Knot symmetries and the fundamental quandle

TL;DR

This paper explores the connection between knot symmetries and automorphisms of the fundamental quandle, showing that all quandle automorphisms are induced by knot homeomorphisms, enabling symmetry analysis from knot diagrams.

Contribution

It establishes a direct link between knot symmetries and quandle automorphisms, demonstrating that every automorphism arises from a knot homeomorphism.

Findings

Every quandle automorphism is induced by a knot homeomorphism.

The relationship allows symmetry analysis directly from knot diagrams.

The results apply to both automorphisms and anti-automorphisms.

Abstract

We establish a relationship between the knot symmetries and the automorphisms of the knot quandle. We identify the homeomorphisms of the pair $(S^{3},K)$ that induce the (anti)automorphisms of the fundamental quandle $Q(K)$. We show that every quandle (anti)automorphism of $Q(K)$ is induced by a homeomorphism of the pair $(S^{3},K)$. As an application of those results, we are able to explore some symmetry properties of a knot based on the presentation of its fundamental quandle, which is easily derived from a knot diagram.