Knot Invariants and Chern-Simons Theory

TL;DR

This paper reviews the development of Chern-Simons gauge theory and its profound connection to knot invariants, highlighting quantum field theory methods for their computation and discussing open problems in the field.

Contribution

It provides a comprehensive overview of how Chern-Simons theory relates to knot invariants, including both perturbative and non-perturbative approaches, and discusses future research directions.

Findings

Perturbative approaches yield Vassiliev invariants.

Non-perturbative approaches produce polynomial and quantum group invariants.

The paper identifies open problems and future developments in the field.

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. Perturbative 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.