my research interests are low-dimensional topology, knot theory and the algebra associated thereto.

papers >

the first-order genus of a knot
we introduce a geometric invariant of knots in the three-sphere, called the first-order genus, that is derived from certain 2-complexes called gropes, and we show it is computable for many examples. while computing this invariant, we draw some interesting conclusions about the structure of a general Seifert surface for some knots.
available at arXiv: 0712.1010
accepted for publication in Mathematical Proceedings of the Cabmridge Philosophical Society

the non-triviality of the Grope filtrations of the knot and link concordance groups
we consider the Grope filtration of the classical knot concordance group that was introduced in a paper of Cochran, Orr and Teichner. our main result is that successive quotients at each stage in this filtration have infinite rank. we also establish the analogous result for the Grope filtration of the concordance group of string links consisting of more than one component.
available at arXiv: 0804.2661

higher-order genera of knots
For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann $\rho$-invariants, which we call higher-order signatures. The higher-order genera offer a refinement of the Grope filtration of the knot concordance group.
available at arXiv: 0807.0434