Lenhard L. Ng
We resolve a question of Fuchs and Tabachnikov by showing that there is a Legendrian knot in standard contact three-space with zero Maslov number which is not Legendrian isotopic to its mirror. The proof uses the differential graded algebras of Chekanov.
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This paper has been withdrawn by the authors, due to a mistake pointed out by Lenny Ng and Josh Sabloff.
An elementary stabilization of a Legendrian link $L$ in the spherical cotangent bundle $ST^*M$ of a surface $M$ is a surgery that results in attaching a handle to $M$ along two discs away from the image in $M$ of the projection of the link $L$. A virtual Legendrian isotopy is a composition of stabilizations, destabilizations and Legendrian isotopies. In contrast to Legendrian knots, virtual Legendrian knots enjoy the property that there is a bijective correspondence between the virtual Legendrian knots and the equivalence classes of Gauss diagrams. We study virtual Legendrian isotopy classes o…
We classify Legendrian torus knots and figure eight knots in the tight contact structure on the 3-sphere up to Legendrian isotopy. As a corollary to this we also obtain the classification of transversal torus knots and figure eight knots up to transversal isotopy.
Using convex surfaces and Kanda's classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on $T^3$. Some of the knot types considered in this article provide new examples of non transversally simple knot types.
We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in PxR, where P is a punctured Riemann surface. As an application we show that for any integer k and any homology class h in H_1(PxR) there are k Legendrian knots all representing h which are pairwise smoothly isotopic through a formal Legendrian isotopy but which lie in mutually distinct Legendrian isotopy classes.
We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M,v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly the boundary of M is non-empty and v may be tangent to the boundary, but only in a concave fashion, and homotopy should preserve tangency type; (2) framed links L in M, up to framed isotopy; (3) triples (M,v,L), with (M,v) as above and L transversal to v, up to pseudo-Legendrian isotopy (transversality-preserving simultaneous homotopy…
We establish tools to facilitate the computation and application of the Chekanov-Eliashberg differential graded algebra (DGA), a Legendrian-isotopy invariant of Legendrian knots in standard contact three-space. More specifically, we reformulate the DGA in terms of front projection, and introduce the characteristic algebra, a new invariant derived from the DGA. We use our techniques to distinguish between several previously indistinguishable Legendrian knots and links.
Manturov Projection for Virtual Legendrian Knots in $ST^*F$
本文受 Manturov 工作启发，构造了从虚拟 Legendrian 纽结到 ST*S^2 中 Legendrian 纽结的投影 proj，并借此证明经典 Legendrian 纽结的虚拟交叉数等于交叉数、虚拟典范亏格等于典范亏格。
We introduce a theory of virtual Legendrian knots. A virtual Legendrian knot is a cooriented wavefront on an oriented surface up to Legendrian isotopy of its lift to the unit cotangent bundle and stabilization and destablization of the surface away from the wavefront. We show that the groups of Vassiliev invariants of virtual Legendrian knots and of virtual framed knots are isomorphic. In particular, Vassiliev invariants cannot be used to distinguish virtual Legendrian knots that are isotopic as virtual framed knots and have equal virtual Maslov numbers.
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…
We classify the Legendrian torus knots in S^1\times S^2 with its standard tight contact structure up to Legendrian isotopy.
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…
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…
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。
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。
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。
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.
We prove a complete classification theorem for loose Legendrian knots in an oriented 3-manifold, generalizing results of Dymara and Ding-Geiges. Our approach is to classify knots in a $3$-manifold $M$ that are transverse to a nowhere-zero vector field $V$ up to the corresponding isotopy relation. Such knots are called $V$-transverse. A framed isotopy class is simple if any two $V$-transverse knots in that class which are homotopic through $V$-transverse immersions are $V$-transverse isotopic. We show that all knot types in $M$ are simple if any one of the following three conditions hold: $1.$ …
Bounds on Legendrian Double Twist Knots
该研究针对标准紧接触三维球面中的双扭转纽结，利用凸曲面理论的旁路技术与紧接触结构分类，给出K(4,m)族最大瑟斯顿-本内奎恩代表的勒让德同痕类数量上界。
For a Legendrian $(2,n)$ torus knot or link with maximal Thurston-Bennequin number, Ekholm, Honda, and Kálmán constructed $C_n$ exact Lagrangian fillings, where $C_n$ is the $n$-th Catalan number. We show that these exact Lagrangian fillings are pairwise non-isotopic through exact Lagrangian isotopy. To do that, we compute the augmentations induced by the exact Lagrangian fillings $L$ to $\mathbb{Z}_2[H_1(L)]$ and distinguish the resulting augmentations.
On The Cost Function Associated With Legendrian Knots
本文提出衡量拓扑同痕勒让德纽结转化为勒让德同痕障碍的 Cost 函数，证明其度量性质，并用于刻画、定量稳定化定理及多类纽结计算。
Let L be a Legendrian knot in R^3 with the standard contact structure. In [10], a map was constructed from equivalence classes of Morse complex sequences for L, which are combinatorial objects motivated by generating families, to homotopy classes of augmentations of the Legendrian contact homology algebra of L. Moreover, this map was shown to be a surjection. We show that this correspondence is, in fact, a bijection. As a corollary, homotopic augmentations determine the same graded normal ruling of L and have isomorphic linearized contact homology groups. A second corollary states that the cou…
In an earlier paper we introduced rectangular diagrams of surfaces and showed that any isotopy class of a surface in the three-sphere can be presented by a rectangular diagram. Here we study transformations of those diagrams and introduce moves that allow transition between diagrams representing isotopic surfaces. We also introduce more general combinatorial objects called mirror diagrams and various moves for them that can be used to transform presentations of isotopic surfaces to each other. The moves can be divided into two types so that, vaguely speaking, type~I moves commute with type~II …
An L-infinity structure for Legendrian contact homology
本文为三维空间中的 Legendre 纽结或链环构造了扩展 Chekanov-Eliashberg 微分分次代数的 L-无穷代数，并由此在交换 Legendre 接触同调上建立 Poisson 括号，证明其是合痕不变量。
Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact La…
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。
In this article we define Lagrangian concordance of Legendrian knots, the analogue of smooth concordance of knots in the Legendrian category. In particular we study the relation of Lagrangian concordance under Legendrian isotopy. The focus is primarily on the algebraic aspects of the problem. We study the behavior of the classical invariants under this relation, namely the Thurston-Bennequin number and the rotation number, and we provide some examples of non-trivial Legendrian knots bounding Lagrangian surfaces in $D^4$. Using these examples, we are able to provide a new proof of the local Tho…
Characterizing slopes for Legendrian knots
该研究在勒让德纽结同痕类由经典不变量唯一确定的假设下，建立了光滑与接触特征斜率的关联准则，并将其应用于多种纽结的勒让德代表的特征斜率研究。
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 …
Legendrian contact homology (LCH) and its associated differential graded algebra are powerful non-classical invariants of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and noncommutative) information. To recover some of the nonlinear information while preserving computability, we introduce invariant cup and Massey products - and, more generally, an A_\infty structure - on the linearized LCH. We apply the products and A_\infty structure in three ways: to find infinite families of Legendrian knots that are not isotopic to their Le…
Examples are given of prime Legendrian knots in the standard contact 3-space that have arbitrarily many distinct Chekanov polynomials, refuting a conjecture of Lenny Ng. These are constructed using a new `Legendrian tangle replacement' technique. This technique is then used to show that the phenomenon of multiple Chekanov polynomials is in fact quite common. Finally, building on unpublished work of Yufa and Branson, a tabulation is given of Legendrian fronts, along with their Chekanov polynomials, representing maximal Thurston-Bennequin Legendrian knots for each knot type of nine or fewer cros…
Stein traces and characterizing slopes
该研究首次构造出标准切触3-球面中Stein迹等价但不同痕的无穷族Legendrian纽结，提出多种构造方法，同时给出反向结果即部分情形下Stein迹或r-手术型可唯一确定Legendrian同痕型。
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。
This paper provides the complete table of prime knot projections with their mirror images, without redundancy, up to eight double points systematically thorough a finite procedure by flypes. In this paper, we show how to tabulate the knot projections up to eight double points by listing tangles with at most four double points by an approach with respect to rational tangles of J. H. Conway. In other words, for a given prime knot projection of an alternating knot, we show how to enumerate possible projections of the alternating knot. Also to tabulate knot projections up to ambient isotopy, we in…
We use microlocal sheaf theory to show that if two knots have Legendrian isotopic conormal tori, then the knots are isotopic or mirror images.
Totally tangential $\mathbb{C}$-links and electromagnetic knots
本文提出一种算法，为任意链环类型构造实解析 Legendrian 代表元，并通过求解线性方程组获得全纯函数与 Bateman 电磁场的显式候选，同时证明电磁纽结不能被任何紧集无限期约束。
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