Homological algebra of knots and BPS states

Understand

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.