Topics in low-dimensional topology

PI计划通过有效的三角剖分、分支曲面和Heegaard分裂来探索三维流形的几何、拓扑和组合结构之间的相互联系。 该项目的第一部分是为某些三维流形构造几何三角剖分。第二部分是利用分支曲面和叠层研究可约三维流形中纽结和链环的Dehn手术。 一个目标是证明电缆猜想，并给出一个新的证明的性质R定理的纽结。 本项目的第三部分是研究Heegaard分裂的各个方面。 研究与低维拓扑学中的几个主要问题之间有许多有趣的联系。 该项目的结果可能为解决低维拓扑中一些困难但重要的问题铺平道路。 PI计划开发新的工具，并使用他以前工作中的技术，如分支表面和层压，以实现这些目标。三维流形是我们生活的三维空间中建模的对象。 甜甜圈和空间宇宙都是三维流形的例子。 这些对象在物理和其他自然科学的许多背景下自然出现，并模拟许多有趣的现象。 三维流形具有优美的几何和拓扑性质，这些性质被编码在一定的组合信息中。 PI将通过一些称为有效三角剖分、分支曲面和Heegaard分裂的几何对象来探索这些组合信息。 该研究针对低维拓扑和纽结理论中的几个核心问题，这些问题对其他科学研究领域（如DNA的拓扑结构）具有潜在的影响。

The PI plans to explore the interconnections between geometric, topological and combinatorial structures of 3-manifolds through efficient triangulations, branched surfaces and Heegaard splittings. The first part of the project is to construct geometric triangulations for certain 3-manifolds. The second part of the research is to use branched surfaces and laminations to study Dehn surgery on knots and links in reducible 3-manifolds. One goal is to prove the Cabling Conjecture and to give a new proof of the Property R theorem for knots. The third part of the project is to study various aspects of Heegaard splittings. There are many interesting connections between the research and several major questions in low-dimensional topology. Results from this project may pave the road toward the resolution of some difficult but important conjectures in low-dimensional topology. The PI plans to develop new tools and use techniques from his previous work, such as branched surfaces a nd laminations, to achieve these goals.Three-manifolds are objects modeled on the 3-dimensional space that we are living in. A donut and the spatial universe are both examples of 3-manifolds. These objects arise naturally in many contexts in physical and other natural sciences and model many interesting phenomena. Three-manifolds have beautiful geometric and topological properties and these properties are encoded in certain combinatorial information. The PI will explore such combinatorial information through some geometric objects called efficient triangulation, branched surface and Heegaard splitting. The research targets several central questions in low-dimensional topology and knot theory, which has potential impact on other areas of scientific investigations, such as the topological structures of DNA.