Structure groups and second homology groups of linear Alexander quandles

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 Zₙ. By using explicit rewriting techniques, we show that the structure group of such a quandle injects into Zᵐ Zₙ 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.