Knot theory from the perspective of field and string theory

Abstract

I present a summary of the recent progress made in field and string theory which has led to a reformulation of quantum-group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be described in topological terms. The approach opens a new point of view in the theory of knot and link invariants. During the last years the theory of knot and link invariants has experienced important progress. The confluence of Chern-Simons gauge theory and string theory has led to a very powerful new approach which provides a topological interpretation for the integer coefficients of a reformulated version of quantum-group polynomial invariants. The main goal of this short note is to present a summary of these recent developments. Chern-Simons gauge theory is a topological quantum field theory whose action is built out of a Chern-Simons term involving as gauge field a gauge connection associated to a group G on a three-manifold M. Its natural observables are Wilson loops, W K R , where K is a loop and R a representation of the gauge group. The vacuum expectation values of products of these operators are topological invariants which are related to quantum-group polynomial invariants. Given a link L of L components, K1, K2, . . . , KL, one computes