  It is known that knot homologies admit a physical description as spaces of
open BPS states. We study operators and algebras acting on these spaces. This
leads to a very rich story, which involves wall crossing phenomena, algebras of
closed BPS states acting on spaces of open BPS states, and deformations of
Landau-Ginzburg models.
  One important application to knot homologies is the existence of "colored
differentials" that relate homological invariants of knots colored by different
representations. Based on this structure, we formulate a list of properties of
the colored HOMFLY homology that categorifies the colored HOMFLY polynomial. By
calculating the colored HOMFLY homology for symmetric and anti-symmetric
representations, we find a remarkable "mirror symmetry" between these
triply-graded theories.
