Torus HOMFLYPT as the Hall–Littlewood polynomials

Abstract

We show that the HOMFLYPT polynomials for the torus knots T[m, n] in all fundamental representations are equal to the Hall–Littlewood polynomials in a representation which depends on m, and with a 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 the Hall–Littlewood–MacDonald duality of character expansion of superpolynomials found in Mironov et al (2012 J. Geom. Phys. 62 148–55). 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 HOMFLYPT 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.