My research is in the field of low-dimensional topology, particularly knot theory. Knot theory is a mathematical discipline that began in the late nineteenth century as the study of smooth embeddings of the circle into . In the intervening period, knot theory has expanded into a much broader field which encompasses the study of a variety of knot-like objects using techniques from many other mathematical disciplines. Within mathematics, knot theory has been connected to the general topological study of manifolds via surgery, as well as representation theory and—more broadly—category theory. Outside of mathematics, knot theory has been related to quantum physics and, more recently, to biology (particularly regarding protein and DNA structure). My research is focused on the study of quantum invariants and homological theories of knots and links.

In my most recent work, we show that odd Khovanov Homology is functorial up to sign.  We then use this result to show that the odd Khovanov homology of the cable of a link admits an action by the Hecke algebra at .

My other major research has been in the area of shadow diagrams for links embedded in lens spaces , and analogs of classical knot theory invariants for lens space knots arising through these diagrams.

For more detailed information about my recent and planned research please look at my up-to-date Research Statement.