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**Hecke algebra representations of braid groups and link polynomials.**
*(English)*
Zbl 0631.57005

A 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 |

### Keywords:

Jones polynomial; representations of the braid group; Hecke algebras; 2- variable polynomial; torus knots; closed 3- and 4-braids; braid index; bridge number
Full Text:
DOI

### Online Encyclopedia of Integer Sequences:

Number of n-crossing links with alternating braids of 3 strands.Number of n-crossing knots with alternating braids of 3 strands.

Number of n-crossing 2 component links with alternating braids of 3 strands.

Number of n-crossing 3 component links with alternating braids of 3 strands.