Topological Recursion for the extended Ooguri-Vafa partition function of colored HOMFLY-PT polynomials of torus knots

TL;DR

This paper demonstrates that topological recursion applied to the spectral curve of colored HOMFLY-PT polynomials of torus knots reproduces the extended Ooguri-Vafa partition function, linking knot invariants with topological recursion.

Contribution

It proves the application of topological recursion to this spectral curve and connects it to the broader program of understanding weighted double Hurwitz numbers.

Findings

Topological recursion reproduces the n-point functions of the extended Ooguri-Vafa partition function.

Generalizes previous results by Brini-Eynard-Marino and Borot-Eynard-Orantin.

Links spectral curve topological recursion to hypergeometric KP tau-functions.

Abstract

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 case fits into the program of Alexandrov-Chapuy-Eynard-Harnad of establishing the topological recursion for general weighted double Hurwitz numbers partition functions (a.k.a. KP tau-functions of hypergeometric type).