An infinite family of Legendrian torus knots distinguished by cube number

Legendrian 非简单 Trefoil 的 Whitehead 双曲线

Legendrian Non-simple Whitehead Doubles Of The Trefoil

研究显示 Trefoil 的 Whitehead 双曲线在标准接触 3 球面中为 Legendrian 非简单，利用结 Floer 同调和手术三角形证明其非简单性。

高维空间中非同源的单位余正则包

Legendrian non-isotopic unit conormal bundles in high dimensions

本文研究了高维空间中非同源的Legendrian子流形，通过定义非经典不变量区分无法通过经典不变量区分的单位余正则包。

Legendrian Contact Homology in Seifert Fibered Spaces

We define a differential graded algebra associated to Legendrian knots in Seifert fibered spaces with transverse contact structures. This construction is distinguished from other combinatorial realizations of contact homology invariants by the existence of orbifold points in the Reeb orbit space of the contact manifold. These orbifold points are images of the exceptional fibers of the Seifert fibered manifold, and they play a key role in the definitions of the differential and the grading, as well as the proof of invariance. We apply the invariant to distinguish Legendrian knots whose homology…

Maximal Page Crossing Numbers of Legendrian Surfaces in Closed Contact 5-Manifolds

We introduce a new Legendrian isotopy invariant for any closed orientable Legendrian surface $L$ embedded in a closed contact $5$-manifold $(M, ξ)$ which admits an "admissable" open book $(B, f)$ (supporting $ξ$) for $L$. We show that to any such $L$ and a fixed page $X$, one can assign an integer $M\mathcal{P}_{X}(L)$, called "Relative Maximal Page Crossing Number of $L$ with respect to $X$", which is invariant under Legendrian isotopies of $L$. We also show that one can extend this to a page-free invariant, i.e., one can assign an integer $M\mathcal{P}_{(B,f)}(L)$, called "Absolute Maximal P…

Legendrian结在厚ened凸曲面中的不变量

Invariants of Legendrian knots in thickened convex surfaces

本文定义了Legendrian结在厚ened凸曲面中的微分格雷厄姆代数，该代数基于分界集Γ的数据，通过计算Reeb弦的计数来生成，并证明其微分平方为零且在Legendrian同伦下不变，用于区分传统不变量无法区分的Legendrian结。

Finite Order Invariants of Legendrian, Transverse, and Framed Knots in Contact 3-manifolds

We show that for a big class of contact manifolds the groups of order $\leq n$ invariants (with values in an arbitrary Abelian group) of Legendrian, of transverse and of framed knots are canonically isomorphic. On the other hand for an arbitrary cooriented contact structure on $S^1\times S^2$ with the nonzero Euler class of the contact bundle we construct examples of Legendrian homotopic Legendrian knots $K_1$ and $K_2$ such that they realize isotopic framed knots but can be distinguished by finite order invariants of Legendrian knots in $S^1\times S^2$. We construct similar examples for a big…