Combinatorial structure of colored HOMFLY-PT polynomials for torus knots

Abstract

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.

40 pages; section 10 addressing the quantum curve was added, as well as some remarks regarding Meixner polynomials thanks to T.Koornwinder

Cited by in corpus (15)

• Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type
• Explicit closed algebraic formulas for Orlov-Scherbin $n$-point functions
• Multiplicity-free $U_q(s{l}_N)$ 6-j symbols: relations, asymptotics, symmetries
• Distinguishing Mutant Knots
• Quantization of Harer-Zagier formulas
• Combinatorics of Bousquet-Mélou-Schaeffer numbers in the light of topological recursion
• A new symmetry of the colored Alexander polynomial
• Harer-Zagier formulas for knot matrix models
• Genus expansion of matrix models and $\hslash$ expansion of KP hierarchy
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• Log topological recursion through the prism of $x-y$ swap
• $x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm
• Are Maxwell knots integrable?
• Combinatorics of KP hierarchy structural constants
• Algebraic structures of Vassiliev invariants for knot families