论文标题
激发表面上的激发和结的激发和结
Instanton homology and knot detection on thickened surfaces
论文作者
论文摘要
假设$σ$是一个零属的面向面向的表面（可能带有边界），而L是$（-1,1）\timesς$的内部链接。我们证明，当且仅当l是$ \ {0 \} \timesς$的嵌入式结中，且仅当l是同位素的同位素时，l等级为2。结果，APS同源物检测到$（-1,1）\时代$中的UNNENOT。这是广义Khovanov同源性的第一个检测结果，该结果在无限的多种流形家族中有效，它部分解决了Arxiv中的猜想：2005.12863。我们的证明与Instanton同源性获得的先前检测结果不同，因为在这种情况下，Kronheimer-Mrowka的光谱序列的第二页与APS同源性并非同构。我们还表征了产品流形中的所有链接，这些链接具有最小的插入式同源性，这可能具有独立的兴趣。
Suppose $Σ$ is a compact oriented surface (possibly with boundary) that has genus zero, and L is a link in the interior of $(-1,1)\timesΣ$. We prove that the Asaeda-Przytycki-Sikora (APS) homology of L has rank 2 if and only if L is isotopic to an embedded knot in $\{0\}\timesΣ$. As a consequence, the APS homology detects the unknot in $(-1,1)\timesΣ$. This is the first detection result for generalized Khovanov homology that is valid on an infinite family of manifolds, and it partially solves a conjecture in arxiv:2005.12863. Our proof is different from the previous detection results obtained by instanton homology because in this case, the second page of Kronheimer-Mrowka's spectral sequence is not isomorphic to the APS homology. We also characterize all links in product manifolds that have minimal sutured instanton homology, which may be of independent interest.
