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arXiv · 1203.0667

Torus HOMFLY as the Hall-Littlewood Polynomials

A. Mironov·A. Morozov·Sh. Shakirov
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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.

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A. Mironov, A. Morozov, Sh. Shakirov. 2012-03-03. Torus HOMFLY as the Hall-Littlewood Polynomials. https://doi.org/10.1088/1751-8113/45/35/355202

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