Finiteness of canonical quotients of Dehn quandles of surfaces

TL;DR

This paper investigates the finiteness properties of canonical quotients of Dehn quandles associated with surfaces, providing classifications and size computations for various quotients, and establishing connections with Artin and Coxeter quandles.

Contribution

It offers a detailed description of the 2-quandle of Dehn quandles and classifies all finite n-quandles for surfaces of genus greater than two, extending known results to a new context.

Findings

The 2-quandle of a surface's Dehn quandle is explicitly described.

All finite n-quandles are classified for surfaces of genus > 2, with some exceptions.

The smallest non-trivial quandle quotient of a surface's Dehn quandle is computed.

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.