Hao Wu
We use the Ozsváth-Szabó contact invariants to distinguish between tight contact structures obtained by Legendrian surgeries on stabilized Legendrian links in tight contact 3-manifolds. We also discuss the implication of our result on the tight contact structures on the Brieskon homology spheres $-Σ(2,3,6n-1)$.
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We obtain some obstructions to existence of Legendrian surgeries between tight lens spaces. We also study Legendrian surgeries between overtwisted contact manifolds.
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.
In this paper, we study contact surgeries along Legendrian links in the standard contact 3-sphere. On one hand, we use algebraic methods to prove the vanishing of the contact Ozsváth-Szabó invariant for contact $(+1)$-surgery along certain Legendrian two-component links. The main tool is a link surgery formula for Heegaard Floer homology developed by Manolescu and Ozsváth. On the other hand, we use contact-geometric argument to show the overtwistedness of the contact 3-manifolds obtained by contact $(+1)$-surgeries along Legendrian two-component links whose two components are linked in some sp…
In this note, we classify Stein fillings of an infinite family of contact 3-manifolds up to diffeomorphism. Some contact 3-manifolds in this family can be obtained by Legendrian surgeries on $(S^3,ξ_{std})$ along certain Legendrian 2-bridge knots. We also classify Stein fillings, up to symplectic deformation, of an infinite family of contact 3-manifolds which can be obtained by Legendrian surgeries on $(S^3,ξ_{std})$ along certain Legendrian twist knots. As a corollary, we obtain a classification of Stein fillings of an infinite family of contact hyperbolic 3-manifolds up to symplectic deforma…
We give new tightness criteria for positive surgeries along knots in the 3-sphere, generalising results of Lisca and Stipsicz, and Sahamie. The main tools will be Honda, Kazez and Matic's, Ozsvath and Szabo's Floer-theoretic contact invariants. We compute the Ozsvath and Szabo's invariant of positive contact surgeries along Legendrian knots in the 3-sphere in terms of the classical invariants of the knot. We also combine a Legendrian cabling construction with contact surgeries to get results about rational contact surgeries.
We show that every tight contact structure on any of the lens spaces $L(ns^2-s+1,s^2)$ with $n\geq 2$, $s\geq 1$, can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot $T(s,-(sn-1))$ in the tight or an overtwisted contact structure on the 3-sphere.
In this paper, sufficient conditions for contact $(+1)$-surgeries along Legendrian knots in contact rational homology 3-spheres to have vanishing contact invariants or to be overtwisted are given. They can be applied to study contact $(\pm1)$-surgeries along Legendrian links in the standard contact 3-sphere. We also obtain a sufficient condition for contact $(+1)$-surgeries along Legendrian two-component links in the standard contact 3-sphere to be overtwisted via their front projections.
We prove various results on contact structures obtained by contact surgery on a single Legendrian knot in the standard contact three--sphere. Our main tool are the contact Ozsvath--Szabo invariants.
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.
By examining knot Floer homology, we extend a result of Ozsváth and Stipsicz and show further infinitely many Legendrian and transversely non-simple knot types among two-bridge knots. We give sufficient conditions of Legendrian and transverse non-simplicity on the continued fraction expansion of the corresponding rational number.
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.
This paper completely answers the question of when contact (r)-surgery on a Legendrian knot in the standard contact structure on the 3-sphere yields a symplectically fillable contact manifold for r in (0,1]. We also give obstructions for other positive r and investigate Lagrangian fillings of Legendrian knots.
On Contact Round Surgeries on $(\mathbb{S}^3,ξ_{st})$ and Their Diagrams
本文研究了在$(\mathbb{S}^3,ξ_{st})$上的接触圆手术及其图示，证明了所有闭合连通接触3流形均可通过此类手术获得，并探讨了其与接触$(\pm1)$-手术图示的联系。
Contact $(+1)$-surgeries and algebraic overtwistedness
通过在柔 fillable 接触流形中进行接触$(+1)$-手术，若Legendrian球的同调类未被填充消去，则得到代数非紧致接触流形。该方法可推广至更一般接触流形，通过$(+1)$-手术生成非紧致流形，从而证明非紧致接触流形的接触同调消失。
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.
On contact cosmetic surgery
本文证明接触整容手术猜想对所有非平凡Legendrian结成立，但可能排除Lagrangian切片结，并讨论Legendrian平凡结的接触整容手术。
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.
Legendrian Non-simple Whitehead Doubles Of The Trefoil
研究显示 Trefoil 的 Whitehead 双曲线在标准接触 3 球面中为 Legendrian 非简单，利用结 Floer 同调和手术三角形证明其非简单性。
We study Legendrian and transverse realizations of the negative torus knots $T_{(p,-q)}$ in all contact structures on the $3$-sphere. We give a complete classification of the strongly non-loose transverse realizations and the strongly non-loose Legendrian realizations with the Thurston-Bennequin invariant smaller than $-pq$. Additionally, we show that the strongly non-loose transverse realizations $T$ are classified by their non-zero invariants $\mathfrak T(T)$ in the minus version of the knot Floer homology. However, not all the elements of $HFK^-(T_{(p,q)})$ can be realized. Along the way, w…
Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsvath-Szabo contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston-Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than the three-sphere with negative maximal Thurston-Bennequin invariant. Perhaps more interesting, …
By using grid homology theory, we give an explicit algorithm for computing Ozsváth-Stipsicz-Szabó's $Υ$-invariant and the $d$-invariant of Dehn surgeries along knots in $S^3$. As its application, we compute the two invariants for all prime knots with up to 11 crossings.
We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus g must have slope 2g-1, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston-Bennequin numbers of cables.
Algebraic overtwistedness of contact $(+1)$-surgeries via augmented Floer theories
该研究通过增广弗洛理论，证明满足特定Thurston-Bennequin数条件的勒让德n-球面的接触(+1)-手术所得流形接触同调消失，还扩展了三维情形下勒让德纽结接触手术的代数过度扭转性结果。
On the Ozsváth-Szabó $d$-invariants for almost simple linear graphs
本文改进了Karakurt和Şavk对Brieskorn同调3球体的$d$-不变量公式，并给出了不等式等于或严格成立的无限例，进而推导了结同调群的结论。
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…
Knots that share four surgeries
本文通过构造共享四个不同斜率 Dehn 手术的纽结对，以及具有接触同胚接触手术的非同位素 Legendrian 纽结对，分别反驳了纽结理论和接触几何中关于手术共享唯一性的两个猜想。
On the Legendrian invariant in knot lattice homology
研究纽结格同调中的勒让德不变量，通过将正规曲面奇点链环的接触不变量转换到格同调理论，利用亚历山大分次定义元素\(\mathcal{L}(L)\)，证明其在爆破下不变，并借助链复形的滤过链同伦给出勒让德不变量的部分组合描述。
We compute the Heegaard Floer homology of $S^3_1(K)$ (the (+1) surgery on the torus knot $T_{p,q}$) in terms of the semigroup generated by $p$ and $q$, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsvath--Szabo d-invariant. We relate the result to known knot invariants of $T_{p,q}$ as the genus and the Levine--Tristram signatures. Furthermore, we emphasize the striking resemblance between Heegaard Floer homologies of (+1) and (-1) surgeries on torus knots. This relation is best seen at the level of $τ$ functions.
We study the Ozsváth-Szabó-Thurston transverse invariant in combinatorial link Floer homology for certain transverse cables $\mathscr{L}_{p,q}$ of transverse link $L$ in $S^3$. Transverse cables $\mathscr{L}_{p,q}$ are constructed from the grid diagram of $L$. The main result is $\hatθ(\mathscr{L}_{p,q})=0$ if and only if $\hatθ(L)=0$ for $\frac{q}{p}$ sufficiently large. We also prove a similar result for invariants of Legendrian knots. Our proof uses an inclusion map $i$ of certain grid complexes associated to $L$ and $L_{p,q}$. We use these results to generate many infinite families of exam…
We introduce a generalization of the Lisca-Ozsváth-Stipsicz-Szabó Legendrian invariant $\mathfrak L$ to links in every rational homology sphere, using the collapsed version of link Floer homology. We represent a Legendrian link $L$ in a contact 3-manifold $(M,ξ)$ with a diagram $D$, given by an open book decomposition of $(M,ξ)$ adapted to $L$, and we construct a chain complex $cCFL^-(D)$ with a special cycle in it denoted by $\mathfrak L(D)$. Then, given two diagrams $D_1$ and $D_2$ which represent Legendrian isotopic links, we prove that there is a map between the corresponding chain complex…
Contact cosmetic surgery on Legendrian knots in integer homology sphere $L$-spaces
研究整数同调球L-空间中Legendrian纽结的接触整容手术，证明除可能的拉格朗日切片纽结外，所有非平凡Legendrian纽结均满足接触整容手术猜想。
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.
In this paper, we introduce a rational $τ$ invariant for rationally null-homologous knots in contact 3-manifolds with nontrivial Ozsváth-Szabó contact invariants. Such an invariant is an upper bound for the sum of rational Thurston-Bennequin invariant and the rational rotation number of the Legendrian representatives of the knot. In the special case of Floer simple knots in L-spaces, we can compute the rational $τ$ invariants by correction terms.
Negative contact surgery on Legendrian non-simple knots
研究Legendrian非简单结的不同LOSS不变量的负有理接触r-手术产生不同接触3流形，首次展示相同经典不变量但所有负有理数r手术不同的例子，并推广至某些两桥结家族。
We give an explicit formula to compute the rotation number of a nullhomologous Legendrian knot in contact (1/n)-surgery diagrams along Legendrian links and obtain a corresponding result for the self-linking number of transverse knots. Moreover, we extend the formula by Ding-Geiges-Stipsicz for computing the d3-invariant to (1/n)-surgeries.
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.
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 …
In arXiv:1611.09927, we constructed a well-defined Lagrangian Floer invariant for any closed, oriented $3$-manifold $Y$ via the symplectic geometry of so-called traceless $\mathrm{SU}(2)$-character varieties. This invariant, $\mathrm{SI}(Y)$, which we refer to as the symplectic instanton homology of $Y$, was also shown to satisfy an exact triangle for Dehn surgeries on knots which is typical of Floer-theoretic invariants of $3$-manifolds. In this article, we demonstrate further structural properties of this symplectic instanton homology. For example, Floer theories are expected to roughly sati…
On contact surgery and knot Floer invariants
本文建立了接触手术与Legendrian纽结Floer不变量之间的非消失关系，利用缝合Floer同调及极限构造，证明了接触不变量非零蕴含相应Legendrian不变量非零。
Naturality of Legendrian LOSS invariant under positive contact surgery
本文推广了LOSS不变量在正接触手术下的自然性，刻画了有理零调情形下的Spin^c结构，并由此构造出由LOSS不变量区分的无限多Legendrian非简单纽结新例子。
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。
We utilize the Ozsvath-Szabo contact invariant to detect the action of involutions on certain homology spheres that are surgeries on symmetric links, generalizing a previous result of Akbulut and Durusoy. Potentially this may be useful to detect different smooth structures on 4-manifolds by cork twisting operation.
In two previous papers, the two first-named authors introduced a notion of contact r-surgery along Legendrian knots in contact 3-manifolds. They also showed how (at least in principle) to convert any contact r-surgery into a sequence of contact plus or minus 1 surgeries, and used this to prove that any (closed) contact 3-manifold can be obtained from the standard contact structure on the 3-sphere by a sequence of such surgeries. In the present paper, we give a shorter proof of that result and a more explicit algorithm for turning a contact r-surgery into plus or minus 1 surgeries. We use this …
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.
We introduce a generalization of the Ozsváth-Szabó $τ$-invariant to links by studying a filtered version of link grid homology. We prove that this invariant remains unchanged under strong concordance and we show that it produces a lower bound for the slice genus of a link. We show that this bound is sharp for torus links and we also give an application to Legendrian link invariants in the standard contact 3-sphere.
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.
Goodman surgery and projectively Anosov flows
本文将Goodman手术推广至射影Anosov流范畴，，证明了任何接触Anosov流均可通过手术构造新流，并在双曲3-流形上生成新例子，同时将双接触手术与经典接触手术关联并推导出紧性结果。
We define combinatorial invariants of Legendrian and transverse links in universally tight lens spaces using grid diagrams, generalizing [OST08] and prove that they are equivalent to the invariants defined in [BVVV13] and [LOSS09]. We use these combinatorial invariants to characterize index one grid diagrams for knots in lens spaces which admit surgeries to the 3-sphere and discuss a potential application to the Berge conjecture.
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…
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,…
Contact surgery numbers of Sigma(2,3,11) and L(4m+3,4)
本文分类了 Brieskorn 球面 Sigma(2,3,11) 与透镜空间 L(4m+3,4) 上接触手术数为一的接触结构，证明了存在无穷多无法由单次有理接触手术得到的非同痕结构，并提供了计算 Euler 类的算法公式及 Legendrian 未知结有理手术的紧性分类。
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 …
Contact surgery numbers
本文研究了接触手术数目的定义与性质，通过不等式联系其他不变量，并分类了若干流形上的接触结构，证明了某些流形上存在无限多个非同源的接触结构无法通过单次有理接触手术得到。
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