Dehn quandles of surfaces and their bounded cohomology
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918346047356928 |
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| author | Kapari, Pankaj Saraf, Deepanshi Singh, Mahender |
| author_facet | Kapari, Pankaj Saraf, Deepanshi Singh, Mahender |
| contents | We introduce new families of quandles that serve as invariants for classifying closed orientable surfaces. These families generalize the classical Dehn quandle and are defined, respectively, on isotopy classes of unoriented closed curves and on integral weighted multicurves. We establish their fundamental algebraic properties and construct a natural quandle covering that relates them. We then analyze their metric properties, showing that these quandles are unbounded with respect to the quandle metric. Next, we compute their second bounded quandle cohomology, proving it to be infinite-dimensional. We also establish a version of the Gromov Mapping Theorem, showing that the natural map from an abelian quandle extension onto the original quandle induces an injection on bounded quandle cohomology in every dimension. Finally, inspired by recent developments in quandle rings, we analyze idempotents in the integral quandle rings arising from the classical Dehn quandle of a surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17661 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dehn quandles of surfaces and their bounded cohomology Kapari, Pankaj Saraf, Deepanshi Singh, Mahender Geometric Topology Group Theory Quantum Algebra 57K12, 57K20 We introduce new families of quandles that serve as invariants for classifying closed orientable surfaces. These families generalize the classical Dehn quandle and are defined, respectively, on isotopy classes of unoriented closed curves and on integral weighted multicurves. We establish their fundamental algebraic properties and construct a natural quandle covering that relates them. We then analyze their metric properties, showing that these quandles are unbounded with respect to the quandle metric. Next, we compute their second bounded quandle cohomology, proving it to be infinite-dimensional. We also establish a version of the Gromov Mapping Theorem, showing that the natural map from an abelian quandle extension onto the original quandle induces an injection on bounded quandle cohomology in every dimension. Finally, inspired by recent developments in quandle rings, we analyze idempotents in the integral quandle rings arising from the classical Dehn quandle of a surface. |
| title | Dehn quandles of surfaces and their bounded cohomology |
| topic | Geometric Topology Group Theory Quantum Algebra 57K12, 57K20 |
| url | https://arxiv.org/abs/2602.17661 |