Studies in knots and 3-manifolds

这一建议的目的是研究与纽结理论和低维流形有关的问题。长期以来，所有维度的流形都是数学中最重要的对象。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使用的工具有很大不同。我的工作主要是为了促进我们对一些低维度问题的理解。 我的研究的第一个也是最大的部分与学习Dehn手术有关。给出S3中的一个结K，我们对它进行Dehn手术，方法是切下K的管状邻域，然后粘回到另一个实心环面上。尽管这一操作的性质很简单，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。概括地说，我将研究以下形式的问题： (1)在S3中，哪些流形是通过外科手术形成的？ (2)我们能将所有外科手术的纽结归类到一个给定的3-流形上吗？ 这种形式的问题在整个低维拓扑学中自然出现，Dehn手术的结果经常应用于其他低维拓扑学领域，如经典纽结理论。 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)关于具有指定边界的4-流形的拓扑，我们能说些什么？ 我目前研究的最后一个广泛目标是找到计算S3中纽结的光滑切片亏格和拓扑切片亏格的新方法。虽然这些不变量很容易定义，但有许多简单的纽结类别，如环结和两桥结，我们对这些属中的一个或另一个仍然知之甚少。

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.