关于纤维化$3$-流形中链环的辛Khovanov同调

Pseudoholomorphic quilts and Khovanov homology

We further study the symplectic Khovanov homology of Seidel and Smith and its generalization to even tangles. This homology theory is a conjectural geometric model for Khovanov homology. In this paper we uncover structures on symplectic Khovanov homology which have analogues in Khovanov homology. To each elementary (as well as minimal) cobordism between two tangles we associate a homomorphism between the symplectic Khovanov homology groups of the two tangles. We define the symplectic analogues $H_{s}^m$ of Khovanov's arc algebras and equip the symplectic Khovanov homology of an $(m,n)$-tangle …

辛环形Khovanov同调与固定点局部化

Symplectic annular Khovanov homology and fixed point localizations

本文提出了一种新的辛环形Khovanov同调，并建立了与链接Floer同调及轴移动映射相关的谱序列，用于研究结的拓扑性质。

Khovanov同调与Lagrangiancobordisms

Khovanov homology and Lagrangian cobordisms

本文探讨了Khovanov同调与可分解Lagrangiancobordisms之间的关系，通过过滤不变量提供了部分答案及障碍条件。

Khovanov homology and tight contact structures

Using the relation between Khovanov homology and the Heegaard Floer homology of branched double covers, we show how Khovanov homology can be used to establish tightness of branched double covers of certain transverse knots. We give examples of several infinite families of knots whose branched covers are tight for Khovanov-homological reasons, and show that some of these branched covers are not Stein fillable.

An invariant of link cobordisms from symplectic Khovanov homology

Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology groups, corresponding to isotopy classes of smooth link cobordisms in 4D between a fixed pair of links. These morphisms define a functor from the category of links and such cobordisms to the category of abelian groups and group homomorphisms up to a sign ambiguity. This provides an extra structure for symplectic Khovanov homology and more generally an isotopy inva…

Khovanov homology detects the trefoils

We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a bypass exact triangle in sutured instanton homology, proven here; and Kronheimer and Mrowka's spectral sequence relating Khovanov homology with singular instanton knot homology. As a byproduct, we also strengthen a result of Kronheimer and Mrowka on $SU(2)$ representations of the knot group.

关于Khovanov同调中突变不变性的研究

On mutation invariance in Khovanov homology

本文研究了Khovanov同调在Conway突变下的不变性，并通过同调镜像对称方法对相关不变量进行了分类。

端Khovanov同调与异乎寻常的拉格朗日平面

End Khovanov homology and exotic Lagrangian planes

本研究定义了四维空间真嵌入曲面的环境微分同胚不变量——端Khovanov同调，并借助该不变量首次发现了四维空间中的异乎寻常拉格朗日平面与辛平面实例。

Legendrian links and the spanning tree model for Khovanov homology

We use the spanning tree model for Khovanov homology to study Legendrian links. This leads to an alternative proof for Ng's Khovanov bound for the Thurston-Bennequin number and to both a necessary and a sufficient condition for this bound to be sharp.

Khovanov同调无法检测的Legendrian简单纽结

Legendrian simple knots not detected by Khovanov homology

本研究构造了Khovanov同调无法检测的Legendrian简单纽结，推翻了Chernov和Maguire的相关猜想。

Khovanov homology, open books, and tight contact structures

We define the reduced Khovanov homology of an open book (S,h), and we identify a distinguished "contact element" in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,h). Our construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for several contact structures which are not Stein-fillable. Lastly, we investigate a comultiplication structure on the reduced Khovanov homology o…

A Legendrian Thurston-Bennequin bound from Khovanov homology

We establish an upper bound for the Thurston-Bennequin number of a Legendrian link using the Khovanov homology of the underlying topological link. This bound is sharp in particular for all alternating links, and knots with nine or fewer crossings.

关于涡旋结、扭结和Legendrian简单结的Khovanov同调猜想

Conjectures on the Khovanov Homology of Torus Knots, Twist Knots, and Legendrian Simple Knots

本文基于Khovanov同调的性质，提出涡旋结和扭结的同调能区分所有Legendrian简单结的猜想，并通过大量数据验证其有效性。