Laminations and Dehn Surgery on Knots

通过使用紧叶理和必要的层压，在研究结的Dehn手术方面取得了巨大的成功。例如，利用紧叶理的方法证明了卫星节的性质R猜想和性质P猜想。首席研究员计划使用层压和分支表面对3流形中的结和连接进行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，并有助于最终解决该领域的一些核心问题。研究者还计划研究3流形中的紧密层合。致密层合是一种重要的层合材料，具有良好的横向结构和许多显著的性能。提出的研究的一个目标是证明，如果一个3-歧管包含一个基本层压，那么它包含一个紧密层压。研究者还将继续研究双曲点环束中基本层合的分类，以及层流3-流形的拓扑刚性。三维流形是以我们生活的三维空间为模型的物体。例如，宇宙是一个三流形，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是将全局复杂的三维流形视为沿二维物体粘合在一起的简单三维块的集合，这种方法已被证明是非常富有成效的。在提议的研究中使用的工具，包括薄片和分支表面，提供了这样有用的二维对象。提出的研究涉及低维拓扑和结理论中的一些核心问题，这些问题不仅影响数学，而且影响物理和其他科学研究。更好地理解3流形可以帮助我们理解宇宙的形状，而结理论被用来研究DNA的结构及其生物学效应。

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.