Hecke algebras, the torus, and knots

Peter Samuelson

Double affine Hecke algebras have recently been used to construct 2-variable polynomials P(K;q,t) for algebraic knots that are conjecturally related to knot homology, and are therefore conjectured to specialize to Witten-Reshetikhin-Turaev invariants. In work with Morton we showed that the Homflypt skein algebra of the torus is isomorphic to the elliptic Hall algebra (i.e. the gl(infinity) DAHA). This identification provides a simple topological interpretation for the formula for P(K;q,t) and gives a proof that P(K;q,t) specializes to the Homfly polynomial of K.

http://scgp.stonybrook.edu/video_portal/video.php?id=1813

Simons- Workshop: Physics and mathematics of knot homologies