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Mutation and the colored Jones polynomial. (English) Zbl 1267.57012

Summary: It is known that the colored Jones polynomials, various 2-cable link polynomials, the hyperbolic volume, and the fundamental group of the double branched cover coincide on mutant knots. We construct examples showing that these criteria, even in various combinations, are not sufficient to determine the mutation class of a knot, and that they are independent in several ways. In particular, we answer negatively the question of whether the colored Jones polynomial determines a simple knot up to mutation.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N70 Cobordism and concordance in topological manifolds
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)

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