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On the Jones polynomial of closed 3-braids. (English) Zbl 0588.57005
In this short paper, the author proves: Proposition 1: There is a family of 3-braids whose closures are not amphicheiral, but have the symmetric Jones polynomials. Proposition 2: There are 3-braids \(\alpha\) and \(\beta\) such that their closure \({\tilde \alpha}\) and \({\tilde \beta}\) have the same Jones polynomials, but \({\tilde \alpha}\neq {\tilde \beta}\). These propositions provide the counterexamples to the conjectures by V. Jones. Since only 3-braids are involved in these propositions, the Jones polynomial can be described in terms of the Alexander polynomial and the exponent sum of a braid. Therefore, the proofs are fairly straightforward.
Reviewer: K.Murasugi

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F36 Braid groups; Artin groups
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References:
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