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A VOLUME FORM ON THE KHOVANOV INVARIANT

JUAN ORTIZ-NAVARRO

Department of Mathematical Sciences, University of Puerto Rico - Mayagüez, USA

https://doi.org/10.1142/S0218216511009844Cited by:0 (Source: Crossref)

Abstract

The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over ℤ is recovered.

Keywords:

AMSC: 57M25, 57M27

References

D. Bar-Natan, Algebr. and Geom. Topology 2(16), 337 (2002), DOI: 10.2140/agt.2002.2.337. Crossref, Google Scholar
D. Bar-Natan, The knot atlas, http://katlas.math.toronto.edu/ . Google Scholar
J. W. Chung and X. S. Lin, Torsion of quasi-isomorphisms, preprint (2006) , arXiv:math.AT/0608459 . Google Scholar
W. Franz, J. Reine Angew. Math. 173, 245 (1935). Google Scholar
V. Jones, Bull. Amer. Math. Soc. 12(1), 103 (1985), DOI: 10.1090/S0273-0979-1985-15304-2. Crossref, Web of Science, Google Scholar
L. H. Kauffman, Topology 26(3), 395 (1987). Crossref, Web of Science, Google Scholar
M. Khovanov, Duke Math. J. 101(3), 359 (1999), DOI: 10.1215/S0012-7094-00-10131-7. Crossref, Web of Science, Google Scholar
E. Lee, Adv. Math. 197, 554 (2005), DOI: 10.1016/j.aim.2004.10.015. Crossref, Web of Science, Google Scholar
K. Reidemeister, Homotopieringe und Linseräume, Hamburger Abhandlungen 11 (1935) . Google Scholar
D. Rolfsen , Knots and Links ( AMS Chelsea Publishing , 2003 ) . Crossref, Google Scholar
A. Shumakovitch, KhoHo version 0.9.3.5, http://www.geometrie.ch/KhoHo . Google Scholar
M. Spivak , Calculus on Manifolds ( Westview Publisher , 1965 ) . Google Scholar
M. Thistlethwaite, The Thistlethwaite link table (1991), http://katlas.math.toronto. edu/wiki/The_Thistlethwaite_Link_Table . Google Scholar
V. Turaev , Introduction to Combinatorial Torsion ( Birkhäuser Publisher , 2001 ) . Crossref, Google Scholar