In this paper, we study the (in)sensitivity of the Khovanov functor to 4-dimensional linking of surfaces. We prove that if $L$L$L$ and $L^{\prime }$$L^{\prime }$ are split links, and $C$$C$ is a cobordism between $L$$L$ and $L^{\prime }$$L^{\prime }$ that is the union of disjoint (but possibly linked) cobordisms between the components of $L$$L$ and the components of $L^{\prime }$$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.

中文翻译：

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分裂链接之间的 Khovanov 同源性和协调性

在本文中，我们研究了 Khovanov 函子对表面的 4 维链接的（不）敏感性。我们证明如果$\mathrm{大号}$大号$L$和${\mathrm{大号}}^{\text{'}}$$L^{\prime }$是拆分链接，并且$C$$C$是之间的协调$\mathrm{大号}$$L$和${\mathrm{大号}}^{\text{'}}$$L^{\prime }$那是组件之间不相交（但可能是链接的）协同关系的联合$\mathrm{大号}$$L$和组件${\mathrm{大号}}^{\text{'}}$$L^{\prime }$, 那么 Khovanov 同源图由$C$$C$完全由各个组件的映射决定$C$$C$并且不检测组件之间的链接。作为推论，我们证明了一个强同伦-带状一致性（即，一个只能用 1 个和 2 个句柄构建的一致性）引起对 Khovanov 同源性的注入，这概括了第二作者和 Zemke 的结果. 此外，我们表明非拆分链接不能与拆分链接一致。