Free fermions and the Alexander-Conway polynomial

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.Work supported in part by NSF grant no. DMS-8822602Work supported in part by the NSF: grant nos. PYI PHY 86-57788 and PHY 90-00386 and by CNRS, France