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.

这个建议的目的是调查有关纽结理论和低维流形的问题。所有维度的流形长期以来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结构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。我的工作主要是为了推进我们对低维问题的理解。我的研究的第一个也是最大的组成部分是与研究德恩手术有关。给定S3中的结K，我们通过切除K的管状邻域并粘回另一个实心环面来对其进行Dehn手术。尽管该操作本质上很简单，但关于它如何改变流形的拓扑和几何形状，我们仍然有很多不了解的地方。广义地说，我将研究以下形式的问题：（1）哪些流形是通过对S3中的纽结进行外科手术而产生的？(2)我们能对所有涌到给定三维流形上的纽结进行分类吗？这种形式的问题自然会出现在整个低维拓扑和德恩手术的结果经常有应用到其他领域的低维拓扑，如经典的纽结理论。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类：（1）哪些3-流形可以嵌入哪些4-流形？(2)关于具有给定边界的4-流形的拓扑，我们能说些什么？我目前的研究的最后一个广泛的目标是找到新的技术来计算光滑切片一般和拓扑切片一般的结在S3。虽然这些不变量很容易定义，但有许多简单的纽结类，例如环面纽结和双桥纽结，我们仍然对这些类中的一个或另一个了解不多。