Noncommutative and Heegaard Floer Methods in Low-Dimensional Topology

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The PI will resolve specific questions about knots, links, and spatial graphs in 3-dimensional space using a combination of topological and algebraic techniques as well as techniques from Heegaard Floer homology and von Neumann algebras. The aim of the first part of the project is to gain a better understanding of the smooth knot concordance group C, by studying a new filtration of the subgroup of topologically slice knots called the bipolar filtration, defining a primary decomposition for the n-solvable quotients, and viewing C as a metric space and studying operators acting on it. Questions about the knot concordance group are closely related to questions about smooth 4-manifolds and thus will provide insight into the mysterious and exciting world of 4-manifolds. In the second project, the PI proposes to establish a higher-order Heegaard Floer homology theory that categorifies the higher-order and twisted Alexander polynomials. The PI will use this theory to define new concordance invariants generalizing the Heegaard Floer tau invariant. In the third project, the PI will investigate the study Legendrian spatial graphs and spatial graph concordance via the recent Graph Floer homology invariant of spatial graph defined by the PI and coauthor. Knot theory is the study of knotted circles or strings in 3-dimensional space. Knots have been studied for over a century, ever since Lord Kelvin (incorrectly) hypothesized that atoms were made of knotted tubes of ether. However, we are still far from a complete classification of them or how they are related to one another. This project will give a better understanding of knots and how they interact in 3- and 4-dimensions. Since the world we live in is 3-dimensional (or 4-dimensional if one considers time), knots and links play a special role in many real life applications. For example, the Jones polynomial of a knot is related to the Pott's model in statistical mechanics. In addition, knot theory plays an role in the study of DNA and cancer research. One can view circular DNA as a knot and certain enzymes, called topoisomerases, manipulate DNA; this manipulation can be viewed as certain simple moves on knots.