Spanning trees and Khovanov homology
By Abhijit Champanerkar, Ilya Kofman. 2006
62 citations recorded by the source corpus.
The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show that 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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/83c950558402dc607027b317d3bb3df8d120414a
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- On knot Floer homology in double branched covers (2007, methodology): “We follow S Wehrli’s argument [19] for Khovanov homology in establishing this result, but see also Champanerkar and Kofman [4] for an alternate approach.”
- On the spectral sequence from Khovanov homology to Heegaard Floer homology (2008, background): “Otherwise, K̃h(L) is supported in two consecutive δ-gradings, δ0 and δ1 = δ0 − 1, corresponding to the two groups K̃h(L0)[a, b] and K̃h(L1)[c, d], respectively [2, 6].”
- A combinatorial spanning tree model for knot Floer homology (2011, background): “2Note that Champanerkar–Kofman [5] and Wehrli [55] independently discovered a different spanning tree model for Khovanov homology.”