Studies in knots and 3-manifolds

本提案的目的是研究有关结理论和低维流形的问题。长期以来，所有维度的流形一直是数学中重要的基础对象。然而，流形的行为及其拓扑结构严重依赖于它们的维度，研究低维（小于4维）问题所需的工具与高维中使用的工具有本质上的不同。我的工作主要是为了提高我们对低维问题的理解。我研究的第一个也是最大的组成部分是研究Dehn手术。给定S3中的一个结K，我们对其进行Dehn手术，通过切割K的管状邻近区域并粘合回另一个实体环面。尽管这个操作的性质很简单，但我们仍然不了解它如何改变流形的拓扑和几何形状。从广义上讲，我将研究以下形式的问题：(1)在S3的一个结上进行手术会产生哪些流形？(2)我们是否可以对所有的结点进行分类？这种形式的问题在低维拓扑中自然出现，Dehn手术的结果经常应用于低维拓扑的其他领域，例如经典的结理论。其次，我将研究涉及3流形和4流形相互作用的问题。这些问题可以分为两类：(1)哪些3流形可以嵌入哪些4流形中？(2)对于具有规定边界的4流形的拓扑结构，我们能说些什么？我目前研究的最后一个广泛目标是寻找计算S3中结的光滑片属和拓扑片属的新技术。虽然这些不变量很容易定义，但有许多简单的结类，如环面结和双桥结，我们对这些属中的一种或另一种仍然知之甚少。

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.