Higher-order Phenomena in Knot Theory

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The principal investigator will study knots up to isotopy and up to concordance using a combination of techniques from the categories of topological and smooth manifolds, including von Neumann rho-invariants and Heegaard Floer homology. The primary goal of this project is to determine structure in the group of smooth concordance classes of topologically slice knots. Using grid diagrams, the principal investigator will work to find a refinement of knot Floer homology which hopefully reflects the non-commutative structure of the fundamental group of the knot exterior. From this refinement, the principal investigator hopes to garner further concordance invariants and apply them to study topologically slice knots. This refinement of knot Floer homology is expected to capture some of the information in the higher-order Alexander modules. The final goal of the project is to develop computational techniques for these modules.The principal investigator hopes to shed light on the structure of knotted curves in space. Knots appear many places in nature. For example, the strands of DNA in cells are knotted, and these knots can affect the replication process. Many of the existing algebraic tools used to study knots are commutative. The principal investigator will develop new non-commutative algebraic tools to study knots and will develop techniques for making computations. By using more general, non-commutative algebraic techniques, the principal investigator hopes to reveal more structure in the set of knots in space.