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FINITENESS OF CANONICAL QUOTIENTS OF DEHN QUANDLES OF SURFACES

Published online by Cambridge University Press:  11 March 2024

NEERAJ K. DHANWANI*
Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S. A. S. Nagar, P. O. Manauli, Punjab 140306, India
MAHENDER SINGH
Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S. A. S. Nagar, P. O. Manauli, Punjab 140306, India e-mail: mahender@iisermohali.ac.in

Abstract

The Dehn quandle of a closed orientable surface is the set of isotopy classes of nonseparating simple closed curves with a natural quandle structure arising from Dehn twists. In this paper, we consider the 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 2, 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 [‘Links with finite n-quandles’, Algebr. Geom. Topol. 17(5) (2017), 2807–2823]. We also compute the size of the smallest nontrivial 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 nontrivial quotient of a braid quandle.

Type
Research Article
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

N.K.D. acknowledges support from the NBHM via grant number 0204/1/2023/R &D-II/606. M.S. is supported by the Swarna Jayanti Fellowship grants DST/SJF/MSA-02/2018-19 and SB/SJF/2019-20/04.

Communicated by Ben Martin

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