Character expansion for HOMFLY polynomials. I. Integrability and difference equations

TL;DR

This paper introduces a novel approach to analyzing HOMFLY polynomials by expanding them into Schur functions, revealing integrable structures and difference equations that relate to knot invariants and algebraic systems.

Contribution

It proposes associating HOMFLY polynomial coefficients with Schur functions and Racah symbols, extending these polynomials to a broader space, and uncovering their integrable and difference equation properties.

Findings

Extended HOMFLY polynomials depend on braid representation.

Torus knot coefficients satisfy Plucker relations and relate to KP tau-functions.

Derived difference equations (A-polynomials) for knot polynomials.

Abstract

We suggest to associate with each knot the set of coefficients of its HOMFLY polynomial expansion into the Schur functions. For each braid representation of the knot these coefficients are defined unambiguously as certain combinations of the Racah symbols for the algebra SU_q. Then, the HOMFLY polynomials can be extended to the entire space of time-variables. The so extended HOMFLY polynomials are no longer knot invariants, they depend on the choice of the braid representation, but instead one can naturally discuss their explicit integrable properties. The generating functions of torus knot/link coefficients are turned to satisfy the Plucker relations and can be associated with tau-function of the KP hierarchy, while generic knots correspond to more involved systems. On the other hand, using the expansion into the Schur functions, one can immediately derive difference equations (A-polynomials) for knot polynomials which play a role of the string equation. This adds to the previously demonstrated use of these character decompositions for the study of beta-deformations from HOMFLY to superpolynomials.