Burak Ozbagci
We construct an open book decomposition compatible with a contact structure given by a rational contact surgery on a Legendrian link in the standard contact $S^3$. As an application we show that some rational contact surgeries on certain Legendrian knots induce overtwisted contact structures.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Let $(M,ξ)$ be a contact 3-manifold. We present two new algorithms, the first of which converts an open book $(Σ,Φ)$ supporting $(M,ξ)$ with connected binding into a contact surgery diagram. The second turns a contact surgery diagram for $(M,ξ)$ into a supporting open book decomposition. These constructions lead to a refinement of a result of Ding-Geiges, which states that every such $(M,ξ)$ may be obtained by contact surgery from $(S^{3},ξ_{std})$, as well as bounds on the support norm and genus of contact manifolds obtained by surgery in terms of classical link data. We then introduce Kirby …
By recent results of Baker--Etnyre--Van Horn-Morris, a rational open book decomposition defines a compatible contact structure. We show that the Heegaard Floer contact invariant of such a contact structure can be computed in terms of the knot Floer homology of its (rationally null-homologous) binding. We then use this description of contact invariants, together with a formula for the knot Floer homology of the core of a surgery solid torus, to show that certain manifolds obtained by surgeries on bindings of open books carry tight contact structures. Possible applications to lens space surgerie…
We describe explicit horizontal open books on some Seifert fibered 3--manifolds. We show that the contact structures compatible with these horizontal open books are Stein fillable and horizontal as well. Moreover we draw surgery diagrams for some of these contact structures.
We construct, somewhat non-standard, Legendrian surgery diagrams for some Stein fillable contact structures on some plumbing trees of circle bundles over spheres. We then show how to put such a surgery diagram on the pages of an open book for $S^3,$ with relatively low genus. Thus we produce open books with low genus pages supporting these Stein fillable contact structures, and in many cases it can be shown that these open books have minimal genus pages.
Using square bridge position, Akbulut-Ozbagci and later Arikan gave algorithms both of which construct an explicit compatible open book decomposition on a closed contact $3$-manifold which results from a contact $(\pm 1)$-surgery on a Legendrian link in the standard contact $3$-sphere. In this article, we introduce the ``generalized square bridge position'' for a Legendrian link in the standard contact $5$-sphere and partially generalize this result to the dimension five via an algorithm which constructs relative open book decompositions on relative contact pairs.
This paper describes a characterization of tightness of closed contact 3-manifolds in terms of supporting open book decompositions. The main result is that tightness of a closed contact 3-manifold is preserved under Legendrian surgery.
Embedded contact knot homology (ECK) is a variation on Embedded contact homology (ECH), defined with respect to an open book decomposition compatible with a contact structure on some 3-manifold, M. The knot in question is given by the (null-homologous) binding of the open book and the chain complex is defined in terms of closed orbits of the Reeb vector field and certain pseudoholomorphic curves in the symplectization of the knot complement. In this thesis we first generalize this construction to the case of rational open book decompositions, allowing us to define ECK for rationally null-homol…
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…
Let $T$ denote a binding component of an open book $(Σ, ϕ)$ compatible with a closed contact 3-manifold $(M, ξ)$. We describe an explicit open book $(Σ', ϕ')$ compatible with $(M, ζ)$, where $ζ$ is the contact structure obtained from $ξ$ by performing a full Lutz twist along $T$. Here, $(Σ', ϕ')$ is obtained from $(Σ, ϕ)$ by a \emph{local} modification near the binding.
We prove that for any contact 3-manifold supported by a spinal open book decomposition with planar pages, there is a universal bound on the Euler characteristic and signature of its minimal symplectic fillings. The proof is an application of the spine removal surgery operation recently introduced in joint work of the authors with Van Horn-Morris.
Twists of contact structures in dimension 3 and higher are studied in this paper from a viewpoint of contact round surgery. Three kinds of new modifications of contact structures which are higher-dimensional generalizations of the $3$-dimensional Lutz twists are introduced. One of the operations makes a contact manifold overtwisted in any dimension. Another one makes it weakly symplectically non-fillable. And the other one makes it strongly symplectically non-fillable. In other words, they make the overtwisted disc, the bordered Legendrian open books, and the Giroux domains, respectively. The …
We describe Milnor open books and Legendrian surgery diagrams for canonical contact structures of links of some rational surface singularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Milnor norm) is strictly greater than the support genus (resp. support norm) of the canonical contact structure, for each member of this family.
We construct a Seifert surface for a given null-homologous transverse link in a contact manifold that is compatible with a planar open book decomposition, then obtain a formula of the self-linking number. It extends Bennequin's self-linking number formula for braids in the standard contact 3-sphere.
We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, we describe horizontal open books on some Seifert fibered 3--manifolds. As another application we describe horizontal open books isomorphic to Milnor open books for some complex surface singularities. Moreover we give examples of tight contact 3--manifolds supported by planar open books. As a cons…
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…
For a nullhomologous Legendrian knot in a closed contact 3-manifold Y we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of Y is mapped by a surgery cobordism to the contact invariant of the result of contact surgery. In addition we characterize the spin-c structure on the cobordism that induces the relevant map. As a consequence we determine necessary and sufficient conditions for the nonvanishing of the contact invariant after rational surgery when Y is the standard 3-sphere, generalizing previou…
If (S,h) is an open book with disconnected binding then we can form a new open book (S',h') by capping off one of the boundary components of S with a disk. We define a U-equivariant map on Heegaard Floer homology which sends c^+(S',h') to c^+(S,h), and we discuss various applications. In particular, we determine the support genera of almost all contact structures compatible with genus one, one boundary component open books. In addition, we compute the 3-dimensional invariant associated to any contact structure with non-vanishing contact invariant which is compatible with a genus one open book …
As an application of the construction of open books on plumbed 3-manifolds, we construct elliptic open books on torus bundles over the circle. In certain cases these open books are compatible with Stein fillable contact structures and have minimal genus.
Trisecting a 4-dimensional book into three chapters
本文提出了一种算法，用于将四维流形的开放书分解转换为三等分图，并展示了其在拓扑构造中的应用和推广作用。
This is a survey on contact open books and contact Dehn surgery. The relation between these two concepts is discussed, and various applications are sketched, e.g. the monodromy of Stein fillable contact 3-manifolds, the Giroux-Goodman proof of Harer's conjecture on fibred links, construction of symplectic caps to fillings (Eliashberg, Etnyre), and detection of non-loose Legendrian knots with the help of contact surgery.
We study an explicit construction of planar open books with four binding components on any three-manifold which is given by integral surgery on three component pure braid closures. This construction is general, indeed any planar open book with four binding components is given this way. Using this construction and results on exceptional surgeries on hyperbolic links, we show that any contact structure of S^3 supports a planar open book with four binding components, determining the minimal number of binding components needed for planar open books supporting these contact structures. In addition,…
A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real structure. We call them real open book decompositions. We show that each real open book carries a real contact structure and two real contact structures supported by the same real open book decomposition are equivariantly isotopic. We also show that every real contact structure on a closed 3-dimensional real manifold is supported by a real open book. Finally, we…
We find a self-linking number formula for a given null-homologous transverse link in a contact manifold that is compatible with either an annulus or a pair of pants open book decomposition. It extends Bennequin's self-linking formula for a braid in the standard contact $3$-sphere.
In the present paper we describe compatible open books for the fibre connected sum along binding components of open books, as well as for the fibre connected sum along multi-sections of open books. As an application the first description provides simple ways of constructing open books supporting all tight contact structures on $T^3$, recovering a result by van Horn-Morris, as well as an open book supporting the result of a Lutz twist along a binding component of an open book, recovering a result by Ozbagci--Pamuk.
We introduce the notion of a nested open book, a submanifold equipped with an open book structure compatible with an ambient open book, and describe in detail the special case of a push-off of the binding of an open book. This enables us to explicitly describe a natural open book decomposition of a fibre connected sum of two open books along their bindings, provided they are diffeomorphic and admit an open book structure themselves. Furthermore, we apply the results to contact open books, showing that the natural open book structure of a contact fibre connected sum of two adapted open books al…
The algorithm given by Akbulut-Ozbagci constructs an explicit open book decomposition on a contact three-manifold described by a contact surgery on a link in the three-sphere. In this article, we will improve this algorithm by using Giroux's contact cell decomposition process. Our algorithm is more economical on choosing the supporting genus of the open book; in particular it gives a good upper bound for the recently defined ``minimal supporting genus invariant'' of contact structures.
On symplectic fillings of spinal open book decompositions I: Geometric constructions
本文介绍了脊柱开放书与接触结构及辛或Stein结构的关系，提出新的辛cobordism构造脊柱移除手术，扩展了Eliashberg、Gay-Stipsicz和第三作者的构造，并应用该手术得到新的非强（有时非弱）辛填充的接触流形例子。
We study open books on three manifolds which are compatible with an overtwisted contact structure. We show that the existence of certain arcs, called sobering arcs, is a sufficient condition for an open book to be overtwisted, and is necessary up to stabilization by positive Hopf-bands. Using these techniques we prove that some open books arising as the boundary of symplectic configurations are overtwisted, answering a question of Gay in Algebr. Geom. Topol. 3 (2003) 569--586.
We prove that transverse links in any contact manifold $(M,ξ)$ can be realized as a sub-binding of a compatible open book decomposition. We define the support genus of a transverse link and prove that the support genus of a transverse knot is zero if there is an overtwisted disk disjoint from it. Next, we find a relationship between the support genus of a transverse link and its Legendrian approximation.
We demonstrate how to combinatorially calculate the EH-class of a compatible contact structure in the sutured Floer homology group of a balanced sutured three manifold which is associated to an abstract partial open book decomposition. As an application we show that every contact three manifold (closed or with convex boundary) can be obtained by gluing tight contact handlebodies whose EH-classes are nontrivial.
We determine parts of the contact homology of certain contact 3-manifolds in the framework of open book decompositions, due to Giroux. We study two cases: when the monodromy map of the compatible open book is periodic and when it is pseudo-Anosov. For an open book with periodic monodromy, we verify the Weinstein conjecture. In the case of an open book with pseudo-Anosov monodromy, suppose the boundary of a page of the open book is connected and the fractional Dehn twist coefficient $c={k\over n}$, where $n$ is the number of prongs along the boundary. If $k\geq 2$, then there is a well-defined …
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…
In this note, we use the recent work of Honda-Kazez-Matic [HKM] to prove that a closed contact 3-manifold admitting a compatible open book decomposition with a nontrivial monodromy which can be presented as a product of left handed Dehn twists is overtwisted.
Heegaard splittings and the tight Giroux Correspondence
该研究引入紧Heegaard分裂概念，借助加细过程建立其与开书的关联，利用Heegaard分裂与凸曲面技术给出紧切触3-流形上Giroux对应的新证明。
We take advantage of the correspondence between fibered links, open book decompositions and contact structures on a closed connected 3-dimensional manifold to determine a mixed link diagram presentation for a particular fibered link $L$ in the lens space $L(p,q)$. Moreover, we construct a diagram for the lift of $L$ in the 3-sphere $S^3$ in order to show that, for $q=1$, $L$ is compatible with the standard contact structure of $L(p,1)$ when $p \leq -1$.
An algorithm to Legendrian realize a curve on a ribbon surface
本文提出在标准接触结构下，通过算法将带状面上的非平凡闭合曲线Legendrian化，并应用于将开书的单调同胚转换为支撑接触流形的接触手术图。
We study contact structures compatible with genus one open book decompositions with one boundary component. Any monodromy for such an open book can be written as a product of Dehn twists around dual non-separating curves in the once-punctured torus. Given such a product, we supply an algorithm to determine whether the corresponding contact structure is tight or overtwisted. We rely on Ozsv{á}th-Szab{ó} Heegaard Floer homology in our construction and, in particular, we completely identify the $L$-spaces with genus one, one boundary component, pseudo-Anosov open book decompositions. Lastly, we r…
Suppose that S is a surface with boundary and that g and h are diffeomorphisms of S which restrict to the identity on the boundary. Let Y_g, Y_h, and Y_{hg} be the three-manifolds with open book decompositions given by (S,g), (S,h), and (S,hg), respectively. We show that the Ozsvath-Szabo contact invariant is natural under a comultiplication map on Heegaard Floer homology. It follows that if the contact invariants associated to the open books (S, g) and (S, h) are non-zero then the contact invariant associated to the open book (S, hg) is also non-zero. We extend this comultiplication to a map …
Liouville hypersurfaces and connect sum cobordisms
本文提出 Liouville 超曲面与 Liouville 连通和操作，推广了 Weinstein 手柄附着与切触手术，并利用其构造了高维奇异切触结构，揭示了特定球面 Dehn 扭转的辛非平凡性及流形的不可填充性。
We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold $(M, ξ, \mathcal{F})$ whose convex boundary is equipped with a signed singular foliation $\mathcal{F}$ closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux Correspondence, as described in earlier work of Licata and Vértesi. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well…
We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set $\mathbb{Z}_{\geq0}\cup\{\infty\}$. It is zero for overtwisted contact structures, $\infty$ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. As an application, we obstruct Stein fillability on contact 3-manifolds with non-vanishing Ozsváth-Szabó contact class.
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。
Quantization of Contact 3-Manifolds and the Reeb Gravitational Field
提出闭接触3-流形的典范几何量子化方案，通过嵌入到C^3构造全纯线丛，证明Reeb向量场在Sasakian条件下是测地线且模拟爱因斯坦引力，并给出区分T^3上紧接触结构的量子不变量。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。