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

2020/05/31 by Boris Bychkov, Anton Kazakov, Dmitry V. Talalaev +1
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Anisotropy #Chiral Potts curve #Geometry #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Physics #Potts model #Quantum mechanics #Statistical physics #Tetrahedron #Theoretical and Computational Physics #math-ph #math.MP #msc:13F60 #msc:14H70 #msc:14M17 #msc:16S99 #msc:82B20

paper · pdf · doi:10.3842/sigma.2021.035

published as SIGMA 17 (2021), 035, 30 pages

arxiv created 2021/04/07 · openalex publication_date 2021/04/07 · arxiv updated 2021/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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

Citations

• Algebraic integrable dynamical systems, 2+1-dimensional models in wholly discrete space-time, and inhomogeneous models in 2-dimensional statistical physics
• Graph polynomials and their applications I: The Tutte polynomial
• The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
• Topological quantum field theory
• Critical percolation in finite geometries
• Solving the 3d Ising Model with the Conformal Bootstrap II. c c -Minimization and Precise Critical Exponents
• Solutions of the functional tetrahedron equation connected with the local Yang -- Baxter equation for the ferro-electric
• Topological quantum field theory
• Graph Polynomials and Their Applications I: The Tutte Polynomial
• Knot theory and statistical mechanics