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. We also suggest an approach to showing
that these knot homologies are functorial. Our technique uses categorifications
of the tensor products of integrable representations of Kac-Moody algebras and
quantum groups, constructed a prequel to this paper.

We give a geometric interpretation of the Jones-Ocneanu trace on the Hecke
algebra, using the equivariant cohomology of sheaves on SL(n). This
construction makes sense for all simple groups, so we obtain a generalization
of the Jones-Ocneanu trace to Hecke algebras of other types. We show that this
trace coincides with a trace defined by Gomi using Lusztig's Fourier transform,
and give a short geometric proof of Gomi's expansion of this trace in terms of
characters.