Studies in knots and 3-manifolds

结和 3 流形的研究

• 批准号: RGPIN-2020-05491
• 负责人: McCoy, Duncan
• 金额: $ 1.89万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759332
• 关键词: Studies; knots; manifolds

The purpose of this proposal is to investigate questions relating to knot theory and low dimensional manifolds. Manifolds of all dimensions have long been objects of fundamental importance in mathematics. However, the behaviour of manifolds and their topology is heavily dependent on their dimension, with the tools required to study questions in low (less than four) dimensions being substantially different to those used in higher dimensions. My work is primarily aimed at advancing our understanding of a number of questions in low dimensions. The first and largest component of my research is related to studying Dehn surgery. Given a knot K in S3, we perform Dehn surgery on it by cutting out a tubular neighbourhood of K and gluing back in another solid torus. Despite the simple nature of this operation, there is still much that we do not understand about how it can change the topology and geometry of a manifold. Broadly speaking, I will be studying questions of the form: (1) Which manifolds arise by surgery on a knot in S3? (2) Can we classify all knots which surger to a given 3-manifold? Questions of this form naturally arise throughout low-dimensional topology and Dehn surgery results frequently have applications to other areas of low dimensional topology, such as classical knot theory. Secondly, I will be working on questions that involve the interactions between 3-manifolds and 4-manifolds. These questions can be categorized into two flavours: (1) Which 3-manifolds can be embedded into which 4-manifolds? (2) What can we say about the topology of 4-manifolds with a prescribed boundary? The last broad aim of my current research is to find new techniques for computing the smooth slice genera and topological slice genera of knots in S3. Although these invariants are simple to define, there are many simple classes of knots, such as torus knots and two-bridge knots for which we still have a poor understanding of one or other of these genera.

该提案的目的是研究与结理论和低维流形相关的问题。长期以来，所有维度的流形一直是数学中具有根本重要性的对象。然而，流形的行为及其拓扑在很大程度上取决于它们的维度，研究低维（小于四维）问题所需的工具与高维中使用的工具有很大不同。我的工作主要旨在增进我们对一些低维问题的理解。我研究的第一个也是最大的部分与研究德恩手术有关。给定 S3 中的结 K，我们通过切掉 K 的管状邻域并粘回另一个实体环面来对其进行 Dehn 手术。尽管此操作本质上很简单，但我们仍然不了解它如何改变流形的拓扑和几何形状。一般来说，我将研究以下形式的问题：(1) S3 中的结手术会产生哪些流形？ (2) 我们能否将所有涌入给定 3 流形的结分类？这种形式的问题在整个低维拓扑中自然出现，并且 Dehn 手术结果经常应用于低维拓扑的其他领域，例如经典结理论。其次，我将研究涉及 3 流形和 4 流形之间相互作用的问题。这些问题可以分为两种类型：（1）哪些 3 流形可以嵌入到哪些 4 流形中？ (2) 对于具有规定边界的 4 流形拓扑，我们能说些什么？我当前研究的最后一个主要目标是找到计算 S3 中结的平滑切片属和拓扑切片属的新技术。尽管这些不变量很容易定义，但有许多简单的结类别，例如环面结和两桥结，我们对这些类别中的一个或其他类别仍然知之甚少。