Mathematics – Geometric Topology
Scientific paper
2010-05-25
Mathematics
Geometric Topology
38 pages, numerous TikZ figures. DVI will not view correctly on all machines, PDF is preferred. This paper is a reworking of m
Scientific paper
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. We also investigate the structure of the categorifications of tensor products more thoroughly; in particular, we show that the projectives categorify Lusztig's canonical tensor of tensor products when the algebra is of symmetric type (the corresponding result for non-symmetric type is false).
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