Heegaard Floer homology is a new and powerful invariant of 3-manifolds and knots in them. Among many other benefits, it detects the knot genus (the least complicated surface whose boundary is the knot), and in particular, whether a knot is non-trivial. It is a homology theory, and like many homology theories the Euler characteristic is interesting: for knots, the Euler characteristic is one of the oldest knot invariants, the Alexander polynomial. Heegaard Floer homology is part of a family of recent homology theories whose Euler characteristic gives other knot polynomials. But it can be hard to compute, and is defined by ad-hoc rules rather than a concise set of properties. In this project, we will develop a theory of bordered homology invariants: extend the Heegaard Floer homology and other homology theories to objects (knots or 3-manifolds) with boundary, so that when a knot or manifold is split into pieces the invariant for the whole can be computed from the invariants for the pieces. Among other benefits, this will make the theory more computable, and give axioms for the theory.
Although knot theory has been studied for many centuries, some of the most elementary questions, such as finding the least complicated surface whose boundary lies on the knot, are still not easy to answer. Heegaard Floer homology is one recent theory that answers this and many other questions in knot theory and topology. However, it is hard to compute even for relatively small knots. In this project, I and my collaborators will extend Heegaard Floer homology (and other related theories) so that they can be computed for pieces of a knot and then built up to the complete knot. This promises to provide a great computational and theoretical tool. Throughout the project, accessibility will be emphasized, and the project will be integrated with, for instance, an undergraduate research program.
