arXiv subjects

An. Morozov

Publications and source records attributed to An. Morozov.

At least 19 recordsLinked to original sources

arXiv2022

Towards matrix model representation of HOMFLY polynomials

We investigate possibilities of generalizing the TBEM eigenvalue matrix model, which represents the non-normalized colored HOMFLY polynomials for torus knots as averages of the corresponding characters. We look for a model of the same type, which is a usual Chern-Simons mixture of the Gaussian potential, typical for Hermitean models, and the sine Vandermonde factors, typical for the unitary ones. We mostly concentrate on the family of twist knots, which contains a single torus knot, the trefoil. It turns out that for the trefoil the TBEM measure is provided by an action of Laplace exponential on the Jones polynomial. This procedure can be applied to arbitrary knots and provides a TBEM-like integral representation for the N=2 case. However, beyond the torus family, both the measure and its lifting to larger N contain non-trivial corrections in \hbar=\log q. A possibility could be to absorb these corrections into a deformation of the Laplace evolution by higher Casimir and/or cut-and-join operators, in the spirit of Hurwitz tau-function approach to knot theory, but this remains a subject for future investigation.

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arXiv2022

Ambiguity in mana and magic definition and knot states

We study the Mana and Magic for quantum states. They have a standard definition through the Clifford group, which is finite and thus classically computable. We introduce a modified Mana and Magic, which keep their main property of classical computability, while making other states classically computable. We also apply these new definitions to the studies of knot states of 2-strand knots.

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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

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.

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arXiv2018

From Topological to Quantum Entanglement

Entanglement is a special feature of the quantum world that reflects the existence of subtle, often non-local, correlations between local degrees of freedom. In topological theories such non-local correlations can be given a very intuitive interpretation: quantum entanglement of subsystems means that there are "strings" connecting them. More generally, an entangled state, or similarly, the density matrix of a mixed state, can be represented by cobordisms of topological spaces. Using a formal mathematical definition of TQFT we construct basic examples of entangled states and compute their von Neumann entropy.

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arXiv2018

A non-torus link from topological vertex

The recently suggested tangle calculus for knot polynomials is intimately related to topological string considerations and can help to build the HOMFLY-PT invariants from the topological vertices. We discuss this interplay in the simplest example of the Hopf link and link $L_{8n8}$. It turns out that the resolved conifold with four different representations on the four external legs, on the topological string side, is described by a special projection of the four-component link $L_{8n8}$, which reduces to the Hopf link colored with two composite representations. Thus, this provides the first explicit example of non-torus link description through the topological vertex. It is not a real breakthrough, because $L_{8n8}$ is just a cable of the Hopf link, still, it can help to intensify the development of the formalism towards more interesting examples.

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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.

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arXiv2018

Tangle blocks in the theory of link invariants

The central discovery of $2d$ conformal theory was holomorphic factorization, which expressed correlation functions through bilinear combinations of conformal blocks, which are easily cut and joined without a need to sum over the entire huge Hilbert space of states. Somewhat similar, when a link diagram is glued from tangles, the link polynomial is a multilinear combination of {\it tangle blocks} summed over just a few representations of intermediate states. This turns to be a powerful approach because the same tangles appear as constituents of very different knots so that they can be extracted from simpler cases and used in more complicated ones. So far this method has been technically developed only in the case of arborescent knots, but, in fact, it is much more general. We begin a systematic study of tangle blocks by detailed consideration of some archetypical examples, which actually lead to non-trivial results, far beyond the reach of other techniques. At the next level, the tangle calculus is about gluing of tangles, i.e. functorial mappings from ${\rm Hom}({\rm tangles})$, and its main advantage is an explicit realization of multiplicative composition structure, which is partly obscured in traditional knot theory.

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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.

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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.

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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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arXiv2017

On the block structure of the quantum R-matrix in the three-strand braids

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 $\mathcal{R}_2 = \mathcal{U} \mathcal{R}_1 \mathcal{U}^{\dagger}$. Since we can always choose the basis so that $\mathcal{R}_1$ is diagonal, the problem is reduced to evaluation of $\mathcal{R}_2$-matrices. This paper is one more step on the road to simplification of such calculations. We found out and proved for some cases that $\mathcal{R}_2$-matrices could be transformed into a block-diagonal ones. The essential condition is that there is a pair of accidentally coinciding eigenvalues among eigenvalues of $\mathcal{R}_1$-matrix. The angle of the rotation in the sectors corresponding to accidentally coinciding eigenvalues from the basis defined by the Racah matrix to the basis in which $\mathcal{R}_2$ is block-diagonal is $\pm \fracπ{4}$.

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arXiv2017

Differential expansion for link polynomials

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 triples of non-coincident representations. Based on the recent achievements in this direction, we conjecture a shape of the differential expansion for symmetrically-colored links and provide a set of examples. Within this study, we use a special framing that is an unusual extension of the topological framing from knots to links. In the particular cases of Whitehead and Borromean rings links, the differential expansions are different from the previously discovered.

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arXiv2017

Gaussian distribution of LMOV numbers

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 LMOV numbers are randomly distributed in genus (!) and are very well parameterized by just three parameters depending on the representation, an integer and the knot. We present an accurate formulation and evidence in support of this new puzzling observation about the old puzzling quantities. It probably implies that the BPS states, counted by the LMOV numbers can actually be composites made from some still more elementary objects.

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arXiv2017

Anomaly in RTT relation for DIM algebra and network matrix models

We discuss the recent proposal of arXiv:1608.05351 about generalization of the RTT relation to network matrix models. We show that the RTT relation in these models is modified by a nontrivial, but essentially abelian anomaly cocycle, which we explicitly evaluate for the free field representations of the quantum toroidal algebra. This cocycle is responsible for the braiding, which permutes the external legs in the q-deformed conformal block and its 5d/6d gauge theory counterpart, i.e. the non-perturbative Nekrasov functions. Thus, it defines their modular properties and symmetry. We show how to cancel the anomaly using a construction somewhat similar to the anomaly matching condition in gauge theory. We also describe the singular limit to the affine Yangian (4d Nekrasov functions), which breaks the spectral duality.

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arXiv2017

Towards topological quantum computer

One of the principal obstacles on the way to quantum computers is the lack of distinguished basis in the space of unitary evolutions and thus the lack of the commonly accepted set of basic operations (universal gates). A natural choice, however, is at hand: it is provided by the quantum R-matrices, the entangling deformations of non-entangling (classical) permutations, distinguished from the points of view of group theory, integrable systems and modern theory of non-perturbative calculations in quantum field and string theory. Observables in this case are (square modules of) the knot polynomials, and their pronounced integrality properties could provide a key to error correction. We suggest to use R-matrices acting in the space of irreducible representations, which are unitary for the real-valued couplings in Chern-Simons theory, to build a topological version of quantum computing.

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arXiv2016

On ambiguity in knot polynomials for virtual knots

We claim that HOMFLY polynomials for virtual knots, defined with the help of the matrix-model recursion relations, contain more parameters, than just the usual $q$ and $A = q^N$. These parameters preserve topological invariance and do not show up in the case of ordinary (non-virtual) knots and links. They are most conveniently observed in the hypercube formalism: then they substitute $q$-dimensions of certain fat graphs, which are not constrained by recursion and can be chosen arbitrarily. The number of these new topological invariants seems to grow fast with the number of non-virtual crossings: 0, 1, 1, 5, 15, 91, 784, 9160, ... This number can be decreased by imposing the factorization requirement for composites, in addition to topological invariance -- still freedom remains. None of these new parameters, however, appear in HOMFLY for Kishino unknot, which thus remains unseparated from the ordinary unknots even by this enriched set of knot invariants.

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arXiv2016

Racah matrices and hidden integrability in evolution of knots

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 \otimes \bar R) \longrightarrow R and (R\otimes \bar R) \otimes R \longrightarrow R with R\otimes (\bar R \otimes R) \longrightarrow R. They are building blocks for the colored HOMFLY polynomials of arbitrary arborescent (double fat) knots. Remarkably, the calculation realizes an unexpected integrability property underlying the evolution matrices.

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