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≠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.
Highlights
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 for categorification of the theory
Still they are helpful for systematic evaluation of entire series of quantum dimensions, including negative ones, which are relevant for virtual link diagrams
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
Summary
In the present text we assume familiarity with the hypercube construction of [16] and just describe the peculiarities coming from the suggestion of [13]. The peculiarity of [13] is that white resolution is associated with the difference between identity and simple crossing (instead of the “horizontal” resolution in [17,16], which does not respect arrows and is applicable only at N = 2). It is this difference that is responsible for factor-spaces and potential negativity. In result we obtain a rather effective and computerizable calculational machinery, which is applied to the study of topological invariance and to evaluation of the first cabled HOMFLY for virtual knots
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