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Combinatorial structure of colored HOMFLY-PT polynomials for torus knots
Natl Res Univ Higher Sch Econ, Fac Math, Usacheva 6, Moscow 119048, Russia;ITEP, Moscow 117218, Russia.
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics. ITEP, Moscow 117218, Russia;Moscow Inst Phys & Technol, Dolgoprudnyi 141701, Russia.
Univ Amsterdam, Korteweg de Vries Inst Math, POB 94248, NL-1090 GE Amsterdam, Netherlands.
ITEP, Moscow 117218, Russia;Moscow Inst Phys & Technol, Dolgoprudnyi 141701, Russia.
2019 (English)In: Communications in Number Theory and Physics, ISSN 1931-4523, E-ISSN 1931-4531, Vol. 13, no 4, p. 763-826Article in journal (Refereed) Published
Abstract [en]

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-Marino 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 (treated as functions of their parameters in which they are indeed non-polynomial). 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, and we prove the quantum spectral curve equation for a natural wave function obtained from the extended Ooguri-Vafa partition function.

Place, publisher, year, edition, pages
INT PRESS BOSTON, INC , 2019. Vol. 13, no 4, p. 763-826
Keywords [en]
HOMFLY-PT polynomials, torus knots, free fermions, Ooguri-Vafa partition function, spectral curve, Chekhov-Eynard-Orantin topological recursion, Hurwitz numbers, Jacobi polynomials
National Category
Geometry
Identifiers
URN: urn:nbn:se:uu:diva-406718DOI: 10.4310/CNTP.2019.v13.n4.a3ISI: 000510479600003OAI: oai:DiVA.org:uu-406718DiVA, id: diva2:1414607
Available from: 2020-03-13 Created: 2020-03-13 Last updated: 2020-03-13Bibliographically approved

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