# Abelian quandles and quandles with abelian structure group

## TL;DR

This paper classifies finite quandles with abelian structure groups, showing they are abelian quandles, and explores their homology, revealing torsion properties relevant to knot invariants and Hopf algebras.

## Contribution

It explicitly describes all finite quandles with abelian structure groups and relates their structure to homology, providing new insights into their algebraic and topological properties.

## Key findings

- Finite quandles with abelian structure groups are exactly abelian quandles.
- The structure group of such quandles is a central extension of a free abelian group by a finite abelian group.
- The second homology group of these quandles is torsion-free, but torsion can occur in more general cases.

## 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.