t_{2}t_{3})(t_{1}-t_{2}-t_{3}+t_{1}t_{2}t_{3})},$
$\displaystyle
a_{3}=\frac{(y_{1}y_{2}y_{3}+1)(y_{3}+y_{2}y_{1})}{(y_{2}+y_{1}y_{3})(y_{1}+y_{3}y_{2})}=\frac{(t_{3}-t_{2}-t_{1}+t_{1}t_{2}t_{3})(t_{3}+t_{2}+t_{1}+t_{1}t_{2}t_{3})}{(t_{1}-t_{2}-t_{3}+t_{1}t_{2}t_{3})(t_{2}-t_{1}-t_{3}+t_{1}t_{2}t_{3})}.$

Using these expressions we can compute

$\displaystyle t_{12}a_{3}+t_{13}a_{2}+a_{1}t_{23}+t_{12}t_{23}t_{13}$
$\displaystyle\qquad{}=\frac{4(t_{3}+t_{2}+t_{1}+t_{1}t_{2}t_{3})^{2}t_{1}t_{2}t_{3}}{(t_{3}-t_{2}-t_{1}+t_{1}t_{2}t_{3})(t_{2}-t_{1}-t_{3}+t_{1}t_{2}t_{3})(t_{1}-t_{2}-t_{3}+t_{1}t_{2}t_{3})},$
$\displaystyle
t_{1}t_{2}(-a_{2}t_{13}-a_{1}t_{23}+a_{3}t_{12}+t_{12}t_{23}t_{13})$
$\displaystyle\qquad{}=\frac{4(t_{3}+t_{2}+t_{1}+t_{1}t_{2}t_{3})^{2}t_{1}t_{2}t_{3}}{(t_{3}-t_{2}-t_{1}+t_{1}t_{2}t_{3})(t_{2}-t_{1}-t_{3}+t_{1}t_{2}t_{3})(t_{1}-t_{2}-t_{3}+t_{1}t_{2}t_{3})}.$

In this way we observe that

$\displaystyle
t_{1}t_{2}(-a_{2}t_{13}-a_{1}t_{23}+a_{3}t_{12}+t_{12}t_{23}t_{13})=a_{3}t_{12}+a_{2}t_{13}+a_{1}t_{23}+t_{12}t_{23}t_{13}.$

This completes the proof. ∎

##  Appendix B For the second proof of the tetrahedron equation

In this Appendix we present some technical part of the proof from Section 4.3.
We present closed formulas for $t_{1}$ and $t_{2}$ variables (see the
discussion after (4.17))

$\displaystyle
t_{1}=\big{(}{-}b_{3}a_{3}+a_{2}b_{2}+a_{1}b_{1}-b_{4}a_{4}+(b_{3}^{2}a_{3}^{2}-2b_{3}a_{3}a_{2}b_{2}-2b_{3}a_{3}a_{1}b_{1}-2b_{3}a_{3}b_{4}a_{4}$
$\displaystyle\hphantom{t_{1}=(}{}+a_{2}^{2}b_{2}^{2}-2a_{2}b_{2}a_{1}b_{1}-2a_{2}b_{2}b_{4}a_{4}+a_{1}^{2}b_{1}^{2}-2a_{1}b_{1}b_{4}a_{4}+b_{4}^{2}a_{4}^{2}+4b_{4}a_{3}a_{1}b_{2}$
$\displaystyle\hphantom{t_{1}=(}{}+4a_{2}b_{1}b_{3}a_{4})^{1/2}\big{)}/(2(-b_{4}a_{3}+a_{2}b_{1})),$
(B.1) $\displaystyle
t_{2}=(a_{2}b_{4}-b_{2}a_{4}-b_{3}a_{1}+a_{3}b_{1}+(a_{2}^{2}b_{4}^{2}-2a_{4}b_{4}a_{2}b_{2}-2a_{2}b_{4}b_{3}a_{1}-2a_{2}b_{1}a_{3}b_{4}$
$\displaystyle\hphantom{t_{2}=(}{}+b_{2}^{2}a_{4}^{2}-2b_{3}a_{4}a_{1}b_{2}--2b_{2}a_{4}a_{3}b_{1}+b_{3}^{2}a_{1}^{2}-2a_{3}b_{3}a_{1}b_{1}+a_{3}^{2}b_{1}^{2}+4a_{3}b_{4}a_{1}b_{2}$
$\displaystyle\hphantom{t_{2}=(}{}+4a_{2}b_{1}b_{3}a_{4})^{1/2})/(2(a_{3}b_{4}-b_{3}a_{4})).$
(B.2)

### Acknowledgements

We are thankful to V. Gorbounov for indicating us the strategy of the first
proof of the tetrahedron equation in the trigonometric case in Section 4.2.
The research was supported by the Russian Science Foundation (project
20-61-46005). The authors thank the anonymous referees for very useful
comments which are improved the paper a lot.

## References

  * [1] Atiyah M., Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 175–186.
  * [2] Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982.
  * [3] Beaudin L., Ellis-Monaghan J., Pangborn G., Shrock R., A little statistical mechanics for the graph theorist, Discrete Math. 310 (2010), 2037–2053, arXiv:0804.2468.
  * [4] Berenstein A., Fomin S., Zelevinsky A., Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122 (1996), 49–149.
  * [5] Biggs N., Interaction models, London Mathematical Society Lecture Note Series, Vol. 30, Cambridge University Press, Cambridge – New York – Melbourne, 1977.
  * [6] Bychkov B., Kazakov A., Talalaev D., Tutte polynomials of vertex-weighted graphs and cohomology of groups, Theoret. and Math. Phys., to appear.
  * [7] Cardy J.L., Critical percolation in finite geometries, J. Phys. A: Math. Gen. 25 (1992), L201–L206, arXiv:hep-th/9111026.
  * [8] El-Showk S., Paulos M.F., Poland D., Rychkov S., Simmons-Duffin D., Vichi A., Solving the 3d Ising model with the conformal bootstrap II. $c$-minimization and precise critical exponents, J. Stat. Phys. 157 (2014), 869–914, arXiv:1403.4545.
  * [9] Ellis-Monaghan J.A., Merino C., Graph polynomials and their applications I: The Tutte polynomial, in Structural Analysis of Complex Networks, Birkhäuser/Springer, New York, 2011, 219–255, arXiv:0803.3079.
  * [10] Ellis-Monaghan J.A., Moffatt I., The Tutte–Potts connection in the presence of an external magnetic field, Adv. in Appl. Math. 47 (2011), 772–782, arXiv:1005.5470.
  * [11] Galashin P., Pylyavskyy P., Ising model and the positive orthogonal Grassmannian, Duke Math. J. 169 (2020), 1877–1942, arXiv:1807.03282.
  * [12] Gorbounov V., Talalaev D., Electrical varieties as vertex integrable statistical models, J. Phys. A: Math. Theor. 53 (2020), 454001, 28 pages, arXiv:1905.03522.
  * [13] Grimmett G., Three theorems in discrete random geometry, Probab. Surv. 8 (2011), 403–441, arXiv:1110.2395.
  * [14] Huang Y.-T., Wen C., Xie D., The positive orthogonal Grassmannian and loop amplitudes of ABJM, J. Phys. A: Math. Theor. 47 (2014), 474008, 48 pages, arXiv:1402.1479.
  * [15] Kashaev R.M., On discrete three-dimensional equations associated with the local Yang–Baxter relation, Lett. Math. Phys. 38 (1996), 389–397, arXiv:solv-int/9512005.
  * [16] Kook W., Reiner V., Stanton D., A convolution formula for the Tutte polynomial, J. Combin. Theory Ser. B 76 (1999), 297–300, arXiv:math.CO/9712232.
  * [17] Korepanov I.G., Algebraic integrable dynamical systems, 2+1-dimensional models in wholly discrete space-time, and inhomogeneous models in 2-dimensional statistical physics, arXiv:solv-int/9506003.
  * [18] Korepanov I.G., Sharygin G.I., Talalaev D.V., Cohomologies of $n$-simplex relations, Math. Proc. Cambridge Philos. Soc. 161 (2016), 203–222, arXiv:1409.3127.
  * [19] Lam T., Pylyavskyy P., Inverse problem in cylindrical electrical networks, SIAM J. Appl. Math. 72 (2012), 767–788, arXiv:1104.4998.
  * [20] Matiyasevich Yu.V., On a certain representation of the chromatic polynomial, arXiv:0903.1213.
  * [21] Sergeev S.M., Solutions of the functional tetrahedron equation connected with the local Yang–Baxter equation for the ferro-electric condition, Lett. Math. Phys. 45 (1998), 113–119, arXiv:solv-int/9709006.
  * [22] Sokal A.D., The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, in Surveys in Combinatorics 2005, London Math. Soc. Lecture Note Ser., Vol. 327, Cambridge University Press, Cambridge, 2005, 173–226, arXiv:math.CO/0503607.
  * [23] Tutte W.T., A ring in graph theory, in Classic Papers in Combinatorics, Modern Birkhäuser Classics, Birkhäuser, Boston, 2009, 124–138.
  * [24] Wu F.Y., Knot theory and statistical mechanics, Rev. Modern Phys. 64 (1992), 1099–1131.
  * [25] Zamolodchikov A.B., Tetrahedra equations and integrable systems in three-dimensional space, Soviet Phys. JETP 52 (1980), 325–336.