Research

My research interests lie in low-dimensional geometric topology. I am especially interested in knot theory and concordance of knots and links, particularly in 3-manifolds other than the 3-sphere. The study of concordance in these more general settings presents many new and interesting complications. The nontriviality of the fundamental group of the 3-manifold permits a wider variety of embeddings. Phenomena which cannot occur in the 3-sphere are detectable using methods from algebraic topology, obstructing not only concordance between two knots or links, but the existence of surfaces with nice properties between them.

For detailed information on what I'm currently thinking about, see my research statement.

I primarily study knot concordance, a notion of equivalence between knots, objects like the one pictured to the left. The knots I study live in objects called manifolds. The surface of the earth is a 2-dimensional manifold; even though it may look flat from where we're standing, we know it has a more interesting global structure. The knots I study live in 3-dimensional manifolds, which again look "flat" locally but may not be globally. Knot concordance is a 4-dimensional relation, studying surfaces that knots bound in certain 4-dimensional manifolds.

Milnor's classical invariants for links in the 3-sphere detect higher-order linking phenomena by studying lower central quotients of link groups and comparing them to those of the unlink. These invariants have been shown to be concordance invariants and have since inpired decades of consequential research connecting many different topological objects. There have been numerous attempts to extend these invariants to other settings. In a forthcoming paper, I define a sequence of concordance invariants for knots and links in closed, orientable 3-manifolds whose vanishing, taken in appropriate contexts, implies the vanishing of all previous versions of Milnor's invariants. These new invariants are defined and can be nontrivial even for empty links; in this case they reduce to invariants of homology cobordism of 3-manifolds.

Poincaré embedding type is a universal homotopy invariant of knot (or link) concordance due to Cha and Orr which includes the homotopy type of the Vogel homology localization of the knot (or link) exterior. Invariants like Milnor's invariants are really detecting information about Poincaré embedding type. This invariant is uninteresting for knots in the 3-sphere, since all such knots have the same embedding type; on the other hand, this invariant is rich for knots in other 3-manifolds.

I studied counterexamples to the 3/4-dimensional s-cobordism theorem, particularly examples of Cappell and Shaneson over quaternionic 3-manifolds. Their work involves constructions of interesting homology cobordisms using knots, and invariants that depend on those knots.

My undergraduate thesis examined spiral knots, a generalization of torus knots, and found recursive formulas for their determinants.