P. Ghiggini, S. Schoenenberger
Recently, there have been several breakthroughs in the classification of tight contact structures. We give an outline on how to exploit methods developed by Ko Honda and John Etnyre to obtain classification results for specific examples of small Seifert manifolds.
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We compute the Ozsváth--Szabó contact invariants for all tight contact structures on the manifolds -Σ(2,3,6n-1).
In this note, we exhibit infinite families of tight non-fillable contact manifolds supported by planar open books with vanishing Heegaard Floer contact invariants. Moreover, we also exhibit an infinite such family where the supported manifold is hyperbolic.
We study some properties of transverse contact structures on small Seifert manifolds, and we apply them to the classification of tight contact structures on a family of small Seifert manifolds.
Whether every hyperbolic 3-manifold admits a tight contact structure or not is an open question. Many hyperbolic 3-manifolds contain taut foliations and taut foliations can be perturbed to tight contact structures. The first examples of hyperbolic 3-manifolds without taut foliations were constructed by Roberts, Shareshian, and Stein, and infinitely many of them do not even admit essential laminations as shown by Fenley. In this paper, we construct tight contact structures on a family of 3-manifolds including these examples. These contact structures are described by contact surgery diagrams and…
We show the equivalence of several notions in the theory of taut foliations and the theory of tight contact structures. We prove equivalence, in certain cases, of existence of tight contact structures and taut foliations.
In this article we classify up to isotopy tight contact structures on Seifert manifolds over the torus with one singular fibre.
Classification of tight contact structures on some Seifert fibered manifolds: I
本文通过 Legendrian 手术构造接触结构获得下界，并利用凸曲面理论获得上界，对具有四条例外纤维的某些 Seifert 纤维化流形上零 Giroux 挠率的紧接触结构进行了分类。
It is a basic question in contact geometry to classify all non-isotopic tight contact structures on a given 3-manifold. If the manifold has a boundary, we need also specify the dividing set on the boundary. In this paper, we answer the classification question completely for the case of a solid torus by writing down a closed formula for the number of non-isotopic tight contact structures with any given dividing set on the boundary of the solid torus. Previously, only a few special cases were known due to work by Honda.
In this article, we prove a generalization of a theorem of Lisca-Matic to Stein cobordisms and develop a method for distinguishing certain Stein cobordisms using rotation numbers. Using these results along with standard techniques from convex surface theory and classifications of tight contact structures on certain 3-manifolds due to Honda, we classify the tight contact structures on a certain class of plumbed 3-manifolds that bound non-simply connected 4-manifolds. Moreover, we give descriptions of the Stein fillings of the Stein fillable contact structures.
We take a first step towards understanding the relationship between foliations and universally tight contact structures on hyperbolic 3-manifolds. If a surface bundle over a circle has pseudo-Anosov holonomy, we obtain a classification of "extremal" tight contact structures. Specifically, there is exactly one contact structure whose Euler class, when evaluated on the fiber, equals the Euler number of the fiber. This rigidity theorem is a consequence of properties of the action of pseudo-Anosov maps on the complex of curves of the fiber and a remarkable flexibility property of convex surfaces i…
Binding Numbers of Tight Contact Structures on $L(n,1)$
研究透镜空间\(L(n,1)\)上紧接触结构的束缚数，借助\(d_3\)不变量、平面单值化分解限制及Durst - Kegel算法得出支持这些结构的平面开卷束缚分量数下界，算出\(L(n,1)\)上普遍紧接触结构束缚数为\(n\)。
In this paper we develop a method for studying tight contact structures on lens spaces. We then derive uniqueness and non-existence statements for tight contact structures with certain (half) Euler classes on lens spaces. We also prove that any lens space admits only finitely many tight contact structures.
Real tight contact structures on lens spaces and surface singularities
本文分类了实心环面上的实紧接触结构，证明 $S^3$ 和 $\mathbb{R}P^3$ 上实紧结构唯一，并给出实透镜空间 $L(p,\pm 1)$ 上实紧结构的计数上下界，通过等变接触手术、实开书分解和实代数曲面奇点构造下界实例。
Tight contact structures on some families of small Seifert fiber spaces
本文提出一种通过比较等价 Thurston–Bennequin 标架的 Legendrian 表示来证明手术后接触结构仍不同的方法，并结合凸曲面理论分类若干小 Seifert 纤维空间族上的紧接触结构。
We exhibit a 3-manifold which admits no tight contact structure.
In this article we present infinitely many 3-manifolds admitting infinitely many universally tight contact structures each with trivial Ozsvath-Szabo contact invariants. By known properties of these invariants the contact structures constructed here are non weakly symplectically fillable.
A contact foliation is a foliation endowed with a leafwise contact structure. In this remark we explain a turbulisation procedure that allows us to prove that tightness is not a homotopy invariant property for contact foliations.
We give a complete proof of the fact that a contact structure that is sufficiently close to a Reebless foliation is universally tight.
We consider the problem of realizing tight contact structures on closed orientable three-manifolds. By applying the theorems of Hofer et al., one may deduce tightness from dynamical properties of (Reeb) flows transverse to the contact structure. We detail how two classical constructions, Dehn surgery and branched covering, may be performed on dynamically-constrained links in such a way as to preserve a transverse tight contact structure.
The Ozsvath-Szabo contact invariant is a complete classification invariant for tight contact structures on small Seifert fibered 3-manifolds which are L-spaces.
We exhibit tight contact structures on 3-manifolds that do not admit any symplectic fillings.
We classify up to isotopy the tight contact structures on small Seifert spaces with $e_0\neq0,-1,-2$. (The first version contains on the $e_0<-2$ case.)
We prove that there is a unique real tight contact structure on the 3-ball with convex boundary up to isotopy through real tight contact structures. We also give a partial classification of the real tight solid tori with the real structure being antipodal map along longitudinal and the identity along meridional direction. For the proofs, we use the real versions of contact neighborhood theorems and the invariant convex surface theory in real contact manifolds.
We determine the closed, oriented Seifert fibered 3-manifolds which carry positive tight contact structures. Our main tool is a new non-vanishing criterion for the contact Ozsvath-Szabo invariant.
We show the existence of tight contact structures on infinitely many hyperbolic three-manifolds obtained via Dehn surgeries along sections of hyperbolic surface bundles over circle.
It has been shown by V. Colin that every tight contact 3-manifold can be written as a connected sum of prime manifolds. Here we prove that the summands in this decomposition are unique up to order and contactomorphism.
Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. We present the first such classification on an infinite family of (mostly) hyperbolic 3-manifolds: surgeries on the figure-eight knot. We also determine which of the tight contact structures are symplectically fillable and which are universally tight.
Tight contact structures on a family of hyperbolic L-spaces
本文对Whitehead链环手术上的紧接触结构进行分类，首次给出无限族双曲L-空间的分类结果，并确定了Stein可填充与几乎过度扭转的结构。
Extending our earlier results, we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi--fillable, thus confirming a conjecture of Ko Honda.
We classify positive, tight contact structures on closed Seifert fibered 3-manifolds with base S^2, three singular fibers and e_0\geq 0.
Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction for a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension $5$, independent on whether the original contact manifold is tight or overtwisted.
We prove that a version of the Thurston-Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere $(M,ξ)$, whenever $ξ$ is tight. More specifically, we show that the self-linking number of a transverse link $T$ in $(M,ξ)$, such that the boundary of its tubular neighbourhood consists of incompressible tori, is bounded by the Thurston norm $||T||_T$ of $T$. A similar inequality is given for Legendrian links by using the notions of positive and negative transverse push-off. We apply this bound to compute the tau-invariant for every strongly quasi-posit…
Let Y(r) be the closed, oriented three-manifold obtained by performing rational r-surgery on the right-handed trefoil knot in the three-sphere. Using contact surgery and the Heegaard Floer contact invariants we construct positive, tight contact structures on Y(r) for every r not equal to 1. This implies, in particular, that the oriented boundaries of the positive E6 and E7 plumbings carry positive, tight contact structures, solving a well-known open problem.
Ozsvath-Szabo contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsvath-Szabo invariant vanishes. We also discuss the relation between these invariants and an invariant on T^3 and construct other examples of new phenomena in Heegaard--Floer theory. Along the way, we prove two conjectures of K Honda, W Kazez and G Matic about their contact topological quant…
We investigate the line between tight and overtwisted for surgeries on fibred transverse knots in contact 3-manifolds. When the contact structure $ξ_K$ is supported by the fibred knot $K \subset M$, we obtain a characterisation of when negative surgeries result in a contact structure with non-vanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure $ξ_{\overline{K}}$ supported by the mirror knot $\overline{K} \subset -M$. We derive several corollaries about the existence of tight contact structures, L-space knots outside $S^3$, non-planar contact …
Let $H\subseteq S^3$ be the two-component Hopf link. After choosing a Legendrian representative of $H$ with respect to the standard tight contact structure on $S^3$ we perform contact $(-1)$-surgery on the link itself. The manifold we get is a lens space together with a tight contact strucure on it, which depends on the chosen Legendrian representative. We classify its minimal symplectic fillings up to homeomorphism (and often up to diffeomoprhism), extending the results of Lisca which covers the case of universally tight structures and the article of Plamenevskaya - Van Horn Morris which desc…
Surgeries on knots and tight contact structures
本文证明对任意纽结和正有理数进行负手术得到的流形总具有紧切触结构，其紧性由非零的Heegaard Floer切触不变量检测。
The support norm $sn(ξ)$ of a contact structure $ξ$ is the minimum of the negative Euler characteristics of the pages of the open books supporting $ξ$. In this paper we prove additivity of the support norm for tight contact structures.
We employ cut and paste contact topological techniques to classify some tight contact structures on the closed, oriented genus-2 surface times the interval. A boundary condition is specified so that the Euler class of the of the contact structure vanishes when evaluated on each boundary component. We prove that there exists a unique, non-product tight contact structure in this case.
We classify tight contact structures on the small Seifert fibered 3--manifold M(-1; r_1, r_2, r_3) with r_i in (0,1) and r_1, r_2 \geq 1/2. The result is obtained by combining convex surface theory with computations of contact Ozsvath--Szabo invariants. We also show that some of the tight contact structures on the manifolds considered are nonfillable, justifying the use of Heegaard Floer theory.
Let p and n be positive integers with p>1, and let E(p,n) be the oriented 3-manifold obtained by performing pn(p-1)-1 surgery on a positive torus knot of type (p, pn+1). We prove that E(2,n) does not carry tight contact structures for any n, while E(p,n) carries tight contact structures for any n and any odd p. In particular, we exhibit the first infinite family of closed, oriented, irreducible 3-manifolds which do not support tight contact structures. We obtain the nonexistence results via standard methods of contact topology, and the existence results by using a quite delicate computation of…
We show that the canonical contact structure on the link of a normal complex singularity is universally tight. As a corollary we show the existence of closed, oriented, atoroidal 3-manifolds with infinite fundamental groups which carry universally tight contact structures that are not deformations of taut (or Reebless) foliations. This answers two questions of Etnyre.
We study weak versus strong symplectic fillability of some tight contact structures on torus bundles over the circle. In particular, we prove that almost all of these tight contact structures are weakly, but not strongly symplectically fillable. For the 3-torus this theorem was established by Eliashberg.
In this article we conjecture a 4-dimensional characterization of tightness: a contact structure is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in Yx[0,1]. An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: if a fibered link L induces a tight contact structure on Y then its fiber surface maximize Euler characteristic amongst all surfaces in Yx[0,1] with boundary L. We provide evidence for both conjectures by proving them for contact structures with non-vanishing Ozsváth-Szabó contact invariant. We…
Tight contact structures without symplectic fillings are everywhere
该研究证明高维流形中同一殆切触类内存在紧但不可辛填充的切触结构，并构造了特定条件下Liouville可填充但Weinstein不可填充的切触结构，推进了切触拓扑与辛几何的关联研究。
We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using earlier work, this establishes a connection between contact topology and the Homfly polynomial. We also show that the contact invariants of our tight contact structures form a basis for sutured F…
Dehn surgery on a knot determines a dual knot in the surgered manifold, the core of the filling torus. We consider duals of knots in $S^3$ that have a lens space surgery. Each dual supports a contact structure. We show that if a universally tight contact structure is supported, then the dual is in the same homology class as the dual of a torus knot.
It was proven in the first author's paper "Contact 3-manifolds twenty years since J. Martinet's work" (Ann. Inst. Fourier, 42(1992), 165--192) that any tight contact structure on the 3-sphere is diffeomorphic to the standard one. It was also claimed there without a proof that similar methods could be used to prove a multi-parametric version: the space of tight contact structures on $S^3$, fixed at a point, is contractible. We prove this result in the current paper.
We prove gluing theorems for tight contact structures. In particular, we rederive (as special cases) gluing theorems due to Colin and Makar-Limanov, and present an algorithm for determining whether a given contact structure on a handlebody is tight. As applications, we construct a tight contact structure on a genus 4 handlebody which becomes overtwisted after Legendrian -1 surgery, and study certain Legendrian surgeries on T^3.
Tight contact structures on toroidal plumbed 3-manifolds
该研究针对无坏顶点的配边3-流形，开展紧切触结构计数、Giroux挠率添加条件分析及Stein图构造算法研究，重点关注低顶点价数的情况并拓展至高顶点价数情形。
We exhibit an infinite family of tight contact structures with the property that none of the supporting open books minimizes the genus and maximizes the Euler characteristic of the page simultaneously, answering a question of Baldwin and Etnyre in arXiv:0910.5021 .
Non-fibered strongly quasipositive links and tightness
本文利用部分开书研究非纤维化链环中强拟正性与紧接触结构的关系，证明强拟正蕴含紧性，并构造反例说明逆命题不成立。
Tight contact structures on hyperbolic homology 3-spheres
本文构造了允许任意多个不同紧切触结构的双曲同调3-球面及其子类，并给出构造双曲L-空间的配方，同时引入几何极限下切触结构的行为研究。
We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.
Tight Contact Structures on Contact Mapping Tori and their Folded Sums
本文基于粘合过程提供了接触映射环面“折叠和”上接触结构的丛理论构造，并证明了在奇数维 $2n+1\geq 7$ 中，若公共接触折叠上的诱导接触形式无可缩 Reeb 轨道，则该折叠接触结构及接触映射环面是紧的。
Tight and overtwisted contact structures
本文围绕三维接触几何中紧与过度扭曲接触结构的二分法展开，探讨其在Eliashberg开创性工作中的起源及他对该理论的重要贡献。
Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact structures for every r not= 2g_s(K)-1, where g_s(K) is the slice genus of K. This implies, in particular, that the Brieskorn spheres -Sigma(2,3,4) and -Sigma(2,3,3) carry tight, positive contact structures. As an application of our main result we show that for each m in N there exists a Seifert fibered rational homology 3-sphere M_m carrying at lea…
We characterize L-spaces which are Seifert fibered over the 2-sphere in terms of taut foliations, transverse foliations and transverse contact structures. We give a sufficient condition for certain contact Seifert fibered 3-manifolds with e_0=-1 to have nonzero contact Ozsvath--Szabo invariants. This yields an algorithm for deciding whether a given small Seifert fibered L-space carries a contact structure with nonvanishing contact Ozsvath--Szabo invariant. As an application, we prove the existence of tight contact structures on some 3-manifolds obtained by integral surgery along a positive tor…
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…
We develop new techniques in the theory of convex surfaces to prove complete classification results for tight contact structures on lens spaces, solid tori, and T^2 X I. Erratum: In this note we seek to remedy errors which appeared in version 2 and were propagated in subsequent papers.
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