Knot Concordance and Metric Spaces

这项研究项目涉及低维拓扑学，它研究称为流形的空间的性质，这些空间在局部上看起来像我们生活的世界。许多成功地用于研究高维流形的数学工具并不适用于四维或更少维的流形，重要的基本问题仍然没有得到解决。这个项目研究的是结节和链节，即三维空间中打结的圆弦。三维流形可以用有框架的链接在数学上表示，其中所谓的框架是修改链接附近空间的指令。因此，研究纽结和链环是理解三维流形的基本工具。在这个项目中，研究人员的目标是更好地了解结作为第四维在时间中移动时的行为。这一研究方向构成了低维拓扑学的一个重要子域，称为和谐性。可以在节点集合上执行具有类似于数字相加的性质的代数运算，并且这导致了称为协调群的代数对象的构造。谐和组是一个极其复杂的实体，尽管许多数学家做出了巨大的努力，但对它的理解却很少。为了了解它的结构，研究人员将引入距离的概念，并研究某些几何运算是如何改变这一距离的。这个项目的主要目标是更好地理解节点和链接的一致性。调查者将通过将协调群研究为具有相关滤子的阿贝尔群，例如n-可解和双极滤子，以及具有离散和非离散度量的度量空间来实现这一点。具体工作如下：(1)证明n-可解滤子的“另一半”是非平凡的；(2)确定链协和群的n-可解滤子的连续商是否是阿贝尔的；(3)理解纽结的导数的Milnor不变量与纽结的调和类之间的关系；(4)将光滑纽结协调群看作度量空间，研究作用于其上的自然几何算子(特别是证明了拓扑片纽结T的子群上存在非离散度量，并证明了双极滤子群的连续商在T上非平凡)；(5)定义纽结复形，研究其代数拓扑。

This research project concerns low-dimensional topology, which studies the properties of spaces called manifolds that locally look like the world we live in. Many of the mathematical tools successfully used to study high-dimensional manifolds do not apply to manifolds in four or fewer dimensions, and important fundamental problems remain unresolved. This project studies knots and links, knotted circular strings in three-dimensional space. Three-dimensional manifolds can be mathematically represented by framed links, where the so-called framings are instructions to modify the space near the link. As a consequence, the study of knots and links is a fundamental tool in understanding three-dimensional manifolds. In this project, the investigator aims to get a better understanding of how knots behave as they move through time as a fourth dimension. This line of investigation constitutes an important subfield of low-dimensional topology known as concordance. An algebraic operation with properties similar to the addition of numbers can be performed on the collection of knots, and this leads to the construction of an algebraic object known as the concordance group. The concordance group is an extremely complicated entity, which is poorly understood despite great efforts by many mathematicians. To gain perspective on its structure, the investigator will introduce a notion of distance and investigate how certain geometric operations change this distance.The primary goal of this project is to gain a better understanding of knot and link concordance. The investigator will do this by studying the concordance group as both an abelian group with associated filtrations, such as the n-solvable and bipolar filtrations, and also as a metric space with discrete and non-discrete metrics. Specific projects are as follows: (1) show that the "other half'" of the n-solvable filtration is nontrivial; (2) determine if successive quotients of the n-solvable filtration of the link concordance group are abelian; (3) understand the relationship between the Milnor invariants of derivatives of a knot and the concordance class of the knot; (4) view the smooth knot concordance group as a metric space and study natural geometric operators acting on it (in particular, show that there is a non-discrete metric on the subgroup of topologically slice knots T and, in addition, show that the successive quotients of the bipolar filtration are nontrivial on T); (5) define the knot complex and study its algebraic topology.