Structure groups and second homology groups of linear Alexander quandles

Abstract

Quandles are self-distributive algebraic structures known as sources of strong knots invariants, but also appearing in other contexts. From any quandle, one can construct two invariants: the structure group and the second quandle homology group. These groups are useful in applications, but hard to compute. In this paper, we focus on Alexander quandles over a cyclic group ${\mathbb{Z}}_n$. By using explicit rewriting techniques, we show that the structure group of such a quandle injects into ${\mathbb{Z}}^m⋉{\mathbb{Z}}_n$ if $m$ is its number of orbits. This allows us to compute its second quandle homology group, and find that the torsion part depends only on $m$ and $n$.

16 pages