Mathematical Sciences: Invariants for Knots and Links in 3-Manifolds

小行星9626140 这个项目是在低维拓扑的一般领域。研究者将试图发展一个理论的有限型不变量的纽结和链接在一个任意的3-流形作为一个推广的琼斯不变量。她还将通过几何方法研究这些不变量，并试图了解它们检测到的结的类型。 低维拓扑学主要关注三维和四维空间的分类，其中两个空间被视为等价的，如果一个空间可以变形到另一个空间上而不撕裂-这被称为同胚变换。这种分类，称为拓扑分类，基本上是在三维和四维以外的维度上进行的。事实证明，这两个维度提供了进一步的技术难题，其中一些是当前非常感兴趣的话题，因为它们与现代物理学的各种思想相结合。

9626140 Kalfagianni This project lies in the general area of low-dimensional topology. The investigator will attempt to develop a theory of finite type invariants for knots and links in an arbitrary 3-manifold as a generalization of the Jones invariants. She will also study these invariants via geometric methods, and try to understand what types of knots they detect. Low-dimensional topology is mainly concerned with classifying spaces of dimensions three and four, where two spaces are treated as equivalent if one space can be deformed onto the other without tearing - this is known as a homeomorphic transformation. Such a classification, known as the topological classification, has been essentially carried out in dimensions other than three and four. It turns out that these two dimensions provide further technical difficulties, some of which are topics of much current interest as they interface with various ideas of modern physics.