arXiv subjects

A. Sleptsov

Publications and source records attributed to A. Sleptsov.

At least 19 recordsLinked to original sources

arXiv2026

Diagrammatic technique for Vogel's universality

In his 1999 preprint "Universal Lie Algebra", P. Vogel put forward a hypothesis on the existence of a universal Lie algebra. Although this hypothesis remains open, it is known that many quantities in Lie theory admit universal descriptions. Remarkably, almost all such universal formulas have been obtained through the representation theory of simple Lie (super)algebras, whereas Vogel's original framework was based on a more abstract diagrammatic algebra. Nevertheless, the diagrammatic approach has received little attention over the past two decades, since the last contributions by P. Vogel and J. Kneissler. In this work, we revive the diagrammatic technique grounded in Vogel's $\Lambda$-algebra and show that it enables truly universal computations. We examine numerous examples and discuss them.

math.QA↗

arXiv2025

Algebraic structures of Vassiliev invariants for knot families

We explore algebraic relations on Vassiliev knot invariants related with correlators in the 3-dimensional Chern--Simons theory. Vassiliev invariants form infinite-dimensional algebra. We focus on $k$-parametric knot families with Vassiliev invariants being polynomials in family parameters. We conjecture that such 1-parametric algebra of Vassiliev invariants is always finitely generated, while in the case of more parameters, we provide example of the knot family with infinite number of generators. Inside a knot family, there appear extra algebraic relations on Vassiliev invariants. We show that there are $\leq k$ algebraically independent Vassiliev invariants for $k$-parametric knot family. However, in all our examples, the number of algebraically independent Vassiliev invariants is exactly $k$, and it is open question if there exists a $k$-parametric knot family with a fewer number of algebraically independent Vassiliev invariants. We also demonstrate that a complete knot invariant of some $k$-parametric knot families consists of $k$ Vassiliev invariants.

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arXiv2025

Vogel's universality and the classification problem for Jacobi identities

This paper is a summary of discussions at the recent ITEP-JINR-YerPhI workshop on Vogel theory in Dubna. We consider relation between Vogel divisor(s) and the old Dynkin classification of simple Lie algebras. We consider application to knot theory and the hidden role of Jacobi identities in the definition/invariance of Kontsevich integral, which is the knot polynomial with the values in diagrams, capable of revealing all Vassiliev invariants -- including the ones, not visible in other approaches. Finally we comment on the possible breakdown of Vogel universality after the Jack/Macdonald deformation. Generalizations to affine, Yangian and DIM algebras are also mentioned. Especially interesting could be the search for universality in ordinary Yang-Mills theory and its interference with confinement phenomena.

hep-th↗

arXiv2022

Tug-the-hook symmetry for quantum 6j-symbols

We introduce a novel symmetry for quantum 6j-symbols, which we call the tug-the-hook symmetry. Unlike other known symmetries, it is applicable for any representations, including ones with multiplicities. We provide several evidences in favour of the tug-the-hook symmetry. First, this symmetry follows from the eigenvalue conjecture. Second, it is shown by several new examples of explicit coincidence of 6j-symbols with multiplicities. Third, the tug-the-hook symmetry for Wilson loops for knots in the 3d Chern-Simons theory implies the tug-the-hook symmetry for quantum 6j-symbols. An important implication of the analysis is the generalization of the tug-the-hook symmetry for the Chern-Simons Wilson loops to the case of links.

hep-th↗

arXiv2021

Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure

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. First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots $T[2,2k+1]$. Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank $N$ and the representation $R$.

hep-th↗

arXiv2021

Chern-Simons perturbative series revisited

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 $ZU(\mathfrak{sl}_N)$ is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved.

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arXiv2021

Combinatorics of KP hierarchy structural constants

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 KP hierarchy. Furthermore, this approach allows us to suggest several interesting directions of the KP deformation via a deformation of these coefficients. We also construct an eigenvalue matrix model, whose correlators fully describe the universal KP coefficients, which allows us to further study their properties and generalizations.

hep-th↗

arXiv2020

Genus expansion of matrix models and $\hbar$ expansion of KP hierarchy

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, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter $\hbar$ are $\tau$-functions of the $\hbar$-KP hierarchy and the expansion in $\hbar$ for the $\hbar$-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the $\hbar$-formulation of the KP hierarchy arXiv:1509.04472, arXiv:1512.07172 with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of $\tau$-functions is straightforward and algorithmic.

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arXiv2020

Distinguishing Mutant Knots

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 antisymmetric representations of $SU(N)$. Some of the mutant knots can be distinguished by the simplest non-symmetric representation $[2,1]$. However there is a class of mutant knots which require more complex representations like $[4,2]$. In this paper we calculate polynomials and differences for the mutant knot polynomials in representations $[3,1]$ and $[4,2]$ and study their properties.

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arXiv2020

A novel symmetry of colored HOMFLY polynomials coming from $\mathfrak{sl}(N|M)$ superalgebras

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 focus are the constraints that such a property imposes on the general group-theoretical structure, namely the $\mathfrak{sl}(N)$ weight system, arising in the perturbative expansion of the invariant. Finally, we demonstrate its tight relation to the eigenvalue conjecture.

hep-th↗

arXiv2020

Difference of mutant knot invariants and their differential expansion

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 differences, it is non-trivial in the case of mutants, which have the non-zero defect. The most effective technical tool, in this case, turns out to be the standard Reshetikhin-Turaev approach.

hep-th↗

arXiv2020

A new symmetry of the colored Alexander polynomial

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 polynomial. We study the constraints this symmetry imposes on the group theoretic structure of the loop expansion and provide solutions to those constraints. The symmetry is a powerful tool for research on polynomial knot invariants and in the end we suggest several possible applications of the symmetry.

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arXiv2019

Perturbative analysis of the colored Alexander polynomial and KP soliton $\tau$-functions

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 $N \rightarrow 0$. The result of the paper is twofold. First, we explain the emergence of Kadomsev-Petviashvily (KP) $\tau$-functions. This result is an extension of what we did in arXiv:1805.02761, where a symbolic correspondence between KP equations and group factors was established. In this paper we prove that integrability of the colored Alexander polynomial is due to it's relation to soliton $\tau$-functions. Mainly, the colored Alexander polynomial is embedded in the action of the KP generating function on the soliton $\tau$-function. Secondly, we use this correspondence to provide a rather simple combinatoric description of the group factors in term of Young diagrams, which is otherwise described in terms of chord diagrams, where no simple description is known. This is a first step providing an explicit description of the group theoretic data of Wilson loops, which would effectively reduce them to a purely topological quantity, mainly to a collection of Vassiliev invariants.

hep-th↗

arXiv2018

Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations

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 quantum Racah coefficients (6j-symbols) of $U_q(sl_2)$. In this paper, we generalise it to links whose components carry different symmetric representations. We illustrate the technique by evaluating multi-colored link polynomials $H_{[r_1],[r_2]}$ for the two-component link L7a3 whose components carry $[r_1]$ and $[r_2]$ colors.

hep-th↗

arXiv2018

Colored Alexander polynomials and KP hierarchy

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 equations. It appears that the solutions of this system induce the KP equations in the Hirota form. The Alexander polynomial is a specialization of the HOMFLY polynomial, and it is a kind of a dual to the double scaling limit, which gives the special polynomial, in the sense that, while the special polynomials provide solutions to the KP hierarchy, the Alexander polynomials provide the equations of this hierarchy. This gives a new connection with integrable properties of knot polynomials and puts an interesting question about the way the KP hierarchy is encoded in the full HOMFLY polynomial.

hep-th↗

arXiv2018

Quantum Racah matrices up to level 3 and multicolored link invariants

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 calculation is made possible by use of the highest weight method. The result allows one to evaluate and investigate colored polynomials for arbitrary 3-strand knots and links and to check the corresponding eigenvalue conjecture. Explicit answers for Racah matrices and colored polynomials for 3-strand knots up to 10 crossings are available at http://knotebook.org. Using the obtained inclusive Racah matrices, we also calculated the exclusive Racah matrices with the help of trick earlier suggested in the case of knots. This method is proved to be effective and gives the exclusive Racah matrices earlier obtained by another method.

hep-th↗

arXiv2018

Eigenvalue hypothesis for multi-strand braids

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 ${\cal R}$-matrix. In this paper, we generalize the hypothesis to higher number of strands in the braid where commuting relations of non-neighbouring $\mathcal{R}$ matrices are also incorporated. By solving these equations, we determine the explicit form for $\mathcal{R}$-matrices and the inclusive Racah matrices in terms of braiding eigenvalues (for matrices of size up to 6 by 6). For comparison, we briefly discuss the highest weight method for four-strand braids carrying fundamental and symmetric rank two $SU_q(N)$ representation. Specifically, we present all the inclusive Racah matrices for representation $[2]$ and compare with the matrices obtained from eigenvalue hypothesis.

hep-th↗

arXiv2018

Checks of integrality properties in topological strings

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 Marino's integrality conjecture up to two boxes in the Young diagram. In this paper, we review the salient aspects of the integrality properties and tabulate explicitly for an arborescent knot and a link. In our knotebook website, we have put these results for over 100 prime knots available in Rolfsen table and some links. The first application of the obtained results, an observation of the Gaussian distribution of the LMOV invariants is also reported.

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