Lectures on Knot Homology and Quantum Curves

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, it allows one to answer questions like Is there a direct relation between Khovanov homology and the A-polynomial of a knot? which would not have been asked otherwise. 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 relation to knot contact homology and knot Floer homology.