Torus HOMFLY as the Hall-Littlewood Polynomials

TL;DR

This paper demonstrates that HOMFLY polynomials for torus knots can be expressed as Hall-Littlewood polynomials, providing a new interpretation of Wilson averages in 3d Chern-Simons theory and revealing deep dualities.

Contribution

It establishes a precise connection between HOMFLY polynomials for torus knots and Hall-Littlewood polynomials, advancing the understanding of knot invariants and their relation to character theory.

Findings

HOMFLY polynomials for torus knots equal Hall-Littlewood polynomials in specific representations.

The relation holds for extended polynomials, with some symmetry breaking.

Special cases reduce to single Hall-Littlewood characters for q=0 and t=0.

Abstract

We show that the HOMFLY polynomials for torus knots T[m,n] in all fundamental representations are equal to the Hall-Littlewood polynomials in representation which depends on m, and with quantum parameter, which depends on n. This makes the long-anticipated interpretation of Wilson averages in 3d Chern-Simons theory as characters precise, at least for the torus knots, and calls for further studies in this direction. This fact is deeply related to Hall-Littlewood-MacDonald duality of character expansion of superpolynomials found in arXiv:1201.3339. In fact, the relation continues to hold for extended polynomials, but the symmetry between m and n is broken, then m is the number of strands in the braid. Besides the HOMFLY case with q=t, the torus superpolynomials are reduced to the single Hall-Littlewood characters in the two other distinguished cases: q=0 and t=0.