Knot theory and matrix integrals

Abstract

This article considers some enumeration problems in knot theory, with a focus on the application of matrix integral techniques. It first reviews the basic definitions of knot theory, paying special attention to links and tangles, especially 2-tangles, before discussing virtual knots and coloured links as well as the bare matrix model that describes coloured link diagrams. It shows how the large size limit of matrix integrals with quartic potential may be used to count alternating links and tangles. The removal of redundancies amounts to renormalization of the potential. This extends into two directions: first, higher genus and the counting of ‘virtual’ links and tangles, and second, the counting of ‘coloured’ alternating links and tangles. The article analyses the asymptotic behaviour of the number of tangles as the number of crossings goes to infinity