Structure groups and second homology groups of linear Alexander quandles

TL;DR

The paper addresses computing the structure group and the second quandle homology $H_2^Q$ for linear Alexander quandles $X=\text{Al}(\mathbb{Z}_n,T)$. It develops a semidirect-product framework with $\mathbb{Z}\ltimes M$ and constructs a surjection $f:\text{As}(X)\to \mathbb{Z}\ltimes M$ along with a weight map $\omega$, then proves an injective embedding $u:\text{As}(X)\to \mathbb{Z}^m\ltimes \mathbb{Z}_n$ (where $m$ is the number of orbits) that yields a normal form for elements of the structure group. From this, the extension is shown central and $H_2^Q(X)$ is computed explicitly in terms of $m$ and $n$, namely $H_2^Q(X)\cong \mathbb{Z}^{(m-1)m}\oplus\left[\gcd(m, n/m)\mathbb{Z}_n\right]^m$. The results provide concrete, computable invariants for linear Alexander quandles and clarify how torsion in $H_2^Q$ depends on the orbit count and the ambient modulus $n$.

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 $\mathbb{Z}^m \ltimes \mathbb{Z}_n$ 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 $n$.

Theorems & Definitions (55)

• Theorem 1: Eisermann eisermannQuandleCoveringsTheir2014
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Proposition 5
• proof
• Lemma 6
• Corollary 7
• proof