Knot invariants and higher representation theory II: the categorification of quantum knot invariants

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by Mazorchuk-Stroppel and Sussan for sl(n). Our technique uses categorifications of the tensor product representations of Kac-Moody algebras and quantum groups, constructed a prequel to this paper. These categories are based on the pictorial approach of Khovanov and Lauda. In this paper, we show that these categories are related by functors corresponding to the braiding and (co)evaluation maps between representations of quantum groups. Exactly as these maps can be used to define quantum invariants attached to any tangle, their categorifications can be used to define knot homologies.