2012/03/03 by А. Миронов, A. Mironov, А. Морозов +2 · 30 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Braid #Character (mathematics) #Combinatorics #Discrete orthogonal polynomials #Duality (order theory) #Geometric and Algebraic Topology #Geometry #Homotopy and Cohomology in Algebraic Topology #Interpretation (philosophy) #Law #Macdonald polynomials #Mathematical physics #Mathematics #Orthogonal polynomials #Philosophy #Pure mathematics #Representation (politics) #Torus #hep-th
paper · pdf · doi:10.1088/1751-8113/45/35/355202
published in Journal of Physics A Mathematical and Theoretical 45(35), 355202 (Institute of Physics) · 9 pages
arxiv created 2012/03/03 · openalex publication_date 2012/08/13 · arxiv updated 2015/06/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.