# zbMATH — the first resource for mathematics

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
##### Keywords:
3-braids; amphicheiral; Jones polynomials; Alexander polynomial
Full Text:
##### References:
 [1] [A] Artin, E.: Theorie der Zopfe. Abh. Math. Semin. Univ. Hamb.4, 47-72 (1925) · JFM 51.0450.01 · doi:10.1007/BF02950718 [2] [Bu] Burau W.: ?ber Zopfgruppen und gleichsinnig verdrillte Verkettungen. Abh. Math. Semin. Hans. Univ.11, 171-178 (1936) · Zbl 0011.17704 · doi:10.1007/BF02940721 [3] [G] Garside, F.: The braid group and other groups. Q. J. Math., Oxf.20, 235-254 (1969) · Zbl 0194.03303 · doi:10.1093/qmath/20.1.235 [4] [H] Hartley, R.: On the classification of 3-braid links. Abh. Math. Semin. Univ. Hamb.50, 108-117 (1980) · Zbl 0446.57003 · doi:10.1007/BF02941419 [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 · doi:10.1090/S0273-0979-1985-15304-2 [7] [Mo] Morton, H.: Infinitely many fibered knots with the same Alexander polynomial. Topology17, 101-104 (1978) · Zbl 0383.57005 · doi:10.1016/0040-9383(78)90016-2 [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 · doi:10.1007/BF02954621 [10] [Sq] Squier, C.: The Burau representation is unitary. Proc. AMS90, (2) 199-202 (1984) · Zbl 0542.20022 · doi:10.1090/S0002-9939-1984-0727232-8 [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 · doi:10.1090/S0273-0979-1985-15361-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.