Heegaard Floer homology: algebraic curves, knot genera, and double null-concordance

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Problems related to finding solutions to polynomials equations arose in ancient Greek mathematics. The advent of coordinate geometry led to a better understanding of the nature of such problems, and the introduction of complex numbers provided an important new perspective. In this way, determining the nature of a curve in the plane defined as the solution set of a polynomial equation with two variables is related to the problem of understanding surfaces in four dimensional space. The Principal Investigator is investigating topological properties of the surfaces that arise. It has been observed that the local properties of these surfaces relate to knots; this project includes continuing research on four-dimensional aspects of classical knot theory. Coupled with his research in knot theory, the Principal Investigator maintains a website devoted to providing students and researchers access to current information about knots. In addition to assisting people working in knot theory (both pure mathematicians and applied), the website serves as an important educational tool, reaching students around the world.A homogenous polynomial equation with three variables defines a complex curve in two-dimensional complex projective space. Recent work of the Principal Investigator, done jointly with Borodzik and Borodzik-Hedden, has restricted the possibilities for singularities in the case that the curve is topologically a sphere or torus. The approach was to understand the Heegaard Floer homology of the boundary of a neighborhood of the curve. Constraints on these homology groups arising from the complement of the curve in turn constrain the singularities the arise. Developing this further is one goal of the Principal Investigator's research. In studying singularities of complex curves, one is led naturally to studying the four-genus of knots. When restricted to torus knots, the topic is well understood. Working with Van Cott, the Principal Investigator is applying a combination of classical techniques and Heegaard Floer theory to understand the four-genus of connected sums of torus knots, beginning with linear combinations of a pair of torus knots. Independent of this work, the Principal Investigator is also trying to extend his previous work with Gilmer concerning the non-orientable four-genus of knots; one goal is to apply Heegaard Floer theory to build new examples of knots with large nonorientable four-ball genus.