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Knot symmetries and the fundamental quandle

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arxiv 1707.04824 v1 pith:BYTQN6PZ submitted 2017-07-16 math.GT

classification math.GT
keywords knotquandlefundamentalantiautomorphismspairsymmetriesable
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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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  1. Good involutions of conjugation subquandles

    math.GT 2025-05 accept novelty 7.0 of 10

    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 Le...

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