Applications of twisted signatures and of Khovanov homology

该项目发生在低维拓扑和纽结理论领域。低维拓扑学是研究流形（局部类似于n维空间的几何对象）及其在四维空间中的相互关系，其中可用的工具箱与高维空间中的工具箱完全不同。纽结理论关注的是流形的相互嵌入，特别是圆嵌入到三维空间中。这个项目的目标是将两个独立的工具应用于纽结理论中的几何问题，这两个工具都相当复杂，并且是三维和四维的特有现象。这两个工具是卡森-戈登不变量，它是亚阿贝尔覆盖的扭曲签名，和霍瓦诺夫同调，它是琼斯多项式的分类。

This project takes place in the fields of low-dimensional topology and knot theory. Low-dimensional topology is the study of manifolds (geometric objects that locally resemble n-dimensional space) and their interrelationships in dimensions up to four, where the available toolbox is quite different from the one in higher dimensions. Knot theory is concerned with the embeddings of manifolds in one another, particularly with embeddings of a circle into three-dimensional space.The goal of this project is to apply two separate tools, both of them rather intricate and endemic to dimensions three and four, to geometric problems in knot theory. These two tools are the Casson-Gordon invariants, which are twisted signatures of metabelian coverings, and Khovanov homology, which is a categorification of the Jones polynomial.