Spanning trees and Khovanov homology

Authors:
Abhijit Champanerkar and Ilya Kofman

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

MSC (2000):
Primary 57M25

DOI:
https://doi.org/10.1090/S0002-9939-09-09729-9

Published electronically:
February 4, 2009

MathSciNet review:
2480298

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9939-09-09729-9

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