Inner products on the Hecke algebra of the braid group

TL;DR

This paper reveals that the Homfly polynomial induces two polynomial-valued inner products on the Hecke algebra of the braid group, connecting algebraic structures with geometric interpretations and inequalities.

Contribution

It introduces two new polynomial-valued inner products on the Hecke algebra linked to the Homfly polynomial, with geometric interpretations involving Legendrian fronts.

Findings

Positive and negative permutation braids form orthonormal bases under these inner products.

Inner products relate to the Morton-Franks-Williams inequality.

Geometric interpretations via Legendrian fronts and rulings.

Abstract

We point out that the Homfly polynomial (that is to say, Ocneanu's trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin's braid group. These bear a close connection to the Morton-Franks-Williams inequality. In these structures, the sets of positive, respectively negative permutation braids become orthonormal bases. In the second case, many inner products can be geometrically interpreted through Legendrian fronts and rulings.