Spanning trees and Khovanov homology

Details Spanning trees and Khovanov homology Spanning trees and Khovanov homology

2006-07-20

arxiv.org/abs/math/0607510v3

Proc. Amer. Math. Soc., 137 (2009), 2157-2167.

Mathematics

Geometric Topology

Results in Version 2 have been split among two papers: The current Version 3 and arXiv:0801.4937[math.GT]

Scientific paper

The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov homology. The spanning trees provide a filtration on the reduced Khovanov complex and a spectral sequence that converges to its homology. For alternating links, all differentials on the spanning tree complex are zero and the reduced Khovanov homology is determined by the Jones polynomial and signature. We prove some analogous theorems for (unreduced) Khovanov homology.

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