Khovanov homology and cobordisms between split links

Abstract

In this paper, we study the (in)sensitivity of the Khovanov functor to 4‐dimensional linking of surfaces. We prove that if L$L$ and L′$L^{\prime }$ are split links, and C$C$ is a cobordism between L$L$ and L′$L^{\prime }$ that is the union of disjoint (but possibly linked) cobordisms between the components of L$L$ and the components of L′$L^{\prime }$, then the map on Khovanov homology induced by C$C$ is completely determined by the maps induced by the individual components of C$C$ and does not detect the linking between the components. As a corollary, we prove that a strongly homotopy–ribbon concordance (that is, 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.