Enumerative geometry of knot homologies

I will explain an intriguing connection between homological invariants of knots and enumerative geometry of
toric Calabi-Yau 3-folds. This connection (motivated from physics) allows to formulate many existent knot
homologies (such as Khovanov homology, knot Floer homology, etc.) in a unified framework based on
counting supersymmetric configurations, called refined BPS states in the physics literature or motivic
Donaldson-Thomas invariants in the math literature. In the opposite direction, it implies certain integrality
and positivity properties for these enumerative invariants. If time permits, I will present some examples
of new predictions for knot homologies based on this connection, studied in a joint work with A.Schwarz
and C.Vafa, with N.Dunfield and J.Rasmussen, and more recently with M.Stosic.