Homological Invariants of Knots and Three-Manifolds

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DMS-0603940Jacob RasmussenThis project is about a class of knot invariants known as ``knot homologies'' which generalize classical invariants such as theAlexander and Jones polynomial. The PI plans to investigate a conjectured relationship between two versions of these invariants. The first version was developed by Khovanov and Rozansky and is combinatorial in nature. The second is known as knot Floer homology, and has its origins in the Heegaard Floer homology of Ozsvath and Szabo. It is not combinatorial, but is known to carry a great deal of geometric information about the knot. Despite the differences in their definitions, these two theories exhibit some truly striking similarities. The project aims to explain these similarities and to use them to get a better understanding of both theories. One posssible application is to find combinatorial analogs of gauge theoretic invariants like the knot Floer homology.The study of knotted curves in three-dimensional space is intimately related to the geometry of three- and four-dimensional spaces themselves. In the last few decades, ideas from physics have led to the development of two major types of invariants for such curves, known as ``quantum'' and ``gauge theoretic'' invariants. Although both types have roots in the quantum theory of fields, there was little sign that they were related. Recently, however, some remarkable similarities between the two have begun to appear. The aim of this project is to better understand the relationship between these two classes of invariants.