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Finiteness of canonical quotients of Dehn quandles of surfaces

Finiteness of canonical quotients of Dehn quandles of surfaces

Authors :
Dhanwani, Neeraj K.
Singh, Mahender
Source :
Journal of the Australian Mathematical Society (2024), 18 pp
Publication Year :
2022

Abstract

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.<br />Comment: 16 pages and 3 figures

Details

Database :
arXiv
Journal :
Journal of the Australian Mathematical Society (2024), 18 pp
Publication Type :
Report
Accession number :
edsarx.2208.12050
Document Type :
Working Paper
Full Text :
https://doi.org/10.1017/S144678872400003X