Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder
We introduce twisted triple crossing diagram maps, collections of points in projective space associated to bipartite graphs on the cylinder, and use them to provide geometric realizations of the cluster integrable systems of Goncharov and Kenyon constructed from toric dimer models. Using this notion...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2025 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2025
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/213198 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder. Niklas Christoph Affolter, Terrence George and Sanjay Ramassamy. SIGMA 21 (2025), 040, 48 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862709407451709440 |
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| author | Affolter, Niklas Christoph George, Terrence Ramassamy, Sanjay |
| author_facet | Affolter, Niklas Christoph George, Terrence Ramassamy, Sanjay |
| citation_txt | Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder. Niklas Christoph Affolter, Terrence George and Sanjay Ramassamy. SIGMA 21 (2025), 040, 48 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We introduce twisted triple crossing diagram maps, collections of points in projective space associated to bipartite graphs on the cylinder, and use them to provide geometric realizations of the cluster integrable systems of Goncharov and Kenyon constructed from toric dimer models. Using this notion, we provide geometric proofs that the pentagram map and the cross-ratio dynamics integrable systems are cluster integrable systems. We show that in appropriate coordinates, cross-ratio dynamics is described by geometric -matrices, which solves the open question of finding a cluster algebra structure describing cross-ratio dynamics.
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| first_indexed | 2026-03-19T09:42:17Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-213198 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T09:42:17Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
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| spelling | Affolter, Niklas Christoph George, Terrence Ramassamy, Sanjay 2026-02-16T16:38:33Z 2025 Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder. Niklas Christoph Affolter, Terrence George and Sanjay Ramassamy. SIGMA 21 (2025), 040, 48 pages 1815-0659 2020 Mathematics Subject Classification: 37J70; 82B20; 13F60 arXiv:2108.12692 https://nasplib.isofts.kiev.ua/handle/123456789/213198 https://doi.org/10.3842/SIGMA.2025.040 We introduce twisted triple crossing diagram maps, collections of points in projective space associated to bipartite graphs on the cylinder, and use them to provide geometric realizations of the cluster integrable systems of Goncharov and Kenyon constructed from toric dimer models. Using this notion, we provide geometric proofs that the pentagram map and the cross-ratio dynamics integrable systems are cluster integrable systems. We show that in appropriate coordinates, cross-ratio dynamics is described by geometric -matrices, which solves the open question of finding a cluster algebra structure describing cross-ratio dynamics. TG thanks Nick Ovenhouse for discussions about networks in a cylinder. SR thanks Ivan Izmestiev for discussions on cross-ratio dynamics during a visit to TU Wien, Anton Izosimov for comments on Newton polygons in the first version of this paper, and Rei Inoue for exchanges on the geometric R-matrix. NA was supported by the Deutsche Forschungsgemeinschaft (DFG) Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics” and by the ENS-MHI chair funded by MHI. NA and SR were partially supported by the Agence Nationale de la Recherche, Grant Number ANR-18-CE40-0033 (ANR DIMERS). SR was also partially supported by the CNRS grant Tremplin@INP, which funded a visit of NA to Paris-Saclay. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder Article published earlier |
| spellingShingle | Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder Affolter, Niklas Christoph George, Terrence Ramassamy, Sanjay |
| title | Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder |
| title_full | Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder |
| title_fullStr | Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder |
| title_full_unstemmed | Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder |
| title_short | Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder |
| title_sort | integrable dynamics in projective geometry via dimers and triple crossing diagram maps on the cylinder |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213198 |
| work_keys_str_mv | AT affolterniklaschristoph integrabledynamicsinprojectivegeometryviadimersandtriplecrossingdiagrammapsonthecylinder AT georgeterrence integrabledynamicsinprojectivegeometryviadimersandtriplecrossingdiagrammapsonthecylinder AT ramassamysanjay integrabledynamicsinprojectivegeometryviadimersandtriplecrossingdiagrammapsonthecylinder |