Аннотация:
We explore several types of functional relations on the family of multivariate Tutte polynomials: the Biggs formula and the star-triangle (Y−Δ) 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.
Образец цитирования:
Boris Bychkov, Anton Kazakov, Dmitry Talalaev, “Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation”, SIGMA, 17 (2021), 035, 30 pp.
\RBibitem{BycKazTal21}
\by Boris~Bychkov, Anton~Kazakov, Dmitry~Talalaev
\paper Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation
\jour SIGMA
\yr 2021
\vol 17
\papernumber 035
\totalpages 30
\mathnet{http://mi.mathnet.ru/sigma1718}
\crossref{https://doi.org/10.3842/SIGMA.2021.035}
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https://www.mathnet.ru/rus/sigma1718
https://www.mathnet.ru/rus/sigma/v17/p35
Эта публикация цитируется в следующих 5 статьяx:
А. А. Казаков, “Дуальность Крамерса–Ванье и полиномы Татта”, ТМФ, 220:2 (2024), 286–297; A. A. Kazakov, “Kramers–Wannier duality and Tutte polynomials”, Theoret. and Math. Phys., 220:2 (2024), 1304–1314
E. Yu. Lerner, “Matroid Variant of Matiyasevich Formula and Its Application”, Lobachevskii J Math, 45:8 (2024), 3912
Dmitry V. Talalaev, Studies in Computational Intelligence, 1008, Advances in Neural Computation, Machine Learning, and Cognitive Research V, 2022, 126
E. Yu. Lerner, S. A. Mukhamedjanova, “Matiyasevich Formula for Chromatic and Flow Polynomials and Feynman Amplitudes”, Lobachevskii J Math, 43:12 (2022), 3552
Д. В. Талалаев, “Уравнение тетраэдров: алгебра, топология и интегрируемость”, УМН, 76:4(460) (2021), 139–176; D. V. Talalaev, “Tetrahedron equation: algebra, topology, and integrability”, Russian Math. Surveys, 76:4 (2021), 685–721