The link concordance invariant from Lee homology

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 $2^{\left|L\right|}$. The basic properties of the $s$–invariant all extend to the case of links; in particular, any orientable cobordism $\Sigma$ between links induces a map between their corresponding vector spaces which is filtered of degree $\chi \left(\Sigma \right)$. 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 $\lfloor k∕2\rfloor$. In particular, no quasi-alternating link is concordant to a split link.