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## Combinatorial structure of colored HOMFLY-PT polynomials for torus knots

Working papers by Cornell University. Series math "arxiv.org". 2017. Vol. 1712. No. 08614. P. 1-38.

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-Mari\~no 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.

Dunin-Barkowski P., Kazaryan M., Popolitov A. et al., Advances in Theoretical and Mathematical Physics 2022 Vol. 26 No. 4 P. 793-833

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 ...

Added: March 20, 2023

Dunin-Barkowski P., Popolitov A., Shadrin S. et al., Communications in Number Theory and Physics 2019 Vol. 13 No. 4 P. 763-826

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 ...

Added: August 18, 2020

Kazaryan M., Zograf P., Letters in Mathematical Physics 2015 Vol. 105 No. 8 P. 1057-1084

We compute the number of coverings of CP1∖{0,1,∞} with a given monodromy type over ∞ and given numbers of preimages of 0 and 1. We show that the generating function for these numbers enjoys several remarkable integrability properties: it obeys the Virasoro constraints, an evolution equation, the KP (Kadomtsev–Petviashvili) hierarchy and satisfies a topological recursion ...

Added: January 19, 2016

Dunin-Barkowski P., Lewanski D., Popolitov A. et al., Journal of London Mathematical Society 2015 Vol. 92 No. 3 P. 547-565

In this paper, we present an example of a derivation of an ELSV-type formula using the methods of topological recursion. Namely, for orbifold Hurwitz numbers we give a new proof of the spectral curve topological recursion, in the sense of Chekhov, Eynard and Orantin, where the main new step compared to the existing proofs is ...

Added: November 16, 2015

Dunin-Barkowski P., Orantin N., Popolitov A. et al., International Mathematics Research Notices 2018 Vol. 2018 No. 18 P. 5638-5662

We prove, in a purely combinatorial way, the spectral curve topological recursion for the problem of enumeration of bi-colored maps, which are dual objects to dessins d'enfant. Furthermore, we give a proof of the quantum spectral curve equation for this problem. Then we consider the generalized case of 4-colored maps and outline the idea of ...

Added: December 22, 2016

Dunin-Barkowski P., Kramer R., Popolitov A. et al., Journal of Geometry and Physics 2019 Vol. 137 P. 1-6

We give a new proof of the cut-and-join equation for the monotone Hurwitz numbers, derived first by Goulden, Guay-Paquet, and Novak. The main interest in this particular equation is its close relation to the quadratic loop equation in the theory of spectral curve topological recursion, and we recall this motivation giving a new proof of ...

Added: February 20, 2019

Bychkov B., Dunin-Barkowski P., Kazaryan M. et al., / Cornell University. Series math "arxiv.org". 2021. No. 2106.08368.

We study a duality for the n-point functions in VEV formalism that we call the ordinary vs fully simple duality. It provides an ultimate generalisation and a proper context for the duality between maps and fully simple maps observed by Borot and Garcia-Failde. Our approach allows to transfer the algebraicity properties between the systems of n-point functions ...

Added: April 20, 2022

Dunin-Barkowski P., Norbury P., Orantin N. et al., Journal of the Institute of Mathematics of Jussieu 2019 Vol. 18 No. 3 P. 449-497

We apply the spectral curve topological recursion to Dubrovin's universal Landau-Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold. ...

Added: December 22, 2016

Dunin-Barkowski Petr, Kazarian Maxim, Orantin N. et al., Advances in Mathematics 2015 Vol. 279 P. 67-103

In this paper we give a new proof of the ELSV formula. First, we refine an argument of Okounkov and Pandharipande in order to prove (quasi-)polynomiality of Hurwitz numbers without using the ELSV formula (the only way to do that before used the ELSV formula). Then, using this polynomiality we give a new proof of ...

Added: September 24, 2015

Dunin-Barkowski P., Kramer R., Popolitov A. et al., Annales Scientifiques de l'Ecole Normale Superieure 2023 Vol. 56 No. 4 P. 1199-1229

We prove the 2006 Zvonkine conjecture that expresses Hurwitz numbers with completed cycles in terms of intersection numbers with the Chiodo classes via the so-called r-ELSV formula, as well as its orbifold generalization, the so-called qr-ELSV formula. ...

Added: October 5, 2023

Providence : American Mathematical Society, 2018

This volume contains the proceedings of the 2016 AMS von Neumann Symposium on Topological Recursion and its Influence in Analysis, Geometry, and Topology, which was held from July 4–8, 2016, at the Hilton Charlotte University Place, Charlotte, North Carolina.
The papers contained in the volume present a snapshot of rapid and rich developments in the emerging ...

Added: February 20, 2019

Bufetov A. I., Известия РАН. Серия математическая 2015 Т. 79 № 6 С. 18-64

В статье рассматриваются бесконечные аналоги детерминантных мер ...

Added: October 16, 2015

Bychkov B., Dunin-Barkowski P., Kazaryan M. et al., / Cornell University. Series math "arxiv.org". 2020. No. 2012.14723.

We study the n-point differentials corresponding to Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions), with an emphasis on their ℏ2-deformations and expansions.
Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We ...

Added: April 20, 2022

Dunin-Barkowski P., , in : Oberwolfach reports. Vol. 13. Issue 1.: Zürich : European Mathematical Society Publishing house, 2016. P. 410-412.

Report on "Dubrovin's superpotential as a global spectral curve" (joint with Paul Norbury, Nicolas Orantin, Alexandr Popolitov, Sergey Shadrin) ...

Added: December 22, 2016

Omelchenko A., Grishanov S., Meshkov V., Textile Research Journal 2009 Vol. 79 No. 8 P. 702-713

This paper proposes a new systematic approach for the description and classification of textile structures based on topological principles. It is shown that textile structures can be considered as a specific case of knots or links and can be represented by diagrams on a torus. This enables modern methods of knot theory to be applied ...

Added: September 11, 2018

Smirnov A., Matveenko S., Semenova E., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2015 Vol. 11

In the article, we describe three-phase finite-gap solutions of the focusing nonlinear Schrödinger equation and Kadomtsev-Petviashvili and Hirota equations that exhibit the behavior of almost-periodic ''freak waves''. We also study the dependency of the solution parameters on the spectral curves. ...

Added: October 15, 2015

Netay I. V., Glutsyuk A., / Cornell University. Series arXiv "math". 2019.

The paper deals with a three-parameter family of special dou- ble confluent Heun equations that was introduced and studied by V. M. Buchstaber and S. I. Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are l, λ, μ ∈ R. Buchstaber and Tertychnyi described those parameter values, for which the ...

Added: November 12, 2019

Dunin-Barkowski P., Norbury P., Orantin N. et al., , in : Proceedings of Symposia in Pure Mathematics. Vol. 100: Topological Recursion and its Influence in Analysis, Geometry, and Topology.: Providence : American Mathematical Society, 2018. P. 297-331.

Hurwitz spaces parameterizing covers of the Riemann sphere can
be equipped with a Frobenius structure. In this review, we recall the construction of such Hurwitz Frobenius manifolds as well as the correspondence
between semisimple Frobenius manifolds and the topological recursion formalism. We then apply this correspondence to Hurwitz Frobenius manifolds by
explaining that the corresponding primary invariants can ...

Added: February 20, 2019

Bychkov B., Dunin-Barkowski P., Shadrin S., European Journal of Combinatorics 2020 Vol. 90 P. 103184

In this paper we prove, in a purely combinatorial-algebraic way, a structural quasi-polynomiality property for the Bousquet-Mélou–Schaeffer numbers. Conjecturally, this property should follow from the Chekhov–Eynard–Orantin topological recursion for these numbers (or, to be more precise, the Bouchard–Eynard version of the topological recursion for higher order critical points), which we derive in this paper from ...

Added: September 22, 2020

Bychkov B., Dunin-Barkowski P., Kazaryan M. et al., Communications in Mathematical Physics 2023 Vol. 402 P. 665-694

We study a duality for the n-point functions in VEV formalism that we call the ordinary vs fully simple duality. It provides an ultimate generalisation and a proper context for the duality between maps and fully simple maps observed by Borot and Garcia-Failde. Our approach allows to transfer the algebraicity properties between the systems of n-point functions ...

Added: June 29, 2023

Dunin-Barkowski P., Mulase M., Norbury P. et al., Journal fuer die reine und angewandte Mathematik 2017 Vol. 2017 No. 726 P. 267-289

We construct the quantum curve for the Gromov–Witten theory of the complex projective line. ...

Added: March 3, 2015

Gorsky Eugene, Rasmussen J., Oblomkov A., Experimental Mathematics 2013 Vol. 22 No. 3 P. 265-281

We conjecture that the stable Khovanov homology of torus knots can be described as the Koszul homology of an explicit irregular sequence of quadratic polynomials. The corresponding Poincaré series turns out to be related to the Rogers–Ramanujan identity. ...

Added: December 9, 2014

Dunin-Barkowski P., , in : Proceedings of Symposia in Pure Mathematics. Vol. 100: Topological Recursion and its Influence in Analysis, Geometry, and Topology.: Providence : American Mathematical Society, 2018. P. 231-295.

We describe a way of producing local spectral curves for arbitrary
semisimple cohomological field theories (and Gromov-Witten theories in particular) and global spectral curves for semisimple cohomological field theories
satisfying certain conditions. By this we mean that applying the topological
recursion procedure on the spectral curve reproduces the total potential of the
corresponding cohomological field theory. ...

Added: February 20, 2019

Glutsyuk A., Netay I. V., Journal of Dynamical and Control Systems 2020 Vol. 26 P. 785-820

The paper deals with a three-parameter family of special dou- ble confluent Heun equations that was introduced and studied by V. M. Buchstaber and S. I. Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are l, λ, μ ∈ R. Buchstaber and Tertychnyi described those parameter values, for which the ...

Added: October 19, 2020