  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.
