  Besides offering a friendly introduction to knot homologies and quantum
curves, the goal of these lectures is to review some of the concrete
predictions that follow from the physical interpretation of knot homologies. In
particular, this interpretation allows one to pose questions that would not
have been asked otherwise, such as, "Is there a direct relation between
Khovanov homology and the A-polynomial of a knot?" We will explain that the
answer to this question is "yes," and introduce a certain deformation of the
planar algebraic curve defined by the zero locus of the A-polynomial. This
novel deformation leads to a categorified version of the Generalized Volume
Conjecture that completely describes the "color behavior" of the colored sl(2)
knot homology, and eventually to a similar version for the colored HOMFLY
homology. Furthermore, this deformation is strong enough to distinguish
mutants, and its most interesting properties include relations to knot contact
homology and knot Floer homology.
