# Commutator theory for racks and quandles

## TL;DR

This paper extends commutator theory from universal algebra to racks and quandles, providing new tools to analyze their algebraic properties and applying these to derive non-existence and non-colorability results.

## Contribution

It introduces a novel application of commutator theory to racks and quandles, linking congruence properties to subgroup structures and deriving significant algebraic and topological consequences.

## Key findings

- No connected involutory quandles of order 2^k exist
- No Latin quandles of order ≡ 2 mod 4 exist
- Knots with trivial Alexander polynomial are not colorable by Latin quandles

## Abstract

We adapt the commutator theory of universal algebra to the particular setting of racks and quandles, exploiting a Galois connection between congruences and certain normal subgroups of the displacement group. Congruence properties such as abelianness and centrality are reflected by the corresponding relative displacement groups, and so do the global properties, solvability and nilpotence. To show the new tool in action, we present three applications: non-existence theorems for quandles (no connected involutory quandles of order $2^k$, no latin quandles of order $\equiv2\pmod4$), a non-colorability theorem (knots with trivial Alexander polynomial are not colorable by latin quandles), and a strengthening of Glauberman's results on Bruck loops of odd order.