  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.
