Transverse contact structures on Seifert 3-manifolds

Tight contact structures on some small Seifert fibered 3--manifolds

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.

Tight contact structures on Seifert surface complements

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…

Ozsvath-Szabo invariants and tight contact three-manifolds, III

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…

Tight small Seifert fibered manifolds with $e_0=-2$

In this paper we provide the classification of tight contact structures on some small Seifert fibered manifolds. As an application of this classification, combined with work of Lekili in \cite{L2010}, we obtain infinitely many counterexamples to a question of Honda-Kazez-Matić that asks whether a right-veering, non-destabilizable open book necessarily supports a tight contact structure.

Tight contact structures on some bounded Seifert manifolds with minimal convex boundary

We classify positive tight contact structures, up to isotopy fixing the boundary, on the manifolds $N=M(D^{2}; r_1, r_2)$ with minimal convex boundary of slope $s$ and Giroux torsion 0 along $\partial N$, where $r_1,r_2\in (0,1)\cap\mathbb{Q}$, in the following cases: (1) $s\in(-\infty, 0)\cup[2, +\infty)$; (2) $s\in[0, 1)$ and $r_1,r_2\in [1/2,1)$; (3) $s\in[1, 2)$ and $r_1,r_2\in(0,1/2)$; (4) $s=\infty$ and $r_1=r_2=1/2$. We also classify positive tight contact structures, up to isotopy fixing the boundary, on $M(D^2;1/2,1/2)$ with minimal convex boundary of arbitrary slope and Giroux torsio…

A topological characterization of symplectic fillings of Seifert 3-manifolds

In this paper, we investigate a relation between rational blowdown surgery and minimal symplectic fillings of a given Seifert 3-manifold with a canonical contact structure. Consequently, we determine a necessary and sufficient condition for a minimal symplectic filling of a Seifert 3-manifold satisfying certain conditions to be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous surface singularity. Furthermore, as an application of the main results, we prove that every minimal symplectic filling of a large family of Seifert 3-mani…

Rational Seifert Surfaces in Seifert Fibered Spaces

Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a rational Seifert surface for the link. In the case when this condition is satisfied, we generalize Seifert's algorithm to explicitly construct a rational Seifert surface for any rationally null-homologous knot. As an application of the techniques developed in the paper, we derive closed formulae for the rational Thurston-Bennequin and rotation numbers of a Legendri…

负定Seifert纤维空间上的可填充结构

Fillable structures on negative-definite Seifert fibred spaces

本文研究负定星形连接空间上的可填充接触结构，证明此类Seifert纤维空间具有唯一的负最大扭数，并通过晶格上同调的Alexander滤子计算该扭数，进一步展示这些流形上的负扭紧结构由复表面奇点的最小解析中的Stein结构诱导。

勒让德缆线空间与塞弗特纤维化链环

Spaces of Legendrian cables and Seifert fibered links

本文确定了标准接触3-球面中具最大瑟斯顿-本内昆不变量的若干勒让德纽结和链环嵌入空间的同伦型，给出充分正缆线的递归公式，证明相关接触结构空间的同伦等价性与可缩性，还发现满足单射h-原理的无穷多新分支。

小Seifert纤维空间的填充性

Fillability of small Seifert fibered spaces

本文研究小Seifert纤维空间上切触结构的填充性，通过分析平面开书的单值因子化，刻画了$M(-1;r_1,r_2,r_3)$上零扭转切触结构中的可填充结构。

通过Legendrian手术研究规范接触结构的Gompf θ-不变量

The Gompf $θ$-Invariant of Canonical Contact Structures via Legendrian Surgery

本文通过Legendrian手术描述规范接触结构，并推导出Seifert纤维化情况下的Gompf θ-不变量的闭式公式，同时证明了任意塑形树的递归公式，展示了规范接触结构在θ值上严格最小化，并排除了大量Stein可填充的接触理化球面的Sympetic有理同调球填充。

Geodesible contact structures on 3--manifolds

In this paper, we study and almost completely classify contact structures on closed 3--manifolds which are totally geodesic for some Riemannian metric. Due to previously known results, this amounts to classifying contact structures on Seifert manifolds which are transverse to the fibers. Actually, we obtain the complete classification of contact structures with negative (maximal) twisting number (which includes the transverse ones) on Seifert manifolds whose base is not a sphere, as well as partial results in the spherical case.

关于小 Seifert 3-流形的辛填充

On symplectic fillings of small Seifert $3$-manifolds

本文分类了满足特定条件的小 Seifert 3-流形的最小辛填充，并证明它们可由加权齐次复曲面奇点最小解消经有理爆破收缩得到。

Symplectic fillings of Seifert fibered spaces

We give finiteness results and some classifications up to diffeomorphism of minimal strong symplectic fillings of Seifert fibered spaces over S^2 satisfying certain conditions, with a fixed natural contact structure. In some cases we can prove that all symplectic fillings are obtained by rational blow-downs of a plumbing of spheres. In other cases, we produce new manifolds with convex symplectic boundary, thus yielding new cut-and-paste operations on symplectic manifolds containing certain configurations of symplectic spheres.

互补腿与辛有理球体

Complementary legs and symplectic rational balls

研究证明了某些Seifert纤维空间不辛界于有理同调球体，并分类了具有辛有理同调球体填充的定向球面3流形接触结构。

Singularity links with exotic Stein fillings

In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifert fibered 3-manifolds with many singular fibers and observe that these 3-manifolds are singularity links. In addition, we prove that the contact structures induced by the Stein fillings are the canonical contact structures on these singularity links. As a consequence, we verify a…

Heegaard Floer接触不变量的帽子版和加号版不等价

The hat and plus version of the Heegaard Floer contact invariant are not equivalent

研究揭示了3流形上接触结构的特性，展示存在紧致且零扭转的接触结构，其中Heegaard Floer不变量非零而加号不变量为零。

Heegaard Floer同调与最大扭转数

Heegaard Floer homology and maximal twisting numbers

本文通过调整Ozsváth-Szabó的完整路径算法，建立了Seifert纤维空间上负扭转紧致接触结构与Heegaard Floer同调群之间的对应关系，揭示了这些结构的分类及其接触不变量c^+的区分。

Contact Ozsvath-Szabo Invariants and Giroux Torsion

In this paper we prove a vanishing theorem for the contact Ozsvath--Szabo invariants of certain contact 3--manifolds having positive Giroux torsion. We use this result to establish similar vanishing results for contact structures with underlying 3--manifolds admitting either a torus fibration over the circle or a Seifert fibration over an orientable base. We also show -- using standard techniques from contact topology -- that if a contact 3--manifold (Y,ξ) has positive Giroux torsion then there exists a Stein cobordism from (Y,ξ) to a contact 3--manifold (Y,ξ') such that (Y,ξ) is obtained from…

接触手术与横结的 Cosmetic 问题及结补集问题

Cosmetic contact surgeries along transverse knots and the knot complement problem

研究了标准接触3球体中沿横结的Cosmetic接触手术，证明了非平凡手术的排除，并展示了横结由外皮接触同构类型确定的结论。

Contact surgeries and the transverse invariant in knot Floer homology

We study naturality properties of the transverse invariant in knot Floer homology under contact (+1)-surgery. This can be used as a calculational tool for the transverse invariant. As a consequence, we show that the Eliashberg-Chekanov twist knots E_n are not transversely simple for n odd and n>3.

Transverse knots, branched double covers and Heegaard Floer contact invariants

Given a transverse link in the standard contact 3-sphere, we study the contact manifold that arises as a branched double cover of the sphere. We give a contact surgery description of such manifolds, which allows to determine the Heegaard Floer contact invariants for some of them. By example of the knots of Birman--Menasco, we show that these contact manifolds may fail to distinguish between non-isotopic transverse knots. We also investigate the relation between the Heegaard Floer contact invariants of the branched double covers and the Khovanov homology, in particular, the transverse link inva…

Ozsvath-Szabo invariants and tight contact three-manifolds, I

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…

Combinatorial knot contact homology and transverse knots

We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and produces a three-variable knot polynomial related to the A-polynomial. We provide a number of computations of transverse homology that demonstrate its effectiveness in distinguishing transverse knots, including knots that cannot be distinguished by the Heegaard Floer transverse invariants or other previous invariants.

Tight Contact Structures via Admissible Transverse Surgery

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 …

On Transverse Knots and Branched Covers

We study contact manifolds that arise as cyclic branched covers of transverse knots in the standard contact 3-sphere. We discuss properties of these contact manifolds and describe them in terms of open books and contact surgeries. In many cases we show that such branched covers are contactomorphic for smoothly isotopic transverse knots with the same self-linking number. These pairs of knots include most of the non-transversely simple knots of Birman-Menasco and Ng-Ozsvath-Thurston.

Transverse Surgery on Knots in Contact 3-Manifolds

We study the effect of surgery on transverse knots in contact 3-manifolds. In particular, we investigate the effect of such surgery on open books, the Heegaard Floer contact invariant, and tightness. The overarching theme of this paper is to show that in many contexts, surgery on transverse knots is more natural than surgery on Legendrian knots. Besides reinterpreting surgery on Legendrian knots in terms of transverse knots, our main results on are in two complementary directions: conditions under which inadmissible transverse surgery (\textit{cf.\@} positive contact surgery on Legendrian knot…

Transverse universal links

We show that there exists a transverse link in the standard contact structures on the 3-sphere such that all contact 3-manifolds are contact branched covers over this transverse link.

Filtrations on the knot contact homology of transverse knots

We construct a new invariant of transverse links in the standard contact structure on R^3. This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link. Here the knot contact homology of a link in R^3 is the Legendrian contact homology DGA of its conormal lift into the unit cotangent bundle S^*R^3 of R^3, and the filtrations are constructed by counting intersections of the holomorphic disks of the DGA differential with two conormal lifts of the contact structure. We also present a combinatorial formula for the filtered DGA in terms of b…

三维中S¹不变切触形式的收缩不等式

Systolic inequalities for S1-invariant contact forms in dimension three

该研究在切触几何领域，针对欧拉数非零的Seifert丛，证明了适用于所有底层圆周作用不变切触形式的通用收缩不等式，即用切触体积一致上界闭Reeb轨道的最短周期。

Surface singularities and planar contact structures

We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its minimal symplectic fillings, and moreover, fillings cannot contain symplectic surfaces of positive genus. Applying these obstructions to canonical contact structures on links of normal surface singularities, we show that links of isolated singularities of surfaces in the complex 3-space are planar only in the case of $A_n$-singularities. In general, we characterize completely planar links of normal surface singularities (in terms of thei…

Quasi-regular Sasakian and K-contact structures on Smale-Barden manifolds

Smale-Barden manifolds are simply-connected closed 5-manifolds. It is an important and difficult question to decide when a Smale-Barden manifold admits a Sasakian or a K-contact structure. The known constructions of Sasakian and K-contact structures are obtained mainly by two techniques. These are either links (Boyer and Galicki), or semi-regular Seifert fibrations over smooth orbifolds (Kollár). Recently, the second named author of this article started the systematic development of quasi-regular Seifert fibrations, that is, over orbifolds which are not necessarily smooth. The present work is …

Admissible transverse surgery does not preserve tightness

We produce the first examples of closed, tight contact 3-manifolds which become overtwisted after performing admissible transverse surgeries. Along the way, we clarify the relationship between admissible transverse surgery and Legendrian surgery. We use this clarification to study a new invariant of transverse knots - namely, the range of slopes on which admissible transverse surgery preserves tightness - and to provide some new examples of knot types which are not uniformly thick. Our examples also illuminate several interesting new phenomena, including the existence of hyperbolic, universall…

高维切触流形中除子的稳定化

Stabilization of divisors in high-dimensional contact manifolds

该研究针对维数不小于5的切触流形中的余维2切触子流形定义稳定化操作，推导其两条核心性质，并证明大量横截链环具有非简单性，推进了高维切触拓扑领域的相关研究。

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…

透镜空间中的接触映射类群与有理解结

The contact mapping class group and rational unknots in lens spaces

本文通过单参数凸曲面理论分类透镜空间紧接触结构中的勒让德与横截有理解结，进而确定了标准接触结构的接触映射类群。

Lutz twist and contact surgery

For any knot T transverse to a given contact structure on a 3-manifold, we exhibit a Legendrian two-component link such that T equals the transverse push-off of one of the link components and contact (+1)-surgery on the link has the same effect as a Lutz twist along T.

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…

Contact structures, deformations and taut foliations

Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields is in general not path connected. Similar methods also show that the space of representations of the fundamental group of a hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Euler class is in general not path connected. As an important step along the way we resolve the question of which universally tight contact structures on Seif…

Fibered Transverse Knots and the Bennequin Bound

We prove that a nicely fibered link (by which we mean the binding of an open book) in a tight contact manifold $(M,ξ)$ with zero Giroux torsion has a transverse representative realizing the Bennequin bound if and only if the contact structure it supports (since it is also the binding of an open book) is $ξ.$ This gives a geometric reason for the non-sharpness of the Bennequin bound for fibered links. We also note that this allows the classification, up to contactomorphism, of maximal self-linking number links in these knot types. Moreover, in the standard tight contact structure on $S^3$ we cl…

Some applications of Menke's JSJ decomposition for symplectic fillings

We apply Menke's JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces. We show that exact symplectic fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a Weinstein 2-handle to an exact filling of a lens space. For large families of contact structures on Seifert fibered spaces over $S^2$, we reduce the problem of classifying exact symplectic structures to the …

Sutured Floer homology and invariants of Legendrian and transverse knots

Using contact-geometric techniques and sutured Floer homology, we present an alternate formulation of the minus and plus version of knot Floer homology. We further show how natural constructions in the realm of contact geometry give rise to much of the formal structure relating the various versions of Heegaard Floer homology. In addition, to a Legendrian or transverse knot K in a contact manifold (Y,ξ), we associate distinguished classes EHL(K) in the minus-version of knot floer homology and EHIL(K) in the plus version, which are each invariant under Legendrian or transverse isotopies of K. Th…

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.

Engel流形中的非同源横环面

Non-isotopic transverse tori in Engel manifolds

本文在Engel流形中构造了非同源但可光滑同源的横环面家族，并引入了类似自链接数的同调不变量。

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。

On the transverse invariant for bindings of open books

Consider a transverse knot which is the binding of an open book for the ambient contact manifold. In this paper, we show that the transverse invariants defined by Lisca, Ozsvath, Stipsicz, and Szabo (LOSS) are nonvanishing for such transverse knots. This is true regardless of whether or not the ambient contact structure is tight. We also prove a vanishing theorem for LOSS's Legendrian and transverse invariants. As a corollary, we show that if (T,π) is an open book with connected binding, then the complement of T has no Giroux torsion.

Transversely holomorphic flows and contact circles on spherical 3-manifolds

Motivated by the moduli theory of taut contact circles on spherical 3-manifolds, we relate taut contact circles to transversely holomorphic flows. We give an elementary survey of such 1-dimensional foliations from a topological viewpoint. We describe a complex analogue of the classical Godbillon-Vey invariant, the so-called Bott invariant, and a logarithmic monodromy of closed leaves. The Bott invariant allows us to formulate a generalised Gauß-Bonnet theorem. We compute these invariants for the Poincaré foliations on the 3-sphere and derive rigidity statements, including a uniformisation theo…

On the equivalence of Legendrian and transverse invariants in knot Floer homology

Using the grid diagram formulation of knot Floer homology, Ozsvath, Szabo and Thurston defined an invariant of transverse knots in the tight contact 3-sphere. Shortly afterwards, Lisca, Ozsvath, Stipsicz and Szabo defined an invariant of transverse knots in arbitrary contact 3-manifolds using open book decompositions. It has been conjectured that these invariants agree where they are both defined. We prove this fact by defining yet another invariant of transverse knots, showing that this third invariant agrees with the two mentioned above.

Transverse invariants and right-veering

A closed braid naturally gives rise to a transverse link in the standard contact 3-space. We study the effect of the dynamical properties of the braid monodromy, such as right-veering, on the contact-topological properties of the transverse link and its transverse invariants in knot Floer and Khovanov homologies. In particular, we show that a 3-braid is right-veering if and only if the (hat-version of) the Heegaard Floer transverse invariant is non-zero. For higher-index braids, we show that this invariant is non-zero whenever the braid monodromy has the fractional Dehn twist coefficient C>1. …

On knots in overtwisted contact structures

We prove that each overtwisted contact structure has knot types that are represented by infinitely many distinct transverse knots all with the same self-linking number. In some cases, we can even classify all such knots. We also show similar results for Legendrian knots and prove a "folk" result concerning loose transverse and Legendrian knots (that is knots with overtwisted complements) which says that such knots are determined by their classical invariants (up to contactomorphism). Finally we discuss how these results partially fill in our understanding of the "geography" and "botany"' probl…

On transverse Hopf links

We classify transverse Hopf links in the standard contact 3-space up to transverse isotopy in terms of their components' self-linking number.

Contact surgery and transverse invariants

We derive new existence results for tight contact structures on certain 3-manifolds which can be presented as surgery along specific knots in S^3. Indeed, we extend our earlier results on knots with maximal Thurston-Bennequin number being equal to 2g-1 to knots for which the maximal self-linking number satisfies the same equality. In the argument (using contact surgery) we define an invariant for transverse knots in contact 3-manifolds under the assumption that either the knot is null-homologous or the 3-manifold has no S^1xS^2-factor in its prime decomposition, and we study its properties usi…

Relative Framing of Transverse knots

It is well-known that a knot in a contact manifold $(M,C)$ transverse to a trivialized contact structure possesses the natural framing given by the first of the trivialization vectors along the knot. If the Euler class $e_C\in H^2(M)$ of $C$ is nonzero, then $C$ is nontrvivializable and the natural framing of transverse knots does not exist. We construct a new framing-type invariant of transverse knots called relative framing. It is defined for all tight $C$, all closed irreducible atoroidal $M$, and many other cases when $e_C\neq 0$ and the classical framing invariant is not defined. We show …

接触拓扑中的导数链环

Derivative links in contact topology

本文把R-链环和导数链环理论引入接触拓扑，用于刻画Lagrangian与辛可切片性，并证明Thurston-Bennequin不变量为-1的Legendrian结的切片性等价于存在紧的横截或Legendrian R-链环导数。