Double affine Hecke algebras and refined Jones polynomials of torus knots

2014/04/23 Wed 16:30 - 17:30
Ivan Cherednik
UNC at Chapel Hill

Using Hecke algebras and Verlinde algebras for calculating the Jones and
HOMFLYPT polynomials of torus knots is generally well understood, though
the explicit formulas attract a lot of attention even for the simplest knots and
are used in the theory of A-polynomials as well as in Number Theory. I will
define the DAHA-Jones (refined) polynomials of torus knots for any root
systems and any weights (practically from scratch). They generalize those
based on Quantum Groups, which was checked for types A-C-D by now.
In type A, the DAHA-superpolynomials will be introduced, presumably
coinciding with the stable Khovanov-Rozansky polynomials for sl(N) and
with those obtained via the BPS states in the M5 theory (String Theory).
If time permits, type C will be briefly dicussed, including some latest
developments in the rank one case