An invariant of link cobordisms from Khovanov homology

Jacobsson, Magnus

doi:10.2140/agt.2004.4.1211

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An invariant of link cobordisms from Khovanov homology

Magnus Jacobsson

Algebraic & Geometric Topology 4 (2004) 1211–1251

DOI: 10.2140/agt.2004.4.1211

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

Volume 4, issue 2 (2004)