Laminations and Dehn Surgery on Knots

在研究Dehn手术中，通过使用紧绷的叶和必要的薄片，已经取得了巨大的成功。例如，性质R猜想和关于卫星结的性质P猜想，都是用张叶的方法证明的。首席研究人员计划使用分层和分支表面研究三维流形中的节点和链接的Dehn手术。该项目的成功将促进对三维和四维流形拓扑学的理解，并有助于最终解决这一领域的一些核心问题。这位研究人员还计划研究3-流形中的紧致叠层。致密层压是一种重要的层压材料，具有非常好的横向结构和许多显著的性能。拟议研究的一个目标是证明，如果一个3-流形包含一个本质层，则它包含一个紧层。研究人员还将继续研究双曲穿透环丛中本质层的分类，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，宇宙是一个三维流形，最近的研究表明，宇宙可能具有一种有趣的拓扑结构。研究三维流形的一种自然的几何方法是将全球复杂的三维流形视为沿着二维对象粘合在一起的简单三维碎片的集合，这种方法已被证明是非常有成效的。在拟议的研究中使用的工具，包括分层和分支表面，提供了这样有用的二维对象。所提出的研究涉及低维拓扑和纽结理论中的一些核心问题，这些问题不仅影响到数学，也影响到物理和其他科学研究。更好地理解三维流形可能有助于我们理解宇宙的形状，而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.