Algebraic and Geometric Topology 4 (2004), paper no. 53, pages 1211-1251.

An invariant of link cobordisms from Khovanov homology

Magnus Jacobsson

Abstract. In [Duke Math. J. 101 (1999) 359-426], Mikhail Khovanov constructed a homology theory for oriented links, whose graded Euler characteristic is the Jones polynomial. He also explained how every link cobordism between two links induces a homomorphism between their homology groups, and he conjectured the invariance (up to sign) of this homomorphism under ambient isotopy of the link cobordism. In this paper we prove this conjecture, after having made a necessary improvement on its statement. We also introduce polynomial Lefschetz numbers of cobordisms from a link to itself such that the Lefschetz polynomial of the trivial cobordism is the Jones polynomial. These polynomials can be computed on the chain level.

Keywords. Khovanov homology, link cobordism, Jones polynomial

AMS subject classification. Primary: 57Q45. Secondary: 57M25.

DOI: 10.2140/agt.2004.4.1211

E-print: arXiv:math.GT/0206303

Submitted: 24 January 2004. (Revised: 18 November 2004.) Accepted: 8 December 2004. Published: 21 December 2004.

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Magnus Jacobsson
Istituto Nazionale di Alta Matematica (INdAM), Citta Universitaria
P.le Aldo Moro 5, 00185 Roma, Italy

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