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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Spanning trees and Khovanov homology


Authors: Abhijit Champanerkar and Ilya Kofman
Journal: Proc. Amer. Math. Soc. 137 (2009), 2157-2167
MSC (2000): Primary 57M25
Published electronically: February 4, 2009
MathSciNet review: 2480298
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Abstract: 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.


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Additional Information

Abhijit Champanerkar
Affiliation: Department of Mathematics, College of Staten Island, The City University of New York, Staten Island, New York 10314
Email: abhijit@math.csi.cuny.edu

Ilya Kofman
Affiliation: Department of Mathematics, College of Staten Island, The City University of New York, Staten Island, New York 10314
Email: ikofman@math.csi.cuny.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09729-9
PII: S 0002-9939(09)09729-9
Received by editor(s): May 24, 2007
Received by editor(s) in revised form: August 12, 2008
Published electronically: February 4, 2009
Additional Notes: The first author was supported by NSF grant DMS-0455978
The second author was supported by grants NSF DMS-0456227 and PSC-CUNY 60046-3637
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2009 American Mathematical Society