The link concordance invariant from Lee homology

Pardon, John

doi:10.2140/agt.2012.12.1081

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The link concordance invariant from Lee homology

John Pardon

Algebraic & Geometric Topology 12 (2012) 1081–1098

DOI: 10.2140/agt.2012.12.1081

Abstract

We use the knot homology of Khovanov and Lee to construct link concordance invariants generalizing the Rasmussen $s$ –invariant of knots. The relevant invariant for a link is a filtration on a vector space of dimension $2^{| L |}$ . The basic properties of the $s$ –invariant all extend to the case of links; in particular, any orientable cobordism $Σ$ between links induces a map between their corresponding vector spaces which is filtered of degree $χ (Σ)$ . A corollary of this construction is that any component-preserving orientable cobordism from a $Kh$ –thin link to a link split into $k$ components must have genus at least $⌊ k ∕ 2 ⌋$ . In particular, no quasi-alternating link is concordant to a split link.

Keywords

Khovanov homology, link concordance, link cobordism, Rasmussen s-invariant, slice genus

Mathematical Subject Classification 2010

Primary: 57M25, 57M27, 57Q60

References

Publication

Received: 25 July 2011

Revised: 9 February 2012

Accepted: 14 February 2012

Published: 7 May 2012

Authors

John Pardon
Department of Mathematics Stanford University 450 Serra Mall Building 380 Stanford CA 94305 USA
http://math.stanford.edu/~pardon/

Volume 12, issue 2 (2012)