The link concordance invariant from Lee homology

TL;DR

This paper extends the Rasmussen s-invariant from knots to links using Lee homology, providing new link concordance invariants and insights into cobordisms and link properties.

Contribution

It introduces a link concordance invariant based on Lee homology, generalizing the s-invariant and analyzing its implications for link cobordisms and properties.

Findings

Defines a filtration-based invariant for links

Establishes cobordism maps with filtered degree equal to Euler characteristic

Shows no quasi-alternating link is concordant to a split link

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 $\Sigma$ between links induces a map between their corresponding vector spaces which is filtered of degree $\chi(\Sigma)$. 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 $\lfloor\frac k2\rfloor$. In particular, no quasi-alternating link is concordant to a split link.