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Introduction

## Publications

Publications (67)

We have recently proposed [1] a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with SU(N) gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values.
First, we discu...

We investigate the structural constants of the KP hierarchy, which appear as universal coefficients in the paper of Natanzon–Zabrodin arXiv:1509.04472 . It turns out that these constants have a combinatorial description in terms of transport coefficients in the theory of flow networks. Considering its properties we want to point out three novel dir...

We have recently proposed arXiv:2105.11565 a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with $SU(N)$ gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values....

A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with SU(N) gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra ZU(slN) is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional re...

We present a novel symmetry of the colored HOMFLY polynomial. It relates pairs of polynomials colored by different representations at specific values of N and generalizes the previously known “tug-the-hook” symmetry of the colored Alexander polynomial (Mishnyakov et al. in Annales Henri Poincaré, 2021. https://doi.org/10.1007/s00023-020-00980-8, ar...

A group-theoretical structure in the perturbative expansion for the Wilson loops in the 3d Chern-Simons theory with $SU(N)$ gauge group is developed in symmetric approach. An image of a map called $\mathfrak{sl}_N$ weight system from the algebra of chord diagrams to the center of the universal enveloping algebra $ZU(\mathfrak{sl}_N)$ is studied. El...

The eigenvalue hypothesis claims that any quantum Racah matrix for finite-dimensional representations of \(U_q(sl_N)\) is uniquely determined by eigenvalues of the corresponding quantum \(\mathcal {R}\)-matrices. If this hypothesis turns out to be true, then it will significantly simplify the computation of Racah matrices. Also, due to this hypothe...

In this paper we study the group theoretic structures of colored HOMFLY polynomials in a specific limit. The group structures arise in the perturbative expansion of SU(N) Chern-Simons Wilson loops, while the limit is N→0. The result of the paper is twofold. First, we explain the emergence of Kadomsev-Petviashvily (KP) τ-functions. This result is an...

Following Natanzon-Zabrodin, we explore the Kadomtsev-Petviashvili hierarchy as an infinite system of mutually consistent relations on the second derivatives of the free energy with some universal coefficients. From this point of view, various combinatorial properties of these coefficients naturally highlight certain non-trivial properties of the K...

We present a new conjectural symmetry of the colored Alexander polynomial, that is the specialization of the quantum \(\mathfrak {sl}_N\) invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the existence of the symmetry by studying the loop expansion and the character expansion of the colored HOMFLY-PT pol...

Knot theory is actively studied both by physicists and mathematicians as it provides a connecting centerpiece for many physical and mathematical theories. One of the challenging problems in knot theory is distinguishing mutant knots. Mutant knots are not distinguished by colored HOMFLY-PT polynomials for knots colored by either symmetric and or ant...

A bstract
We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ -functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brez...

A closed form expression for multiplicity-free quantum 6-j symbols (MFS) was proposed in [1] for symmetric representations of Uq(slN), which are the simplest class of multiplicity-free representations. In this paper we rewrite this expression in terms of q-hypergeometric series Φ34. We claim that it is possible to express any MFS through the 6-j sy...

We prove that topological recursion applied to the spectral curve of colored HOMFLY-PT polynomials of torus knots reproduces the n-point functions of a particular partition function called the extended Ooguri-Vafa partition function. This generalizes and refines the results of Brini-Eynard-Marino and Borot-Eynard-Orantin. We also discuss how the st...

We study $\hbar$ expansion of the KP hierarchy following Takasaki-Takebe arXiv:hep-th/9405096 considering several examples of matrix model $\tau$-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevi...

Knot theory is actively studied both by physicists and mathematicians as it provides a connecting centerpiece for many physical and mathematical theories. One of the challenging problems in knot theory is distinguishing mutant knots. Mutant knots are not distinguished by colored HOMFLY-PT polynomials for knots colored by either symmetric and or ant...

We present a novel symmetry of the colored HOMFLY polynomial. It relates pairs of polynomials colored by different representations at specific values of $N$ and generalizes the previously known "tug-the-hook" symmetry of the colored Alexander polynomial. As we show, the symmetry has a superalgebra origin, which we discuss qualitatively. Our main fo...

We evaluate the differences of HOMFLY-PT invariants for pairs of mutant knots colored with representations of SL(N), which are large enough to distinguish between them. These mutant pairs include the pretzel mutants, which require at least the representation, labelled by the Young diagram [4, 2]. We discuss the differential expansion for the differ...

We evaluate the differences of HOMFLY-PT invariants for pairs of mutant knots colored with representations of $SL(N)$, which are large enough to distinguish between them. These mutant pairs include the pretzel mutants, which require at least the representation, labeled by the Young diagram $[4,2]$. We discuss the differential expansion for the diff...

We present a new "tug-the-hook" symmetry of the colored Alexander polynomial which is the specialization of the quantum $\mathfrak{sl}_N$ invariant widely known as the colored HOMFLY-PT polynomial. In the perturbative expansion of the Alexander polynomial, this symmetry is realized as a property of the group theoretical data of the invariant. Mainl...

A closed form expression for multiplicity-free quantum 6-j symbols (MFS) was proposed in arXiv:1302.5143 for symmetric representations of $U_q(sl_N)$, which are the simplest class of multiplicity-free representations. In this paper we rewrite this expression in terms of q-hypergeometric series ${}_4\Phi_3$. We claim that it is possible to express a...

The eigenvalue hypothesis claims that any quantum Racah matrix for finite-dimensional representations of $U_q(sl_N)$ is uniquely determined by eigenvalues of the corresponding quantum $\cal{R}$-matrices. If this hypothesis turns out to be true, then it will significantly simplify the computation of Racah matrices. Also due to this hypothesis variou...

Obtaining HOMFLY-PT polynomials \(H_{R_1,\ldots ,R_l}\) for arbitrary links with l components colored by arbitrary SU(N) representations \(R_1,\ldots ,R_l\) is a very complicated problem. For a class of rank r symmetric representations, the [r]-colored HOMFLY-PT polynomial \(H_{[r_1],\ldots ,[r_l]}\) evaluation becomes simpler, but the general answ...

In this paper we elaborate on the statement given in arXiv:1805.02761. Mainly, we study the relation between the colored Alexander polynomial and the famous KP hierarchy. We explain and prove this relation by exploring the fact that the dispersion equations of the one soliton $\tau$-function are equivalent to the system of equations arising due to...

In the present paper we discuss the eigenvalue conjecture, suggested in 2012, in the particular case of Uq(sl2). The eigenvalue conjecture provides a certain symmetry for Racah coefficients and we prove that the eigenvalue conjecture is provided by the Regge symmetry for Uq(sl2), when three representations coincide. This in perspective provides us...

This is a review of (q-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal polynomials, and consider their various generalizations. The review also includes the orthogonal polynomials into...

We discuss the relation between knot polynomials and the KP hierarchy. Mainly, we study the scaling 1-hook property of the coloured Alexander polynomial: ARK(q)=A[1]K(q|R|) for all 1-hook Young diagrams R. Via the Kontsevich construction, it is reformulated as a system of linear equations. It appears that the solutions of this system induce the KP...

Obtaining colored HOMFLY-PT polynomials for knots from 3-strand braid carrying arbitrary $SU(N)$ representation is still tedious. For a class of rank $r$ symmetric representations, $[r]$-colored HOMFLY-PT $H_{[r]}$ evaluation becomes simpler. Recently it was shown that $H_{[r]}$, for such knots from 3-strand braid, can be constructed using the quan...

We discuss the relation between knot polynomials and the KP hierarchy. Mainly, we study the scaling 1-hook property of the coloured Alexander polynomial: $\mathcal{A}^\mathcal{K}_R(q)=\mathcal{A}^\mathcal{K}_{[1]}(q^{\vert R\vert})$ for all 1-hook Young diagrams $R$. Via the Kontsevich construction, it is reformulated as a system of linear equation...

This paper is a next step in the project of systematic description of colored knot and link invariants started in previous papers. In this paper, we managed to explicitly find the inclusive Racah matrices, i.e. the whole set of mixing matrices in channels $R_1\otimes R_2\otimes R_3\longrightarrow Q$ with all possible $Q$, for $|R|\leq 3$. The calcu...

We rewrite the (extended) Ooguri-Vafa partition function for colored HOMFLY-PT polynomials for torus knots in terms of the free-fermion (semi-infinite wedge) formalism, making it very similar to the generating function for double Hurwitz numbers. This allows us to perform a key step towards a purely combinatorial proof of topological recursion for...

Quantum $\mathcal{R}$-matrices are the building blocks for the colored HOMFLY polynomials. In the case of three-strand braids with an identical finite-dimensional irreducible representation $T$ of $SU_q(N)$ associated with each strand one needs two matrices: $\mathcal{R}_1$ and $\mathcal{R}_2$. They are related by the Racah matrices $\mathcal{R}_2...

This is a review of ([Formula: see text]-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal polynomials, and consider their various generalizations. The review also includes the orthogonal...

Computing polynomial form of the colored HOMFLY-PT for non-arborescent knots obtained from three or more strand braids is still an open problem. One of the efficient methods suggested for the three-strand braids relies on the eigenvalue hypothesis which uses the Yang-Baxter equation to express the answer through the eigenvalues of the ${\cal R}$-ma...

Explicit expressions are found for the 6j symbols in symmetric representations of quantum suN through appropriate hypergeometric Askey–Wilson (q-Racah) polynomials. This generalizes the well-known classical formulas for Uq(su2) and provides a link to conformal theories and matrix models.

The differential expansion is one of the key structures reflecting group theory properties of colored knot polynomials, which also becomes an important tool for evaluation of non-trivial Racah matrices. This makes highly desirable its extension from knots to links, which, however, requires knowledge of the $6j$-symbols, at least, for the simplest t...

Explicit expressions are found for the $6j$ symbols in symmetric representations of quantum $\mathfrak{su}_N$ through appropriate hypergeometric Askey-Wilson (q-Racah) polynomials. This generalizes the well-known classical formulas for $U_q(\mathfrak{su}_2)$ and provides a link to conformal theories and matrix models.

Recent advances in knot polynomial calculus allowed us to obtain a huge variety of LMOV integers counting degeneracy of the BPS spectrum of topological theories on the resolved conifold and appearing in the genus expansion of the plethystic logarithm of the Ooguri-Vafa partition functions. Already the very first look at this data reveals that the L...

Tests of the integrality properties of a scalar operator in topological strings on a resolved conifold background or orientifold of conifold backgrounds have been performed for arborescent knots and some non-arborescent knots. The recent results on polynomials for those knots colored by SU(N) and SO(N) adjoint representations are useful to verify M...

This paper is a next step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the $\textit{inclusive}$ Racah matrices, i.e. the whole set of mixing matrices in channels $R^{\otimes 3}\longrightarrow Q$ with all possible $Q$, for $R=[3,3]$. The case $R=[3,3]$...

We construct a general procedure to extract the exclusive Racah matrices S and \bar S from the inclusive 3-strand mixing matrices by the evolution method and apply it to the first simple representations R =[1], [2], [3] and [2,2]. The matrices S and \bar S relate respectively the maps (R\otimes R)\otimes \bar R\longrightarrow R with R\otimes (R \ot...

We describe the inclusive Racah matrices for the first non-(anti)symmetric rectangular representation R = [2, 2] for quantum groups U
q
(sl
N
). Most of them have sizes 2, 3, and 4 and are fully described by the eigenvalue hypothesis. Of two 6 × 6 matrices, one is also described in this way, but the other one corresponds to the case of degenerate e...

This paper is a new step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R^3->Q with all possible Q, for R=[3,1]. The calculation is made possible by the use of a newly-develope...

Arborescent knots are the ones which can be represented in terms of double
fat graphs or equivalently as tree Feynman diagrams. This is the class of knots
for which the present knowledge is enough for lifting topological description
to the level of effective analytical formulas. The paper describes the origin
and structure of the new tables of colo...

We compute Vassiliev invariants up to order six for arbitrary pretzel knots,
which depend on $g+1$ parameters $n_1,\ldots,n_{g+1}$. These invariants are
symmetric polynomials in $n_1,\ldots,n_{g+1}$ whose degree coincide with their
order. We also discuss their topological and integer-valued properties.

This paper starts a systematic description of colored knot polynomials,
beginning from the first non-(anti)symmetric representation R=[2,1]. The
project involves several steps: (i) parametrization of big families of knots a
la arXiv:1506.00339, (ii) evaluating Racah/mixing matrices for various numbers
of strands in various representations a la arXi...

With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g + 1 two strand braids, parallel or antiparallel, and depend on g + 1 integer numbers. We demonstrate that they possess a pronounced new structure: are decomposed into...

A very simple expression is conjectured for arbitrary colored Jones and
HOMFLY polynomials of a rich $(g+1)$-parametric family of Pretzel knots and
links. The answer for the Jones and HOMFLY polynomials is fully and explicitly
expressed through the Racah matrix of U_q(SU_N), and looks related to a modular
transformation of toric conformal block.

We discuss a connection of HOMFLY polynomials with Hurwitz covers and represent a generating function for the HOMFLY polynomial of a given knot in all representations as Hurwitz partition function, i.e. the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group characters (cut-and-join eigenvalues). The ge...

We discuss relation between knot theory and topological quantum field theory. Also it is considered a theory of superpolynomial invariants of knots which generalizes all other known theories of knot invariants. We discuss a possible generalization of topological quantum field theory with the help of superpolynomial invariants.

Recently it was shown that the (Ooguri-Vafa) generating function of HOMFLY
polynomials is the Hurwitz partition function, i.e. that the dependence of the
HOMFLY polynomials on representation is naturally captured by symmetric group
characters (cut-and-join eigenvalues). The genus expansion and expansion
through Vassiliev invariants explicitly demon...

In the genus expansion of the HOMFLY polynomials their representation
dependence is naturally captured by symmetric group characters. This
immediately implies that the Ooguri-Vafa partition function (OVPF) is a Hurwitz
tau-function. In the planar limit involving factorizable special polynomials,
it is actually a trivial exponential tau-function. In...

In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d
Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way
on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa
partition function becomes a trivial KP tau-function. We study higher genus
corrections to this formula in the form...

Currently there are two proposed ansatze for NSR superstring measures: the
Grushevsky ansatz and the OPSMY ansatz, which for genera g<=4 are known to
coincide. However, neither the Grushevsky nor the OPSMY ansatz leads to a
vanishing two point function in genus four, which can be constructed from the
genus five expressions for the respective ansatz...

In arXiv:1106.4305 extended superpolynomials were introduced for the torus
links T[m,mk+r], which are functions on the entire space of time variables and,
at expense of reducing the topological invariance, possess additional algebraic
properties, resembling those of the matrix model partition functions and the
KP/Toda tau-functions. Not surprisingl...

We solve the regularized Knizhnik-Zamolodchikov equation and find an explicit
expression for the Drinfeld associator. We restrict to the case of the
fundamental representation of $gl(N)$. Several tests of the results are
presented. It can be explicitly seen that components of this solution for the
associator coincide with certain components of WZW...

We review quantum field theory approach to the knot theory. Using holomorphic
gauge we obtain the Kontsevich integral. It is explained how to calculate
Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial
way which can be programmed on a computer. We discuss experimental results and
temporal gauge considerations which lea...

The colored HOMFLY polynomials, which describe Wilson loop averages in
Chern-Simons theory, possess an especially simple representation for torus
knots, which begins from quantum R-matrix and ends up with a trivially-looking
split W representation familiar from character calculus applications to matrix
models and Hurwitz theory. Substitution of Mac...

We suggest a general method of computation of the homology of certain smooth
covers $\hat{\mathcal{M}}_{g,1}(\mathbb{C})$ of moduli spaces
$\mathcal{M}_{g,1}\br{\mathbb{C}}$ of pointed curves of genus $g$. Namely, we
consider moduli spaces of algebraic curves with level $m$ structures. The
method is based on the lifting of the Strebel-Penner strati...

We discuss relations between two different representations of hypothetical holomorphic NSR measures, based on two different ways of constructing the semi-modular forms of weight 8. One of these ways is to build forms from the ordinary Riemann theta constants and another -- from the lattice theta constants. We discuss unexpectedly elegant relations...

We classify projective plane nonsingular curves admitting a 3-term presentation; they exist in any degree, generally constitute 5 birational families and are defined over rational numbers. The Belyi functions on all these curves are presented.

We consider reparameterization-invariant Lagrangian theories with higher derivatives, investigate the geometric structures
behind these theories, and construct the Hamiltonian formalism geometrically. We present the Legendre transformation formula
for such systems, which differs from the usual one. We show that the phase bundle, i.e., the image of...