Some Matrix Integrals related to Knots and Links

TL;DR

This paper explores matrix integrals linked to knot theory, rederives recent results, and discusses generalizations, providing insights into their combinatorial and topological significance.

Contribution

It introduces a detailed analysis of a simple matrix integral model related to knots and links, and discusses extensions to more complex models like the ABAB model.

Findings

Rederived recent results on knot-related matrix integrals

Identified the ABAB model as a key non-trivial example

Discussed potential generalizations of the models

Abstract

The study of a certain class of matrix integrals can be motivated by their interpretation as counting objects of knot theory such as alternating prime links, tangles or knots. The simplest such model is studied in detail and allows to rederive recent results of Sundberg and Thistlethwaite. The second non-trivial example turns out to be essentially the so-called ABAB model, though in this case the analysis has not yet been carried out completely. Further generalizations are discussed. This is a review of work done (in part) in collaboration with J.-B. Zuber.