Dehn surgery, rational open books and knot Floer homology
Dehn surgery, rational open books and knot Floer homology
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DOI：
10.2140/agt.2013.13.1815
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发表时间：
2011-05
影响因子：
0.7
通讯作者：
M. Hedden;O. Plamenevskaya
中科院分区：
文献类型：
--
作者：
M. Hedden;O. Plamenevskaya
根据Baker、Etnyre和Van Horn-Morris最近的研究结果，一个合理的开卷分解定义了一个兼容的接触结构。我们证明了这种接触结构的Heegaard Floer接触不变量可以用它的结Floer同源性（合理的零同源性）来计算。然后，我们使用这种接触不变量的描述，以及手术实体环面核心的结花同调的公式，证明了在开卷书的装订上进行手术获得的某些流形具有紧密的接触结构。Dehn手术是在三维流形中切除嵌入圆（结）的邻域，然后用边界环面的微分同构重新粘合它的过程。这种结构长期以来在3流形的研究中起着重要的作用，并提供了一种完整的结构方法。如果3 -歧管配备了额外的结构，人们可以希望调整手术程序以纳入该结构。这个想法在各种情况下都得到了有效的应用。我们目前的兴趣在于三维接触几何的领域。在这里，沿着Legendrian结的接触手术已经成为研究具有接触结构的3 -流形的宝贵工具。对于接触手术，我们从Legendrian结（一个与接触结构相切的结）开始，并以这样一种方式进行Dehn手术，即结补上的接触结构扩展到手术实体环面上。为了保证扩展是唯一的，需要有一个条件，一个充分条件是斜率在Legendrian坐标系下的形式为1=k；见丁和盖吉斯b[5]和神田b[28]。kD的病例可能是最著名的，通常被称为Legendrian手术。
By recent results of Baker, Etnyre and 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. 57M25, 57M27, 57R17, 57R58 Dehn surgery is the process of excising a neighborhood of an embedded circle (a knot) in a 3‐dimensional manifold and subsequently regluing it with a diffeomorphism of the bounding torus. This construction has long played a fundamental role in the study of 3‐manifolds, and provides a complete method of construction. If the 3‐ manifold is equipped with extra structure, one can hope to adapt the surgery procedure to incorporate this structure. This idea has been fruitfully employed in a variety of situations. Our present interest lies in the realm of 3‐dimensional contact geometry. Here, contact surgery along Legendrian knots has been an invaluable tool for the study of 3‐manifolds equipped with a contact structure. For contact surgery, we start with a Legendrian knot (a knot which is tangent to the contact structure), and perform Dehn surgery in such a way that the contact structure on the knot complement is extended over the surgery solid torus. To guarantee that the extension is unique a condition on the surgery slope is required, and a sufficient condition is that the slope is of the form 1=k with respect to the Legendrian framing; see Ding and Geiges [5] and Kanda [28]. The case when kD 1 is perhaps the most well-known, and is often called Legendrian surgery.