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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
An invariant of link cobordisms from Khovanov homology

Magnus Jacobsson

Algebraic & Geometric Topology 4 (2004) 1211–1251

arXiv: math.GT/0206303

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
Mathematical Subject Classification 2000
Primary: 57Q45
Secondary: 57M25
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Publication
Received: 24 January 2004
Revised: 18 November 2004
Accepted: 8 December 2004
Published: 21 December 2004
Authors
Magnus Jacobsson
Istituto Nazionale di Alta Matematica (INdAM)
Città Universitaria
P.le Aldo Moro 5
00185 Roma
Italy