Communications in Number Theory and Physics

Volume 13 (2019)

Number 4

Combinatorial structure of colored HOMFLY-PT polynomials for torus knots

Pages: 763 – 826



Petr Dunin-Barkowski (Faculty of Mathematics, Higher School of Economics, National Research University, Moscow, Russia; and ITEP, Moscow, Russia)

Aleksandr Popolitov (Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden; ITEP, Moscow, Russia; and Moscow Institute of Physics and Technology, Dolgoprudny, Russia)

Sergey Shadrin (Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands)

Alexey Sleptsov (ITEP, Moscow, 117218, Russia; Institute for Information Transmission Problems, Moscow, Russia; and Moscow Institute of Physics and Technology, Dolgoprudny, Russia)


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 nonpolynomial factors are given by the Jacobi polynomials (treated as functions of their parameters in which they are indeed nonpolynomial). 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.


HOMFLY-PT polynomials, torus knots, free fermions, Ooguri–Vafa partition function, spectral curve, Chekhov–Eynard–Orantin topological recursion, Hurwitz numbers, Jacobi polynomials

2010 Mathematics Subject Classification

Primary 81T45. Secondary 14H81, 14J33, 33C45, 57M27.

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S.S. was supported by the Netherlands Organization for Scientific Research. A.P. and A.S. were supported by the Russian Science Foundation (Grant No. 16-12-10344). P.D.-B. was supported by RFBR grant 16-31- 60044-mol_a_dk and partially supported by RFBR grants 18-01-00461 and 18-31-20046-mol_a_ved. P. D.-B.’s research was carried out within the HSE University Basic Research Program and partially funded by the Russian Academic Excellence Project ‘5-100’.

Received 18 September 2018

Accepted 20 July 2019

Published 6 December 2019