Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation
We explore several types of functional relations on the family of multivariate Tutte polynomials: the Biggs formula and the star-triangle ( − Δ) transformation at the critical point = 2. We deduce the theorem of Matiyasevich and its inverse from the Biggs formula, and we apply this relation to cons...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2021 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2021
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211314 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation. Boris Bychkov, Anton Kazakov and Dmitry Talalaev. SIGMA 17 (2021), 035, 30 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862695562390798336 |
|---|---|
| author | Bychkov, Boris Kazakov, Anton Talalaev, Dmitry |
| author_facet | Bychkov, Boris Kazakov, Anton Talalaev, Dmitry |
| citation_txt | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation. Boris Bychkov, Anton Kazakov and Dmitry Talalaev. SIGMA 17 (2021), 035, 30 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We explore several types of functional relations on the family of multivariate Tutte polynomials: the Biggs formula and the star-triangle ( − Δ) transformation at the critical point = 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 = 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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| first_indexed | 2026-03-18T06:05:41Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211314 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T06:05:41Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bychkov, Boris Kazakov, Anton Talalaev, Dmitry 2025-12-29T11:09:12Z 2021 Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation. Boris Bychkov, Anton Kazakov and Dmitry Talalaev. SIGMA 17 (2021), 035, 30 pages 1815-0659 2020 Mathematics Subject Classification: 82B20; 16T25; 05C31 arXiv:2005.10288 https://nasplib.isofts.kiev.ua/handle/123456789/211314 https://doi.org/10.3842/SIGMA.2021.035 We explore several types of functional relations on the family of multivariate Tutte polynomials: the Biggs formula and the star-triangle ( − Δ) transformation at the critical point = 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 = 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. We are thankful to V. Gorbounov for indicating to us the strategy of the first proof of the tetrahedron equation in the trigonometric case in Section 4.2. The research was supported by the Russian Science Foundation (project 20-61-46005). The authors thank the anonymous referees for their very useful comments, which have improved the paper a lot. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation Article published earlier |
| spellingShingle | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation Bychkov, Boris Kazakov, Anton Talalaev, Dmitry |
| title | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation |
| title_full | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation |
| title_fullStr | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation |
| title_full_unstemmed | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation |
| title_short | Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation |
| title_sort | functional relations on anisotropic potts models: from biggs formula to the tetrahedron equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211314 |
| work_keys_str_mv | AT bychkovboris functionalrelationsonanisotropicpottsmodelsfrombiggsformulatothetetrahedronequation AT kazakovanton functionalrelationsonanisotropicpottsmodelsfrombiggsformulatothetetrahedronequation AT talalaevdmitry functionalrelationsonanisotropicpottsmodelsfrombiggsformulatothetetrahedronequation |