Khovanov homology and cobordisms between split links

TL;DR

This paper investigates how Khovanov homology behaves under certain cobordisms between split links, showing it cannot detect linking between components and establishing new results on ribbon concordance.

Contribution

It proves that Khovanov homology maps for split links depend only on individual components, not linking, and demonstrates that strongly homotopy-ribbon concordances induce injections on Khovanov homology.

Findings

Khovanov maps for split links are determined by component maps

Strongly homotopy-ribbon concordances induce injections on Khovanov homology

Non-split links cannot be ribbon concordant to split links

Abstract

In this paper, we study the (in)sensitivity of the Khovanov functor to four-dimensional linking of surfaces. We prove that if $L$ and $L'$ are split links, and $C$ is a cobordism between $L$ and $L'$ that is the union of disjoint (but possibly linked) cobordisms between the components of $L$ and the components of $L'$, then the map on Khovanov homology induced by $C$ is completely determined by the maps induced by the individual components of $C$ and does not detect the linking between the components. As a corollary, we prove that a strongly homotopy-ribbon concordance (i.e., a concordance whose complement can be built with only 1- and 2-handles) induces an injection on Khovanov homology, which generalizes a result of the second author and Zemke. Additionally, we show that a non-split link cannot be ribbon concordant to a split link.