### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 035, 30 pages      arXiv:2005.10288      https://doi.org/10.3842/SIGMA.2021.035

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

Boris Bychkov ab, Anton Kazakov abc and Dmitry Talalaev abc
a) Faculty of Mathematics, National Research University Higher School of Economics, Usacheva 6, 119048, Moscow, Russia
b) Centre of Integrable Systems, P.G. Demidov Yaroslavl State University, 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

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.

Key words: tetrahedron equation; local Yang-Baxter equation; Biggs formula; Potts model; Ising model.

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