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

Boris Bychkov ; Anton Kazakov ; Dmitry Talalaev

Symmetry, integrability and geometry: methods and applications, Tome 17 (2021)

Résumé

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.

Détail

Keywords: local Yang–Baxter equation, Biggs formula, Potts model, Ising model.
Mots-clés : tetrahedron equation

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