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Khovanov homology and cobordisms between split links
Journal of Topology ( IF 0.8 ) Pub Date : 2022-06-14 , DOI: 10.1112/topo.12244
Onkar Singh Gujral 1 , Adam Simon Levine 2
Affiliation  

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 $L^{\prime }$ 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.

中文翻译:

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 大号 $L$ 大号 大号 ' $L^{\素数}$ 是拆分链接,并且 C $加元 是之间的协调 大号 $L$ 大号 ' $L^{\素数}$ 那是组件之间不相交(但可能是链接的)协同关系的联合 大号 $L$ 和组件 大号 ' $L^{\素数}$ , 那么 Khovanov 同源图由 C $加元 完全由各个组件的映射决定 C $加元 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。
更新日期:2022-06-18
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