Theoretical and Mathematical Physics, 192(1): 1039–1079 (2017)
MATRIX MODEL AND DIMENSIONS AT HYPERCUBE VERTICES
A. Yu. Morozov,
A. A. Morozov,
and A. V. Popolitov
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.
Keywords: Chern–Simons theory, knot theory, virtual knot, matrix model
Knot polynomials  (Wilson-loop averages in Chern–Simons theory  and their various deforma-
tions –) are currently one of the actively developing ﬁelds in theoretical physics. They belong to the
important class of Hurwitz τ-functions  and supposedly have the main properties of the matrix-model
τ-functions , , which currently unify all the known special functions and provide a basis for an
analytic description of Nature.
There are several ways to study and even deﬁne the knot polynomials, coming from diﬀerent overlapping
ﬁelds of physics and mathematics, which must be related to each other: representation theory, topology,
modular transformations of conformal blocks, integrable systems, Morse theory, etc. The most intriguing of
these ﬁelds is the categoriﬁcation program , ﬁrst proposed in this context by Khovanov . The physical
interpretation of this program is a transition to higher-dimensional spaces (where the quantities of interest
are associated with ﬁxed points of some new evolution ). A detailed description of these ideas is beyond
the scope of this paper. Here, we concentrate on technical details needed for building such a formalism and
presenting eﬀective computations in its framework. For this, we rely on the two relatively old advances, the
Institute for Theoretical and Experimental Physics, Moscow, Russia; Institute for Information Transmission
Problems, Moscow, Russia, e-mail: firstname.lastname@example.org, Andrey.Morozov@itep.ru.
National Research Nuclear University MEPhI, Moscow, Russia.
Laboratory of Quantum Topology, Chelyabinsk State University, Chelyabinsk, Russia.
Korteweg–de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands.
This research was performed at the Institute for Information Transmission Problems and supported by a grant
from the Russian Science Foundation (Project No. 14-50-00150).
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i
Matematicheskaya Fizika, Vol. 192, No. 1, pp. 115–163, July, 2017. Original article submitted April 23, 2016.
2017 Pleiades Publishing, Ltd.