  We conjecture the existence of four independent gradings in the colored
HOMFLY homology. We describe these gradings explicitly for the rectangular
colored homology of torus knots and make qualitative predictions of various
interesting structures and symmetries in the colored homology of general knots.
We also give a simple representation-theoretic model for the HOMFLY homology of
the unknot colored by any representation. While some of these structures have a
natural interpretation in the physical realization of knot homologies based on
counting supersymmetric configurations (BPS states, instantons, and vortices),
others are completely new. They suggest new geometric and physical realizations
of colored HOMFLY homology as the Hochschild homology of the category of branes
in a Landau-Ginzburg B-model or, equivalently, in the mirror A-model.
Supergroups and supermanifolds are surprisingly ubiquitous in all aspects of
this work.
