View article

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.

An 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 antisymmetric representations, we find a remarkable “mirror symmetry” between these triply graded theories.