arXiv subjects

Aleksandr Popolitov

Publications and source records attributed to Aleksandr Popolitov.

9 recordsLinked to original sources

arXiv2024

Direct proof of one-hook scaling property for Alexander polynomial from Reshetikhin-Turaev formalism

We prove that normalized colored Alexander polynomial (the $A \rightarrow 1$ limit of colored HOMFLY-PT polynomial) evaluated for one-hook (L-shape) representation R possesses scaling property: it is equal to the fundamental Alexander polynomial with the substitution $q \rightarrow q^{|R|}$. The proof is simple and direct use of Reshetikhin-Turaev formalism to get all required R-matrices.

math-ph↗

arXiv2021

Harer-Zagier formulas for knot matrix models

Knot matrix models are defined so that the averages of characters are equal to knot polynomials. From this definition one can extract single trace averages and generation functions for them in the group rank - which generalize the celebrated Harer-Zagier formulas for Hermitian matrix model. We describe the outcome of this program for HOMFLY-PT polynomials of various knots. In particular, we claim that the Harer-Zagier formulas for torus knots factorize nicely, but this does not happen for other knots. This fact is mysteriously parallel to existence of explicit beta = 1 eigenvalue model construction for torus knots only, and can be responsible for problems with construction of a similar model for other knots.

hep-th↗

arXiv2020

Topological Recursion for the extended Ooguri-Vafa partition function of colored HOMFLY-PT polynomials of torus knots

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 statement of spectral curve topological recursion in this case fits into the program of Alexandrov-Chapuy-Eynard-Harnad of establishing the topological recursion for general weighted double Hurwitz numbers partition functions (a.k.a. KP tau-functions of hypergeometric type).

math-ph↗

arXiv2020

Solving q-Virasoro constraints

We show how q-Virasoro constraints can be derived for a large class of (q,t)-deformed eigenvalue matrix models by an elementary trick of inserting certain q-difference operators under the integral, in complete analogy with full-derivative insertions for beta-ensembles. From free-field point of view the models considered have zero momentum of the highest weight, which leads to an extra constraint T_{-1} Z = 0. We then show how to solve these q-Virasoro constraints recursively and comment on the possible applications for gauge theories, for instance calculation of (supersymmetric) Wilson loop averages in gauge theories on D^2 \cross S^1 and S^3.

hep-th↗

arXiv2020

Quantization of Harer-Zagier formulas

We derive the analogues of the Harer-Zagier formulas for single- and double-trace correlators in the q-deformed Hermitian Gaussian matrix model. This fully describes single-trace correlators and opens a road to $q$-deformations of important matrix models properties, such as genus expansion and Wick theorem.

hep-th↗

arXiv2019

Exact SUSY Wilson loops on $S^3$ from $q$-Virasoro constraints

Using the ideas from the BPS/CFT correspondence, we give an explicit recursive formula for computing supersymmetric Wilson loop averages in 3d $\mathcal{N}=2$ Yang-Mills-Chern-Simons $U(N)$ theory on the squashed sphere $S^3_b$ with one adjoint chiral and two antichiral fundamental multiplets, for specific values of Chern-Simons level $κ_2$ and Fayet-Illiopoulos parameter $κ_1$. For these values of $κ_1$ and $κ_2$ the north and south pole turn out to be completely independent, and therefore Wilson loop averages factorize into answers for the two constituent $D^2 \times S^1$ theories. In particular, our formula provides results for the theory on the round sphere when the squashing is removed.

hep-th↗

arXiv2019

Evolution for Khovanov polynomials for figure-eight-like family of knots

We look at how evolution method deforms, when one considers Khovanov polynomials instead of Jones polynomials. We do this for the figure-eight-like knots (also known as 'double braid' knots, see arXiv:1306.3197) -- a two-parametric family of knots which "grows" from the figure-eight knot and contains both two-strand torus knots and twist knots. We prove that parameter space splits into four chambers, each with its own evolution, and two isolated points. Remarkably, the evolution in the Khovanov case features an extra eigenvalue, which drops out in the Jones (t -> -1) limit.

math-ph↗

arXiv2019

Nimble evolution for pretzel Khovanov polynomials

We conjecture explicit evolution formulas for Khovanov polynomials for pretzel knots in some regions in the windings space. Our description is exhaustive for genera 1 and 2. As previously observed, evolution at T != -1 is not fully smooth: it switches abruptly at the boundaries between different regions. We reveal that this happens also at the boundary between thin and thick knots, moreover, the thick-knot domain is further stratified. For thin knots evolution is governed by the standard T-deformation lambda of the eigenvalues of the R-matrix. Emerging in the thick knots regions are additional Lyapunov exponents, which are multiples of the naive ones. Such frequency doubling is typical for non-linear dynamics, and our observation can signal about a hidden non-linearity of superpolynomial evolution. Since evolution with eigenvalues lambda^2, ..., lambda^g is "faster" than the one with lambda in the thin-knot region, we name it "nimble.

hep-th↗

arXiv2018

Combinatorial structure of colored HOMFLY-PT polynomials for torus knots

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 conjecture the combinatorial meaning of full expansion of the correlation differentials obtained via the topological recursion on the Brini-Eynard-Mariño spectral curve for the colored HOMFLY-PT polynomials of torus knots. This correspondence suggests a structural combinatorial result for the extended Ooguri-Vafa partition function. Namely, its coefficients should have a quasi-polynomial behavior, where non-polynomial factors are given by the Jacobi polynomials. We prove this quasi-polynomiality in a purely combinatorial way. In addition to that, we show that the (0,1)- and (0,2)-functions on the corresponding spectral curve are in agreement with the extension of the colored HOMFLY-PT polynomials data.

math-ph↗