Finiteness of canonical quotients of Dehn quandles of surfaces
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916745421258752 |
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| author | Dhanwani, Neeraj K. Singh, Mahender |
| author_facet | Dhanwani, Neeraj K. Singh, Mahender |
| contents | The Dehn quandle of a closed orientable surface is the set of isotopy classes of non-separating simple closed curves with a natural quandle structure arising from Dehn twists. In this paper, we consider finiteness of some canonical quotients of these quandles. For a surface of positive genus, we give a precise description of the 2-quandle of its Dehn quandle. Further, with some exceptions for genus more than two, we determine all values of $n$ for which the $n$-quandle of its Dehn quandle is finite. The result can be thought of as the Dehn quandle analogue of a similar result of Hoste and Shanahan for link quandles. We also compute the size of the smallest non-trivial quandle quotient of the Dehn quandle of a surface. Along the way, we prove that the involutory quotient of an Artin quandle is precisely the corresponding Coxeter quandle and also determine the smallest non-trivial quotient of a braid quandle. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_12050 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Finiteness of canonical quotients of Dehn quandles of surfaces Dhanwani, Neeraj K. Singh, Mahender Geometric Topology Group Theory Quantum Algebra Primary 57K10, 57K20, Secondary 57K12 The Dehn quandle of a closed orientable surface is the set of isotopy classes of non-separating simple closed curves with a natural quandle structure arising from Dehn twists. In this paper, we consider finiteness of some canonical quotients of these quandles. For a surface of positive genus, we give a precise description of the 2-quandle of its Dehn quandle. Further, with some exceptions for genus more than two, we determine all values of $n$ for which the $n$-quandle of its Dehn quandle is finite. The result can be thought of as the Dehn quandle analogue of a similar result of Hoste and Shanahan for link quandles. We also compute the size of the smallest non-trivial quandle quotient of the Dehn quandle of a surface. Along the way, we prove that the involutory quotient of an Artin quandle is precisely the corresponding Coxeter quandle and also determine the smallest non-trivial quotient of a braid quandle. |
| title | Finiteness of canonical quotients of Dehn quandles of surfaces |
| topic | Geometric Topology Group Theory Quantum Algebra Primary 57K10, 57K20, Secondary 57K12 |
| url | https://arxiv.org/abs/2208.12050 |