On matrix-model approach to simplified Khovanov-Rozansky calculus

TL;DR

This paper develops a matrix-model based approach to simplified Khovanov-Rozansky calculus, enabling systematic evaluation of quantum dimensions for virtual knots and links, and relating fat graph genus to Turaev genus for thin knot detection.

Contribution

It introduces a matrix-model framework for hypercube calculus in knot theory, providing recursive relations and explicit formulas for virtual knot invariants.

Findings

Derived explicit expressions for virtual trefoil and 3.2 knot HOMFLY polynomials.

Established a relation between fat graph genus and Turaev genus.

Demonstrated the method's effectiveness for complex virtual link calculations.

Abstract

Wilson-loop averages in Chern-Simons theory (HOMFLY polynomials) can be evaluated in different ways -- the most difficult, but most interesting of them is the hypercube calculus, the only one applicable to virtual knots and used also for categorification (higher-dimensional extension) of the theory. We continue the study of quantum dimensions, associated with hypercube vertices, in the drastically simplified version of this approach to knot polynomials. At $q=1$ the problem is reformulated in terms of fat (ribbon) graphs, where Seifert cycles play the role of vertices. Ward identities in associated matrix model provide a set of recursions between classical dimensions. For $q \neq 1$ most of these relations are broken (i.e. deformed in a still uncontrollable way), and only few are protected by Reidemeister invariance of Chern-Simons theory. Still they are helpful for systematic evaluation of entire series of quantum dimensions, including negative ones, which are relevant for virtual link diagrams. To illustrate the effectiveness of developed formalism we derive explicit expressions for the 2-cabled HOMFLY of virtual trefoil and virtual 3.2 knot, which involve respectively 12 and 14 intersections -- far beyond any dreams with alternative methods. As a more conceptual application, we describe a relation between the genus of fat graph and Turaev genus of original link diagram, which is currently the most effective tool for the search of thin knots.