A note on the Legendrian isotopy problem

Legendrian Submanifolds in $R^{2n+1}$ and Contact Homology

Contact homology for Legendrian submanifolds in standard contact $(2n+1)$-space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex $n$-space. It provides new invariants of Legendrian isotopy. Using these invariants the theory of Legendrian isotopy is shown to be very rich. For example, infinite families of pairwise non-isotopic Legendrian $n$-spheres and $n$-tori, which are indistinguishable by means of previously known invariants, are constructed. In a sense, the definition of contact homology presented in this paper is a high dimensi…

Differential algebras of Legendrian links

The problem of classification of Legendrian knots (links) up to isotopy in the class of Legendrian embeddings (Legendrian isotopy) naturally leads to the following two subproblems. The first of them is: which combinations of the three classical invariants can be realized by a Legendrian knot? (It is well-known that each invariant by itself can be realized by a Legendrian knot). The first step in this direction was done by Bennequin who proved that the Bennequin number of a Legendrian knot is less than twice its genus. The second subproblem is whether there exist a pair of Legendrian knots whic…

An energy-capacity inequality for Legendrian submanifolds

We prove that the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off is at least the sum of the $\mathbb{Z}_2$-Betti numbers of the submanifold, provided that the contact isotopy is sufficiently small when compared to the smallest Reeb chord on the Legendrian. Moreover, the established invariance enables us to use two different contact forms: one for the count of Reeb chords and another for the measure of the smallest length, under the assumption that there is a suitable symplectic cobordism from the latter to the former. The size of the contact isotopy…

伪Legendrian和Legendrian简单性在3流形中的链接

Pseudo-Legendrian and Legendrian Simplicity of Links in 3-Manifolds

该研究通过构造非简单同胚类的Legendrian链接，展示了在过扭接触结构中，Legendrian同胚与框架同胚、同伦及链接同伦之间的差异，揭示了非简单性现象在相同光滑同胚类中的共存。

勒让德链环与似曾相识时刻

Legendrian links and déjà vu moments

本文提出似曾相识勒让德链环的概念，刻画分量可由正勒让德同痕连接但同痕不可嵌入的情形，并在多种几何背景中构造实例并研究其基本性质。

关于虚拟Legendrian纽结与链的猜想

Conjectures about virtual Legendrian knots and links

本文提出并部分证明了关于virtual Legendrian knots的猜想，包括其与普通Legendrian knots的关系、与广义时空因果性的关联，并在2维M和(2+1)维时空中证明了这些猜想及Arnold的4尖点猜想版本。

Legendrian singular links and singular connected sums

We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian singular links. Moreover, we investigate several phenomena only occur in the Legendrian setup.

Rectangular diagrams of surfaces: representability

We introduce a simple combinatorial way, which we call a rectangular diagram of a surface, to represent a surface in the three-sphere. It has a particularly nice relation to the standard contact structure on $\mathbb S^3$ and to rectangular diagrams of links. By using rectangular diagrams of surfaces we are going, in particular, to develop a method to distinguish Legendrian knots. This requires a lot of technical work of which the present paper addresses only the first basic question: which isotopy classes of surfaces can be represented by a rectangular diagram. Vaguely speaking the answer is …

On the Legendrian and transverse classification of cablings

In this note we study Legendrian and transverse knots in the knot type of a (p,q)-cable of a knot K in 3-sphere. We give two structural theorems that describe when the (p,q)-cable of a Legendrian simple knot type K is also Legendrian simple.

A complete knot invariant from contact homology

We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere, along with a product on a filtered part of this homology. As a corollary, we obtain a new, holomorphic-curve proof of a result of the third author that the Legendrian isotopy class of …

On Legendrian knots and polynomial invariants

It is proved in this note that the analogues of the Bennequin inequality which provide an upper bound for the Bennequin invariant of a Legendrian knot in the standard contact three dimensional space in terms of the lower degree in the framing variable of the HOMFLY and the Kauffman polynomials are not sharp. Furthermore, the relationships between these restrictions on the range of the Bennequin invariant are investigated, which leads to a new simple proof of the inequality involving the Kauffman polynomial.

Notes on the isotopy finiteness

This is the less official, English version of the proof of the fact that every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.

A note on exact Lagrangian cobordisms with disconnected Legendrian ends

We provide in this note two relevant examples of Lagrangian cobordisms. The first one gives an example of two exact Lagrangian submanifolds which cannot be composed in an exact fashion. The second one is an example of an exact Lagrangian cobordism on which all primitive of the Liouville form is not constant on the negative end and such that the positive end is a stabilisation whereas the negative end admits augmentations. These examples emphasise point (i) of the definition of exact Lagrangian cobordisms in [6]. In order to provide such examples we construct Lagrangian immersions with single d…

Legendrian torus knots in lens spaces

In this note, we first classify all topological torus knots lying on the Heegaard torus in lens spaces, and then we study Legendrian representatives of these knots. We classify oriented positive Legendrian torus knots in the universally tight contact structures on the lens spaces up to contactomorphism.

A combinatorial description of the LOSS Legendrian knot invariant

In this note, we observe that the hat version of the Heegaard Floer invariant of Legendrian knots in contact three-manifolds defined by Lisca-Ozsvath-Stipsicz-Szabo can be combinatorially computed. We rely on Plamenevskaya's combinatorial description of the Heegaard Floer contact invariant.

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。

Note on the canonical genus of a knot

We show that every canonical Seifert surface is (up to isotopy) given by a knot diagram in which the (open) Seifert disks are pairwise disjoint.

Notes on the contact Ozsvath--Szabo invariants

We prove various results on contact structures obtained by contact surgery on a single Legendrian knot in the standard contact three--sphere. Our main tool are the contact Ozsvath--Szabo invariants.

On contact surgery

In this note we show that $+1$-contact surgery on distinct Legendrian knots frequently produces contactomorphic manifolds. We also give examples where this happens for $-1$-contact surgery. As an amusing corollary we find overtwisted contact structures that contain a large number of distinct Legendrian knots with the same classical invariants and tight complements.

Doubly slice knots and obstruction to Lagrangian concordance

In this short note we observe that a result of Eliashberg and Polterovitch allows to use the doubly slice genus as an obstruction for a Legendrian knot to be a slice of a concordance from the trivial Legendrian knot with maximal Thurston-Bennequin invariant to itself. This allows to obstruct concordances from the Pretzel knot P(3,-3,-m) when m > 3 to the unknot. Those examples are of interest because the Legendrian contact homology algebra cannot be used to obstruct such a concordance.

同伦与同调：5维流形中的2球面

Homotopy versus isotopy: 2-spheres in 5-manifolds

本文研究了5维流形中2球面的同伦与同调关系，通过Whitney移动消除二重点，证明了在特定条件下同伦等价于同调。

Waldhausen's Theorem

This note is an exposition of Waldhausen's proof of Waldhausen's Theorem: the three-sphere has a single Heegaard splitting, up to isotopy, in every genus. As a necessary step we also give a sketch of the Reidemeister-Singer Theorem.

横截结的邻域与不稳定化

Neighborhoods of transverse knots and destabilizations

本文研究横截结的唯一标准邻域及非紧致Legendrian结的结构定理，揭示了Legendrian结的一般不稳定化结果，并发现无Giroux扭转的流形具有无限多个紧致接触结构。

A 3-Variable Bracket

Kauffman's bracket is an invariant of regular isotopy of knots and links which since its discovery in 1985 it has been used in many different directions: (a) it implies an easy proof of the invariance of (in fact, it is equivalent to) the Jones polynomial; (b) it is the basic ingredient in a completely combinatorial construction for quantum 3-manifold invariants; (c) by its fundamental character it plays an important role in some theories in Physics; it has been used in the context of virtual links; it has connections with many objects other objects in Mathematics and Physics. I show in this n…