My Research
I study low dimensional geometry and topology with an emphasis on knot theory.
The world in which we live is three dimensional. This basic fact implies that we live in some 3-manifold. In this way, the study of 3-manifolds illuminates the nature of the universe. Knotting phenomena in manifolds is so rich that a complete understanding of it would lead to a complete understanding of all 3-manifolds. The broad goal of my research is to study 3-manifolds via knots.
My recent research investigates knots in 3-manifolds from the perspective of bridge position and the related notion of thin position. Thin position has been a particularly useful tool in 3-manifold topology. Specifically, thin position played an integral role in two celebrated results: Gordon and Luecke's solution to the knot complement problem and Thompson's solution to the recognition problem for the 3-sphere. Although thin position is well studied, some of its most basic properties are still not understood. For instance, it is not, in general, known when thin position is equal to bridge position. Additionally, thin position of the connected sum of knots is still unknown. My research attempts to answer these questions and related questions.