Presentations of Dehn quandles

Abstract

Quandles are non-associative algebraic structures with defining axioms modelled on the three Reidemeister moves of planar diagrams of links in the 3-space. The paper gives two approaches to write efficient presentations for a large class of quandles, called Dehn quandles, using presentations of their underlying groups. The first approach gives finite presentations for Dehn quandles of a class of Garside groups and Gaussian groups. The second approach is for general Dehn quandles when the centralisers of generators in their underlying groups are known. Several examples including Dehn quandles of spherical Artin groups, surface groups and mapping class groups of orientable surfaces are given to illustrate the results.