Coloured Alexander polynomials and KP hierarchy

A Mironov,S Mironov,V Mishnyakov,A Morozov,A Sleptsov

doi:10.1016/j.physletb.2018.06.069

A Mironov, S Mironov + Show 3 more

Open Access

https://doi.org/10.1016/j.physletb.2018.06.069

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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: ARK(q)=A[1]K(q|R|) 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.

Highlights

Nowadays knot theory is of great interest in mathematical physics
While the special polynomials provide solutions to the KP hierarchy, the Alexander polynomials induce the equations of the KP hierarchy
Our main result is proving that KPn and Alne are literally the same polynomial spaces. This means that the group factors, which appear from the basic property (10) of the Alexander polynomial, mysteriously correspond to equations of the KP hierarchy

Summary

Introduction

Nowadays knot theory is of great interest in mathematical physics. This is due to the fact that knot invariants appear in various physical problems such as quantum field theories [1,2,3], quantum groups [4], lattice models [5], CFT [6], topological strings [7], quantum computing [8] etc. Explicit calculations and the detailed proof will be presented elsewhere This observation gives another example of integrable properties of knot polynomials, and argues in favour of use of the term “dual” in discussing the two limits of the HOMFLY polynomial. The results can be summarized in the following diagram: a→1 Another well-known perturbative expansion of the HOMFLY polynomial is the loop expansion [9], which is based on the gauge invariance of Chern-Simons theory. The theory is gauge invariant, the two object are equal, the Kontsevich integral is a perturbative expansion and the HOMFLY polynomial is not This construction gives a perturbative description of the HOMFLY polynomial with arbitrary variables q, a. This is our main result in this paper, while another our result is the number of different KP equations of each order which coincides with the number of solutions of the Alexander equations

Alexander polynomial

Alexander system of equations

The KP hierarchy

The main results

Discussion