Functional relations on anisotropic Potts models: from Biggs formula to Zamolodchikov equation.

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, 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, extend the latter to the case of valency 2 points and show that the Biggs formula and the star-triangle transformation commute.