Abelian quandles and quandles with abelian structure group
Victoria Lebed, Arnaud Mortier
TL;DR
The paper classifies finite quandles with abelian structure groups by proving such quandles are necessarily abelian and are parametrisable as filtered-permutation (FP) quandles via explicit matrices. It describes the structure group $G(X,\triangleleft)$ as a central extension of $\mathbb{Z}^r$ by a finite abelian group $G'(X,\triangleleft)$, which equals the commutator subgroup; the abelian case corresponds to $G'(X,\triangleleft)$ being trivial. A deep link to rack homology is developed through path maps, showing $H_2(X,\triangleleft)$ decomposes with torsion pieces controlled by $G'(X,\triangleleft)$. Consequently, abelian-structure quandles have torsion-free $H_2$, while general abelian quandles can exhibit torsion in $H_2$. The paper also gives concrete criteria for $r=3$ and analyzes examples such as $U_{m,n}$ and its variants, illustrating how $G'(X,\triangleleft)$ and $H_2$ interact in practice and highlighting implications for knot invariants and Hopf algebras.
Abstract
Sets with a self-distributive operation (in the sense of $(a \triangleleft b) \triangleleft c = (a \triangleleft c) \triangleleft (b \triangleleft c))$, in particular quandles, appear in knot and braid theories, Hopf algebra classification, the study of the Yang-Baxter equation, and other areas. An important invariant of quandles is their structure group. The structure group of a finite quandle is known to be either "boring" (free abelian), or "interesting" (non-abelian with torsion). In this paper we explicitly describe all finite quandles with abelian structure group. To achieve this, we show that such quandles are abelian (i.e., satisfy $(a \triangleleft b) \triangleleft c = (a \triangleleft c) \triangleleft b)$; present the structure group of any abelian quandle as a central extension of a free abelian group by an explicit finite abelian group; and determine when the latter is trivial. In the second part of the paper, we relate the structure group of any quandle to its 2nd homology group $H_2$. We use this to prove that the $H_2$ of a finite quandle with abelian structure group is torsion-free, but general abelian quandles may exhibit torsion. Torsion in $H_2$ is important for constructing knot invariants and pointed Hopf algebras.