Laminations and Dehn Surgery on Knots

在研究Dehn手术通过使用拉紧的叶状结构和基本的叠层来治疗结方面已经取得了巨大的成功。 例如，性质R猜想和性质P猜想的卫星结，已证明使用的方法拉紧叶理。 主要研究者计划使用叠层和分支表面研究3-流形中的结和链接的Dehn手术。 该项目的成功将促进对三维和四维流形拓扑的理解，并有助于该领域一些中心问题的最终解决方案。 研究人员还计划研究3-流形中的紧密叠层。 紧密层压是一种基本的层压材料，具有非常好的横向结构和许多显着的性能。 该研究的一个目标是证明，如果一个3流形包含一个本质层，那么它包含一个紧层。 调查员也将继续他的工作分类的基本层压在双曲punctured-torus束，并对拓扑刚性的层状3-流形。 三维流形是在我们生活的三维空间中建模的物体。 例如，宇宙是一个三维流形，最近的研究表明，宇宙可能有一个有趣的拓扑结构。 研究三维流形的一种自然的几何方法，已经被证明是非常富有成效的，是将全局复杂的三维流形看作是沿着沿着二维物体粘在一起的简单三维块的集合。 在拟议的研究中使用的工具，包括叠层和分支表面，提供了这样的有用的二维对象。 该研究涉及低维拓扑学和纽结理论中的一些核心问题，不仅影响数学，而且影响物理学和其他科学研究。 更好地理解三维流形可以帮助我们理解宇宙的形状，纽结理论被用来研究DNA的结构及其生物效应。

There has been tremendous success in studying Dehn surgery on knots through the use of taut foliations and essential laminations. For example, the property R conjecture, and the property P conjecture for satellite knots, have been proved using the methods of taut foliations. The principal investigator plans to study Dehn surgery on knots and links in 3-manifolds using laminations and branched surfaces. The success of the project will advance the understanding of the topology of 3 and 4 dimensional manifolds and contribute to the ultimate solutions of some central problems in this area. The investigator also plans to study tight laminations in 3-manifolds. A tight lamination is an essential lamination with a very nice transverse structure and many remarkable properties. One goal of the proposed research is to prove that if a 3-manifold contains an essential lamination then it contains a tight lamination. The investigator will also continue his work on the classification of essential laminations in hyperbolic punctured-torus bundles, and on the topological rigidity of laminar 3-manifolds. Three-manifolds are objects modeled on the 3-dimensional space that we are living in. For instance, the universe is a 3-manifold and recent study shows that the universe may have an interesting topological structure. A natural geometric way of studying 3-manifolds, which has been proved extremely fruitful, is to view a globally complicated 3-manifold as a collection of simple 3-dimensional pieces glued together along a 2-dimensional object. The tools used in the proposed research, including laminations and branched surfaces, provide such useful 2-dimensional objects. The proposed research is related to some central questions in low-dimensional topology and knot theory, which impact not only mathematics, but also physics and other scientific research. A better understanding of 3-manifolds may help us understand the shape of the universe, and knot theory is used to study the structure of DNA and its biological effect.