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arXiv · 2005.10288

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

Boris Bychkov·Anton Kazakov·Dmitry Talalaev
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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.

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Boris Bychkov, Anton Kazakov, Dmitry Talalaev. 2020-05-20. Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation. https://doi.org/10.3842/sigma.2021.035

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