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

Abstract

We explore several types of functional relations on the family of multivariate Tutte polynomials: the Biggs formula and the star-triangle (𝑌 − Δ) 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.