Matrix model and dimensions at hypercube vertices

Abstract

We consider correlation functions in the Chern–Simons theory (knot polynomials) using an approach in which each knot diagram is associated with a hypercube. The number of cycles into which the link diagram is decomposed under different resolutions plays a central role. Certain functions of these numbers are further interpreted as dimensions of graded spaces associated with hypercube vertices, but finding these functions is a somewhat nontrivial problem. It was previously suggested to solve this problem using the matrix model technique by analogy with topological recursion. We develop this idea and provide a wide collection of nontrivial examples related to both ordinary and virtual knots and links. The most powerful version of the formalism freely connects ordinary knots/links with virtual ones. Moreover, it allows going beyond the limits of the knot-related set of (2, 2)-valent graphs.