Legendrian and Transversal Knots

Legendrian and transverse cables of positive torus knots

In this paper we classify Legendrian and transverse knots in the knot types obtained from positive torus knots by cabling. This classification allows us to demonstrate several new phenomena. Specifically, we show there are knot types that have non-destabilizable Legendrian representatives whose Thurston-Bennequin invariant is arbitrarily far from maximal. We also exhibit Legendrian knots requiring arbitrarily many stabilizations before they become Legendrian isotopic. Similar new phenomena are observed for transverse knots. To achieve these results we define and study "partially thickenable" t…

Legendrian knots, transverse knots and combinatorial Floer homology

Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.

Legendrian satellites

In this paper we study Legendrian knots in the knot types of satellite knots. In particular, we classify Legendrian Whitehead patterns and learn a great deal about Legendrian braided patterns. We also show how the classification of Legendrian patterns can lead to a classification of the associated satellite knots if the companion knot is Legendrian simple and uniformly thick. This leads to new Legendrian and transverse classification results for knots in the 3-sphere with its standard contact structure as well as a more general perspective on some previous classification results.

An atlas of Legendrian knots

We present an atlas of Legendrian knots in standard contact three-space. This gives a conjectural Legendrian classification for all knots with arc index at most 9, including alternating knots through 7 crossings and nonalternating knots through 9 crossings. Our method involves a computer search of grid diagrams and applies to transverse knots as well. The atlas incorporates a number of new, small examples of phenomena such as transverse nonsimplicity and non-maximal non-destabilizable Legendrian knots, and gives rise to new infinite families of transversely nonsimple knots.

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.

On invariants for Legendrian knots

Suppose that L is a null--homologous Legendrian knot in a contact 3--manifold. We determine the connection between the sutured invariant of the complement of L and the Legendrian invariant defined by Lisca, Ozsvath, Stipsicz and Szabo. In particular, we derive a vanishing theorem for the Legendrian invariant in the presence of Giroux torsion in the complement of the knot, and reprove several known properties of the Legendrian invariant from this perspective.

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。

Non-Orientable Lagrangian Cobordisms between Legendrian Knots

In the symplectization of standard contact $3$-space, $\mathbb R \times \mathbb R^3$, it is known that an orientable Lagrangian cobordism between a Legendrian knot and itself, also known as an orientable Lagrangian endocobordism for the Legendrian knot, must have genus $0$. We show that any Legendrian knot has a non-orientable Lagrangian endocobordism, and that the crosscap genus of such a non-orientable Lagrangian endocobordism must be a positive multiple of $4$. The more restrictive exact, non-orientable Lagrangian endocobordisms do not exist for any exactly fillable Legendrian knot but do e…

普遍紧透镜空间中的勒让德负环面纽结

Legendrian Negative Torus Knots in Universally Tight Lens Spaces

本文利用凸 Heegaard 分解刻画普遍紧透镜空间中的勒让德负环面纽结，并结合既有正环面结果完成全部勒让德环面纽结的粗等价分类。

Studying uniform thickness I: Legendrian simple iterated torus knots

We prove that the class of topological knot types that are both Legendrian simple and satisfy the uniform thickness property (UTP) is closed under cabling. An immediate application is that all iterated cabling knot types that begin with negative torus knots are Legendrian simple. We also examine, for arbitrary numbers of iterations, iterated cablings that begin with positive torus knots, and establish the Legendrian simplicity of large classes of these knot types, many of which also satisfy the UTP. In so doing we obtain new necessary conditions for both the failure of the UTP and Legendrian n…

Exact Lagrangian caps of Legendrian knots

We prove that any Legendrian knot in $(S^3,ξ_{std})$ bounds an exact Lagrangian surface in $\mathbb{R}^4\setminus B^4$ after a sufficient number of stabilizations. In order to show this, we construct a family combinatorial moves on knot projections with some additional data that correspond to Lagrangian cobordisms between knots.

Contact Surgeries on the Legendrian Figure-Eight Knot

We show that all positive contact surgeries on every Legendrian figure-eight knot in $(S^3, ξ_{\rm{std}})$ result in an overtwisted contact structure. The proof uses convex surface theory and invariants from Heegaard Floer homology.

Smoothly non-isotopic Lagrangian disk fillings of Legendrian knots

In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors.

Invariants of Legendrian and transverse knots in monopole knot homology

We use the contact invariant defined in [2] to construct a new invariant of Legendrian knots in Kronheimer and Mrowka's monopole knot homology theory (KHM), following a prescription of Stipsicz and Vértesi. Our Legendrian invariant improves upon an earlier Legendrian invariant in KHM defined by the second author in several important respects. Most notably, ours is preserved by negative stabilization. This fact enables us to define a transverse knot invariant in KHM via Legendrian approximation. It also makes our invariant a more likely candidate for the monopole Floer analogue of the "LOSS" in…

Satellites of Legendrian knots and representations of the Chekanov-Eliashberg algebra

We study satellites of Legendrian knots in R^3 and their relation to the Chekanov-Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in R^3 and augmentations of its DGA by showing that the DGA has finite-dimensional representations if and only if there exist certain rulings of satellites of the knot. We derive several consequences of this result, notably that the question of existence of ungraded finite-dimensional representations for the DGA of a Legendrian knot depends only on the topological typ…

关于与勒让德纽结相关的代价函数

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。

Vassiliev invariants of Legendrian, of transverse and framed knots in contact 3-manifolds

We show that for a large class of contact 3-manifolds the groups of Vassiliev invariants of Legendrian and of framed knots are canonically isomorphic. As a corollary, we obtain that the group of finite order Arnold's $J^+$-type invariants of wave fronts on a surface $F$ is isomorphic to the group of Vassiliev invariants of framed knots in the spherical cotangent bundle $ST^*F$ of $F$. On the other hand we construct the first examples of contact manifolds for which Vassiliev invariants of Legendrian knots can distinguish Legendrian knots that realize isotopic framed knots and are homotopic as L…

The nonuniqueness of Chekanov polynomials of Legendrian knots

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…

Knots and Contact Geometry II: Connected Sums

We study the behavior of Legendrian and transverse knots under the operation of connected sums. As a consequence we show that there exist Legendrian knots that are not distinguished by any known invariant. Moreover, we classify Legendrian knots in some non-Legendrian simple knot types.

Legendrian and transverse twist knots

In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the $m(5_2)$ knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least $n$ different Legendrian representatives with maximal Thurston--Bennequin number of the twist knot $K_{-2n}$ with crossing number $2n+1$. In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that $K_{-2n}$ has exactly $\lceil\frac{n^2}2\rceil$ Legendrian representatives with maximal Thurston--Bennequin number, and $\lceil\frac{n}{2}\rceil$ t…

Heegaard Floer invariants of Legendrian knots in contact three--manifolds

We define invariants of null--homologous Legendrian and transverse knots in contact 3--manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non--loose knots in overtwisted 3--spheres. Moreover, we apply the invariants to find transversely non--simple knot types in many overtwisted contact 3--manifolds.

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.

ST*F 中虚拟 Legendrian 纽结的 Manturov 投影

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。

Cabling and transverse simplicity

We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type K, we analyze the Legendrian knots in knot types obtained from K by cabling, in terms of Legendrian knots in the knot type K. As a corollary of this analysis, we show that the (2,3)-cable of the (2,3)-torus knot is not transversely simple and moreover classify the transverse knots in this knot type. This is the first classification of transverse knots in a non-transversely-simple knot type. We also classify Legendrian knots in this knot type and exhibit the first example of a Legendrian knot that does …

来自Legendrian双扭结的簇结构：一种比较方法

Cluster structures from Legendrian double twist knots: a comparative approach

本文系统比较了同一光滑纽结类型的不同Legendrian代表元上的簇代数，证明了双扭结家族中这些簇结构的等价性，并刻画了Maslov-m精确Lagrangian填充的存在性。

The correspondence between augmentations and rulings for Legendrian knots

We strengthen the link between holomorphic and generating-function invariants of Legendrian knots by establishing a formula relating the number of augmentations of a knot's contact homology to the complete ruling invariant of Chekanov and Pushkar.

Loose Legendrian and Pseudo-Legendrian Knots in 3-Manifolds

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: $1.$ …

Invariants of Legendrian knots in circle bundles

Let E be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in E. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian knots in the standard contact 3-space using ideas from Eliashberg, Givental, and Hofer's contact homology. The main difficulty lies in dealing with what are ostensibly 1-parameter families of generators for the DGA; these are solved using ``Morse-Bott'' techniques. As an applicat…

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。

关于 Legendrian 纽结的 Jones 多项式推广及其范畴化

On a generalization of Jones polynomial and its categorification for Legendrian Knots

本文通过新型 skein 关系推广 Jones 多项式，并构造相应的 Khovanov 型范畴化，用于刻画 Legendrian 纽结。

Virtual Legendrian Isotopy

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…

Legendrian grid number one knots and augmentations of their differential algebras

In this article we study the differential graded algebra (DGA) invariant associated to Legendrian knots in tight lens spaces. Given a grid number one diagram for a knot in L(p, q), we show how to construct a special Lagrangian diagram suitable for computing the DGA invariant for the Legendrian knot specified by the diagram. We then specialize to L(p, p - 1) and show that for two families of knots, the existence of an augmentation of the DGA depends solely on the value of p.

Augmentations and Rulings of Legendrian Knots

A connection between holomorphic and generating family invariants of Legendrian knots is established; namely, that the existence of a ruling (or decomposition) of a Legendrian knot is equivalent to the existence of an augmentation of its contact homology. This result was obtained independently and using different methods by Fuchs and Ishkhanov. Close examination of the proof yields an algorithm for constructing a ruling given an augmentation. Finally, a condition for the existence of an augmentation in terms of the rotation number is obtained.

One parameter families of Legendrian torus knots

We compute the Chekanov-Eliashberg contact homology of what we call the Legendrian closure of a positive braid. We also construct an augmentation for each such link diagram. Then we apply the monodromy techniques established in an earlier paper to a certain natural loop in the space L' of positive Legendrian (p,q) torus knots to show that L' has a nontrivial fundamental group, which is mapped non-injectively into the fundamental group of K by the map induced by inclusion (here, K is the space of all smooth positive (p,q) torus knots). This is the first example that Legendrian and classical kno…

On the Support Genus of Legendrian Knots

In this paper, we show that any topological knot or link in $S^1 \times S^2$ sits on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists. As a consequence, any knot or link type in $S^1 \times S^2$ has a Legendrian representative having support genus zero. We also show this holds for some knots and links in the lens spaces $L(p,1)$.

On the Slicing Genus of Legendrian Knots

We apply Heegaard-Floer homology theory to establish generalized slicing Bennequin inequalities closely related to a recent result of T. Mrowka and Y. Rollin proved using Seiberg-Witten monopoles.

Contact homology and one parameter families of Legendrian knots

We consider S^1-families of Legendrian knots in the standard contact R^3. We define the monodromy of such a loop, which is an automorphism of the Chekanov-Eliashberg contact homology of the starting (and ending) point. We prove this monodromy is a homotopy invariant of the loop. We also establish techniques to address the issue of Reidemeister moves of Lagrangian projections of Legendrian links. As an application, we exhibit a loop of right-handed Legendrian torus knots which is non-contractible in the space Leg(S^1,R^3) of Legendrian knots, although it is contractible in the space Emb(S^1,R^3…

Legendrian tori的非收缩环来自结的族

Non-contractible loops of Legendrian tori from families of knots

本文通过结的弦代数，提出一种拓扑方法计算Legendrian接触同调中的monodromy，并构造了一个无限族非收缩的Legendrian环。

Rulings of Legendrian knots as spanning surfaces

Each ruling of a Legendrian link can be naturally treated as a surface. For knots, the ruling is 2-graded if and only if the surface is orientable. For 2-graded rulings of homogeneous (in particular, alternating) knots, we prove that the genus of this surface is at most the genus of the knot. While this is not true in general, we do prove that the canonical genus (a.k.a. diagram genus) of any knot is an upper bound for the genera of its 2-graded rulings.

线性化切触同调中 Legendrian 纽结的挠元

Torsion in linearized contact homology for Legendrian knots

本文构造了 $\mathbb{R}^3$ 中 Legendrian 纽结，其线性化切触同调在整数环上含挠元，从而表明存在非精确填充诱导的增广，但其模 2 约化可诱导。

Rational Symplectic Field Theory for Legendrian knots

We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary number of positive punctures. The construction uses ideas from string topology.

Exceptional Legendrian torus knots

We present classification results for exceptional Legendrian realisations of torus knots. These are the first results of that kind for non-trivial topological knot types. Enumeration results of Ding-Li-Zhang concerning tight contact structures on certain Seifert fibred manifolds with boundary allow us to place upper bounds on the number of tight contact structures on the complements of torus knots; the classification of exceptional realisations of these torus knots is then achieved by exhibiting sufficiently many realisations in terms of contact surgery diagrams. We also discuss a couple of ge…

Monopole Floer homology and Legendrian knots

We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We prove several vanishing results, investigate their behavior under contact surgeries, and use this to construct many examples of non-loose knots in overtwisted 3-manifolds. We also show that these invariants are functorial with respect to Lagrangian concordance.

Lagrangian cobordisms and Legendrian invariants in knot Floer homology

We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb{R}^3$. Our results give new, computable, and effective obstructions to the existence of such cobordisms.

Climbing a Legendrian mountain range without Stabilization

We introduce a new braid-theoretic framework with which to understand the Legendrian and transversal classification of knots, namely a Legendrian Markov Theorem without Stabilization which induces an associated transversal Markov Theorem without Stabilization. We establish the existence of a nontrivial knot-type specific Legendrian and transversal MTWS by enhancing the Legendrian mountain range for the (2,3)-cable of a (2,3)-torus knot provided by Etnyre and Honda, and showing that elementary negative flypes allow us to move toward maximal tb value without having to use Legendrian stabilizatio…

Augmentations and Rulings of Legendrian Knots

For any Legendrian knot in (R^3,ker(dz-ydx)), we show that the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over Z[t,t^{-1}] is equivalent to the existence of a ruling of the front diagram, generalizing results of Fuchs, Ishkhanov, and Sabloff. We also show that any even graded augmentation must send t to -1.

Equivalence classes of augmentations and Morse complex sequences of Legendrian knots

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…

勒让德双扭转纽结的界

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。

Legendrian 纽结补集问题

The Legendrian knot complement problem

研究证明3球面中Legendrian纽结的接触同构类型决定其补集，并通过反例指出该结论不适用于链环，同时提供计算Thurston-Bennequin不变量的新方法。

Legendrian Satellites and Decomposable Concordances

We investigate the ramifications of the Legendrian satellite construction on the relation of Lagrangian cobordism between Legendrian knots. Under a simple hypothesis, we construct a Lagrangian concordance between two Legendrian satellites by stacking up a sequence of elementary cobordisms. This construction narrows the search for "non-decomposable" Lagrangian cobordisms and yields new families of decomposable Lagrangian slice knots. Finally, we show that the maximum Thurston-Bennequin number of a smoothly slice knot provides an obstruction to any Legendrian satellite of that knot being Lagrang…

The Surgery Unknotting Number of Legendrian Links

The surgery unknotting number of a Legendrian link is defined as the minimal number of particular oriented surgeries that are required to convert the link into a Legendrian unknot. Lower bounds for the surgery unknotting number are given in terms of classical invariants of the Legendrian link. The surgery unknotting number is calculated for every Legendrian link that is topologically a twist knot or a torus link and for every positive, Legendrian rational link. In addition, the surgery unknotting number is calculated for every Legendrian knot in the Legendrian knot atlas of Chongchitmate and N…

Legendrian knots and links classified by classical invariants

It is shown that Legendrian (resp. transverse) cable links in the 3-sphere with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston-Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space of the circle with its standard tight contact structure.

均匀粗细结型缆链

Cable links of uniformly thick knot types

本文基于Legendrian手术引入新方法，研究了均匀粗细但非Legendrian简单结型缆链的Legendrian实现，发现稳定化链的新同痕现象，并对充分负斜率缆链及twist knots的负缆进行了分类。

Invariants of Legendrian knots from open book decompositions

In this note, we define a new invariant of a Legendrian knot in a contact manifold using an open book decomposition supporting the contact structure. We define the support genus sg(L) of a Legendrian knot L in a contact 3-manifold (M, ξ) as the minimal genus of a page of an open book of M supporting the contact structure ξsuch that L sits on a page and the framing given by the contact structure and by the page agree. We show any null-homologous loose knot in an overtwisted contact structure has support genus zero. To prove this, we observe any topological knot or link in any 3-manifold M sits …