Free fermions and the Alexander-Conway polynomial

Abstract

We show how the Conway Alexander polynomial arises from theq deformation of (Z 2 graded)sl(n, n) algebras. In the simplestsl(1, 1) case we then establish connection between classical knot theory and its modern versions based on quantum groups. We first shown how the crystal and the fundamental group of the complement of a knot give rise naturally to the Burau representation of the braid group. The Burau matrix is then transformed into theU q sl(1, 1) R matrix by going to the exterior power algebra. Using a det=str identity, this allows us to recover the state model of [K2, 89] as well. We also show how theU q> sl(1, 1) algebra describes free fermions “propagating” on the knot diagram. We rewrite the Conway Alexander polynomial as a Berezin integral, and thus as an apparently new determinant.