An invariant of link cobordisms from Khovanov homology

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.