Knot Invariants and Chern-Simons Theory

Abstract

A brief review of the development of Chern-Simons gauge theory since its relation to knot theory was discovered in 1988 is presented. The presentation is done guided by a dictionary which relates knot theory concepts to quantum field theory ones. From the basic objects in both contexts the quantities leading to knot and link invariants are introduced and analyzed. The quantum field theory approaches that have been developed to compute these quantities are reviewed. Per-turbative approaches lead to Vassiliev or finite type invariants. Non-perturbative ones lead to polynomial or quantum group invariants. In addition, a brief discussion on open problems and future developments is included. In 1988, Edward Witten established the connection between Chern-Simons gauge theory and the theory of knot and link invariants [34]. Since then the theory has been intensively studied, making important progress as a result of the application of standard field theory methods. The development of the theory of knot and link invariants has been also very impressive in the last fifteen years and at some stages has occurred parallel to Chern-Simons gauge theory. There is a natural correspondence between both developments. This 1 Invited lecture delivered at the