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

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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.