Dehn quandles of groups and orientable surfaces

TL;DR

This paper introduces Dehn quandles for groups and surfaces, showing their algebraic properties, automorphisms, and relations to known structures like knot and surface quandles, unifying various quandle constructions.

Contribution

It defines Dehn quandles for groups and surfaces, analyzes their properties, automorphisms, and embeddings, and connects them to existing quandle theories and topological applications.

Findings

Dehn quandles embed naturally into their enveloping groups.

Enveloping groups of Dehn quandles are central extensions or the groups themselves.

Automorphism groups of Dehn quandles are computed for surfaces.

Abstract

Unifying various constructions of quandles including Coxeter quandles, free quandles, knot quandles of prime knots and Dehn quandles of orientable surfaces, we introduce Dehn quandles of groups with respect to their subsets. It turns out that Dehn quandles are precisely the ones that embed naturally into their enveloping groups. We prove that the enveloping group of the Dehn quandle of a given group with respect to its generating set is a central extension of that group, and that enveloping groups of Dehn quandles of Artin groups and link groups with respect to their standard generating sets are the groups themselves. We discuss orderability of Dehn quandles and prove that free involutory quandles are left orderable, whereas certain generalised Alexander quandles are bi-orderable. Specialising to surfaces, we give generating sets for Dehn quandles of orientable surfaces with punctures and compute their automorphism groups. As applications, we recover a result of Niebrzydowski and Przytycki proving that the knot quandle of the trefoil knot is isomorphic to the Dehn quandle of the torus and also extend a result of Yetter on epimorphisms of Dehn quandles of orientable surfaces onto certain involutory homological quandles.