Colored Alexander polynomials and KP hierarchy

TL;DR

This paper explores the connection between colored Alexander knot polynomials and the KP hierarchy, revealing that solutions to a linear system derived from the Alexander polynomial induce KP equations, thus linking knot invariants with integrable systems.

Contribution

It demonstrates that the Alexander polynomial's properties lead to KP hierarchy equations, establishing a novel link between knot polynomials and integrable systems.

Findings

Solutions induce KP equations in Hirota form

Alexander polynomial solutions relate to KP hierarchy

New connection between knot invariants and integrability

Abstract

We discuss the relation between knot polynomials and the KP hierarchy. Mainly, we study the scaling 1-hook property of the coloured Alexander polynomial: $\mathcal{A}^\mathcal{K}_R(q)=\mathcal{A}^\mathcal{K}_{[1]}(q^{\vert R\vert})$ for all 1-hook Young diagrams $R$. Via the Kontsevich construction, it is reformulated as a system of linear equations. It appears that the solutions of this system induce the KP equations in the Hirota form. The Alexander polynomial is a specialization of the HOMFLY polynomial, and it is a kind of a dual to the double scaling limit, which gives the special polynomial, in the sense that, while the special polynomials provide solutions to the KP hierarchy, the Alexander polynomials provide the equations of this hierarchy. This gives a new connection with integrable properties of knot polynomials and puts an interesting question about the way the KP hierarchy is encoded in the full HOMFLY polynomial.