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。研究的第二部分是利用分枝曲面和薄片来研究可约三维流形中纽结和环上的Dehn手术。一个目的是证明缆线猜想，并给出纽结的性质R定理的一个新证明。项目的第三部分是研究Heegaard分裂的各个方面。这项研究与低维拓扑中的几个主要问题之间有许多有趣的联系。这个项目的结果可能会为解决低维拓扑中一些困难但重要的猜想铺平道路。这位PI计划开发新的工具，并使用他之前工作中的技术，如分支曲面和分层，以实现这些目标。三重流形是在我们生活的三维空间上建模的对象。甜甜圈和空间宇宙都是3-流形的例子。这些物体在物理和其他自然科学中的许多背景下自然产生，并模拟了许多有趣的现象。三维流形具有美丽的几何和拓扑性质，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种组合信息。这项研究针对低维拓扑和纽结理论中的几个核心问题，这对其他科学研究领域有潜在的影响，如DNA的拓扑结构。

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.