Hecke algebra representations of braid groups and link polynomials.

*(English)*Zbl 0631.57005A few years ago, the author discovered a new polynomial invariant for knots, which behaved like the classical Alexander polynomial but was different from it [Bull. Am. Math. Soc., New Ser. 12, 103-111 (1985; Zbl 0564.57006)]. This was a little revolution in knot theory. The new invariant was later on generalized to a 2-variable polynomial knot invariant, giving both the Alexander polynomial and the Jones polynomial as specializations [P. Freyd et al., ibid. 12, 239-246 (1985; Zbl 0572.57002)].

In the present article, the author presents his own version of the 2- variable polynomial, based on his original approach through representations of the braid group and Hecke algebras. The article also includes a computation of the 2-variable polynomial for torus knots, special relations for closed 3- and 4-braids, an analysis of values of the Jones (1-variable) polynomial at roots of unity, and some connections between the 2-variable polynomial and the braid index and bridge number of a knot. At the end of the paper, the Jones polynomials of all prime knots of 10 crossings and less are given in a table.

In the present article, the author presents his own version of the 2- variable polynomial, based on his original approach through representations of the braid group and Hecke algebras. The article also includes a computation of the 2-variable polynomial for torus knots, special relations for closed 3- and 4-braids, an analysis of values of the Jones (1-variable) polynomial at roots of unity, and some connections between the 2-variable polynomial and the braid index and bridge number of a knot. At the end of the paper, the Jones polynomials of all prime knots of 10 crossings and less are given in a table.

Reviewer: F.Bonahon

##### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

20F36 | Braid groups; Artin groups |