On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions

TL;DR

This paper explores the genus expansion of knot polynomials, revealing their connection to Hurwitz tau-functions and uncovering a hierarchical structure analogous to KP tau-functions, with implications for understanding their integrability and representation dependence.

Contribution

It demonstrates that the genus expansion of HOMFLY polynomials naturally leads to a hierarchical decomposition of Hurwitz tau-functions, offering new insights into their structure and integrability.

Findings

HOMFLY polynomials' representation dependence is captured by symmetric group characters.

The Ooguri-Vafa partition function is a Hurwitz tau-function.

The genus expansion provides a hierarchical decomposition similar to Takasaki-Takebe expansion.

Abstract

In the genus expansion of the HOMFLY polynomials their representation dependence is naturally captured by symmetric group characters. This immediately implies that the Ooguri-Vafa partition function (OVPF) is a Hurwitz tau-function. In the planar limit involving factorizable special polynomials, it is actually a trivial exponential tau-function. In fact, in the double scaling Kashaev limit (the one associated with the volume conjecture) dominant in the genus expansion are terms associated with the symmetric representations and with the integrability preserving Casimir operators, though we stop one step from converting this fact into a clear statement about the OVPF behavior in the vicinity of q=1. Instead, we explain that the genus expansion provides a hierarchical decomposition of the Hurwitz tau-function, similar to the Takasaki-Takebe expansion of the KP tau-functions. This analogy can be helpful to develop a substitute for the universal Grassmannian description in the Hurwitz tau-functions.