Matrix model and dimensions at hypercube vertices

TL;DR

This paper advances the matrix model approach to calculating dimensions at hypercube vertices in Chern-Simons theory, providing numerous examples and demonstrating the method's versatility in relating ordinary and virtual knots.

Contribution

It develops the matrix model technique for hypercube vertex functions in knot theory, extending its applicability to virtual links and complex graph structures.

Findings

Matrix model method successfully computes dimensions at hypercube vertices.

The formalism can convert between ordinary and virtual knots.

The approach extends beyond (2,2)-valent graphs.

Abstract

In hypercube approach to correlation functions in Chern-Simons theory (knot polynomials) the central role is played by the numbers of cycles, in which the link diagram is decomposed under different resolutions. Certain functions of these numbers are further interpreted as dimensions of graded spaces, associated with hypercube vertices. Finding these functions is, however, a somewhat non-trivial problem. In arXiv:1506.07516 it was suggested to solve it with the help of the matrix model technique, in the spirit of AMM/EO topological recursion. In this paper we further elaborate on this idea and provide a vast collection of non-trivial examples, related both to ordinary and virtual links and knots. Remarkably, most powerful versions of the formalism freely convert ordinary knots/links to virtual and back -- moreover, go beyond the knot-related set of the (2,2)-valent graphs.