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有关的问题。长期以来，各个方面的歧管一直是数学中基本重要性的对象。但是，流形的行为及其拓扑的行为在很大程度上取决于它们的维度，而在低（少于四个）维度中研究问题的工具与在较高维度中使用的问题大不相同。我的工作主要旨在促进我们对低维度中许多问题的理解。我研究的第一个也是最大的组成部分与研究Dehn手术有关。鉴于S3中的一个结K，我们通过切出一个k的管状邻居并将其粘在另一个坚固的圆环中，对其进行Dehn手术。尽管此操作的性质很简单，但我们仍然不了解它如何改变多种多样的拓扑和几何形状。从广义上讲，我将研究形式的问题：（1）在S3中的结上进行手术而产生的多种流动性？ （2）我们可以将所有哪个手术的结为给定的3个策略分类？这种形式的问题自然出现在低维拓扑结构和Dehn手术结果中，经常在其他低维拓扑的领域（例如经典结理论）应用。其次，我将研究涉及3个manifolds和4个manifolds之间相互作用的问题。这些问题可以分为两种口味：（1）哪些3个manifolds可以嵌入哪些4个manifolds？ （2）我们对具有规定边界的4个manifolds的拓扑怎么说？我目前的研究的最后一个广泛目的是找到用于计算S3中打结的平滑切片属和拓扑切片属的新技术。尽管这些不变式易于定义，但是有许多简单的结，例如圆环结和两桥结，我们对这些属的一个或另一个属的理解仍然很差。