发表机构
Max Planck Institute for Mathematics in Bonn（波恩马克斯·普朗克数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究正链环与$0$-充分链环的约化偶与奇Khovanov同调，证明正链环第一同调群由Seifert图决定，并揭示这些同调可检测纤维性，且近极值量子次数下稳定同伦类型为球谱楔和。
AI 中文摘要
在本文中，我们研究约化偶与奇Khovanov同调，特别关注正链环与$0$-充分链环。对于正链环，我们证明第一约化同调群仅支撑在单一量子次数上，在该次数上它是自由阿贝尔群，并且其秩由任何正图表的Seifert图决定。作为推论，约化偶与奇Khovanov同调以及未约化奇Khovanov同调能检测正链环中的纤维性，这扩展了先前关于未约化偶Khovanov同调的已知结果。我们进一步证明，在这三种理论中，挠群不以与未约化偶Khovanov同调中挠群相同的方式检测纤维性。我们还证明，对于$q\geq p$的正纽结的$(p,q)$-缆绳表现出相同的行为，尽管此类缆绳本身未必是正的。我们还研究了在$0$-充分链环的极值及近极值量子次数下，与约化偶和奇Khovanov同调相关的稳定同伦类型。在近极值量子次数下，我们证明约化稳定同伦类型是球谱的楔和。这与未约化情形形成对比，在未约化情形中，相应的同伦类型取决于$0$-态图是否为二部图。
英文摘要
In this article, we study reduced even and odd Khovanov homology, with a particular focus on positive and $0$-adequate links. For positive links, we show that the first reduced homology is supported in a single quantum grading, where it is free abelian, and that its rank is determined by the Seifert graph of any positive diagram. As a consequence, reduced even and odd Khovanov homology, as well as unreduced odd Khovanov homology, detect fiberedness among positive links, extending the previously known result for unreduced even Khovanov homology. We further show that the torsion in these three theories does not detect fiberedness in the same way as the torsion in unreduced even Khovanov homology. We further show that $(p,q)$-cables of positive knots with $q\geq p$ exhibit the same behavior, even though such cables need not themselves be positive. We also study the stable homotopy types associated to reduced even and odd Khovanov homologies in the extremal and almost extremal quantum gradings of $0$-adequate links. At the almost extremal quantum grading, we show that the reduced stable homotopy types are wedges of sphere spectra. This contrasts with the unreduced setting, where the corresponding homotopy types depend on whether the $0$-state graph is bipartite.
Comments26 pages
