Interplay between MacDonald and Hall-Littlewood expansions of extended torus superpolynomials

TL;DR

This paper explores the relationship between MacDonald and Hall-Littlewood polynomial expansions of extended torus superpolynomials, providing new formulas and computational methods for these algebraic objects in knot theory.

Contribution

It introduces a novel approach to describe coefficients in superpolynomial expansions using Hall-Littlewood polynomials, enhancing understanding of their algebraic structure.

Findings

Derived explicit formulas for superpolynomials with small parameters

Tested formulas up to 17 strands in braid representations

Established a dual m-evolution description using Hall-Littlewood polynomials

Abstract

In arXiv:1106.4305 extended superpolynomials were introduced for the torus links T[m,mk+r], which are functions on the entire space of time variables and, at expense of reducing the topological invariance, possess additional algebraic properties, resembling those of the matrix model partition functions and the KP/Toda tau-functions. Not surprisingly, being a suitable extension it actually allows one to calculate the superpolynomials. These functions are defined as expansions into MacDonald polynomials, and their dependence on k is entirely captured by the action of the cut-and-join operator, like in the HOMFLY case. We suggest a simple description of the coefficients in these character expansions, by expanding the initial (at k=0) conditions for the k-evolution into the new auxiliary basis, this time provided by the Hall-Littlewood polynomials, which, hence, play a role in the description of the dual m-evolution. For illustration we list manifest expressions for a few first series, mk\pm 1, mk\pm 2, mk\pm 3, mk\pm 4. Actually all formulas were explicitly tested up to m=17 strands in the braid.