Colored knot amplitudes and HallLittlewood polynomials
Abstract
The amplitudes of refined ChernSimons (CS) theory, colored by antisymmetric (or symmetric) representations, conjecturally generate the Lambda^r (or S^r) colored triply graded homology of (n,m) torus knots. This paper is devoted to the generalization of RossoJones formula to refined amplitudes, that involves nontrivial Gammafactors  expansion coefficients in the Macdonald basis. We derive from refined CS theory a linear recursion w.r.t. transformations (n,m) > (n, n+m) and (n,m) > (m,n) that fully determines these factors. Applying this recursion to (n,nk+1) torus knots colored by antisymmetric representations [1^r] we prove that their amplitudes are rectangular [n^r] HallLittlewood polynomials under k units of framing (a.k.a. the BergeronGarsia nabla) operator. For symmetric representations [r], we find the dual  qWhittaker  polynomials. These results confirm and give a colored extension of the observation of arXiv:1201.3339 that triply graded homology of many torus knots has a strikingly simple description in terms of HallLittlewood polynomials.
 Publication:

arXiv eprints
 Pub Date:
 August 2013
 DOI:
 10.48550/arXiv.1308.3838
 arXiv:
 arXiv:1308.3838
 Bibcode:
 2013arXiv1308.3838S
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Quantum Algebra
 EPrint:
 20 pages, 3 figures