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


57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F36 Braid groups; Artin groups
Full Text: DOI EuDML


[1] [A] Artin, E.: Theorie der Zopfe. Abh. Math. Semin. Univ. Hamb.4, 47-72 (1925) · JFM 51.0450.01
[2] [Bu] Burau W.: ?ber Zopfgruppen und gleichsinnig verdrillte Verkettungen. Abh. Math. Semin. Hans. Univ.11, 171-178 (1936) · Zbl 0011.17704
[3] [G] Garside, F.: The braid group and other groups. Q. J. Math., Oxf.20, 235-254 (1969) · Zbl 0194.03303
[4] [H] Hartley, R.: On the classification of 3-braid links. Abh. Math. Semin. Univ. Hamb.50, 108-117 (1980) · Zbl 0446.57003
[5] [J,1] Jones, V.:Braid groups. Hecke algebras and type II1 factors. (MSRI, Berkeley) preprint
[6] [J,2] Jones, V.: A polynomial invariant for knots via von Neumann algebras. Bull. AMS12, 103-111 (1985) · Zbl 0564.57006
[7] [Mo] Morton, H.: Infinitely many fibered knots with the same Alexander polynomial. Topology17, 101-104 (1978) · Zbl 0383.57005
[8] [Mu] Murasugi, K.: On closed 3-braids. memoirs AMS No. 151 (1974). Am. Math. Soc., Providence, R.I.
[9] [Sc] Schreier, O.: Uber die GruppenA aBb=1. Abh. Math. Semin. Univ. Hamb.3, 167-169 (1923) · JFM 50.0070.01
[10] [Sq] Squier, C.: The Burau representation is unitary. Proc. AMS90, (2) 199-202 (1984) · Zbl 0542.20022
[11] [FYHLMO] Freyd and Yetter, Hoste, Lickorish and Millett, Ocneanu: A new polynomial invariant of knots and links. Ball AMS12, 239-246 (April, 1985) · Zbl 0572.57002
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