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

TL;DR

This paper investigates functional relations in multivariate Tutte polynomials, deriving new theorems, exploring transformations like the star-triangle, and establishing the tetrahedron equation, with implications for anisotropic Potts models.

Contribution

It introduces new proofs and extensions of the Biggs formula and star-triangle transformation, and demonstrates their commutation and relation to the tetrahedron equation.

Findings

Derived the inverse of Matiyasevich's theorem from Biggs formula.

Proved the tetrahedron equation for the star-triangle transformation at n=2.

Extended the tetrahedron equation to valency 2 points.

Abstract

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.