Knot Theory

This article is an introduction to knot theory from the point of view of combinatorial topology and the Reidemeister moves, combined with the relationships of knot polynomials such as the Jones polynomial with ideas and techniques in theoretical physics and statistical mechanics. The paper begins with a introduction to Fox coloring, quandle and Alexander polynomial. Then it discusses the Kauffman bracket model of the Jones polynomial and how this is related to Vassiliev invariants. From Vassiliev invariants the paper turns to Lie algebras as background for the construction invariants, quantum link invariants and the work of Witten using Lie algebras and functional integrals to construct new invariants of knots, links and three-manifolds and how Witten's approach is related to Vassiliev invariants.