Good involutions of conjugation subquandles and core quandles are characterized by central-valued functions, with algorithms, enumeration data, new CNS-quandle families, and a category equivalence between racks and Legendrian racks.
Knot symmetries and the fundamental quandle
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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.
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Good involutions of conjugation subquandles
Good involutions of conjugation subquandles and core quandles are characterized by central-valued functions, with algorithms, enumeration data, new CNS-quandle families, and a category equivalence between racks and Legendrian racks.