Knot symmetries and the fundamental quandle

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.