  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.
