Applications of twisted signatures and of Khovanov homology

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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.