Universal tangle invariant and commutants of quantum algebras

Abstract

We construct a universal tangle invariant on a quantum algebra. We show that the invariant maps tangle to commutants of the algebra; every (1,1)-tangle is mapped to a Casimir operator of the algebra; the eigenvalue of the Casimir operator in an irreducible representation of the algebra is a link polynomial for the closure of the tangle. This result is applied to a discussion of the Alexander - Conway polynomial and quantum holonomy in Chern - Simons theory in three dimensions.