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It is known that knot homologies admit a physical description as spaces ofopen BPS states. We study operators and algebras acting on these spaces. Thisleads to a very rich story, which involves wall crossing phenomena, algebras ofclosed BPS states acting on spaces of open BPS states, and deformations ofLandau-Ginzburg models.  One important application to knot homologies is the existence of "coloreddifferentials" that relate homological invariants of knots colored by differentrepresentations. Based on this structure, we formulate a list of properties ofthe colored HOMFLY homology that categorifies the colored HOMFLY polynomial. Bycalculating the colored HOMFLY homology for symmetric and anti-symmetricrepresentations, we find a remarkable "mirror symmetry" between thesetriply-graded theories.

Homological algebra of knots and BPS states