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Combinatorial structure of colored HOMFLY-PT polynomials for torus knots. (English) Zbl 1429.81080
Summary: 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.

##### MSC:
 81T45 Topological field theories in quantum mechanics 14H81 Relationships between algebraic curves and physics 14J33 Mirror symmetry (algebro-geometric aspects) 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 57K45 Higher-dimensional knots and links
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