Research Interest

The area of research that I was interested in is the geometry and topology of 3-manifolds. When investigating these spaces, one question that we can ask is what types of surfaces exist in the 3-manifold. Through Thurston's ground breaking work, it turns out that this purely topological information can be used to extract geometric information. The truly remarkable property that is unique to 3-dimensions is Thurston's Geometrization Theorem, which was proved by Perelman, asserting that geometry and topology are the same.

Specifically, I was interested in using the tools of 3-manifolds to study knots and links. A knot, intuitively speaking, is when we take a piece of string, tie it up in any way, and then glue the two ends of the string together. We can then try to untangle said piece of string without cutting or breaking it anywhere. If, after much trial and error, we cannot fully untangle the knot, then we may give up and ask ourselves, "is this resulting knot as simple as we can possibly make it?" A few follow-up questions we may ask are: 

• What kind of knot did we end up with?

• If we were to do the same process with a second piece of string, except tie the string differently, do we end up with the same knot as the first piece of string?

The role of 3-manifolds come in by taking the complement of a knot or link inside of the 3-sphere obtaining a 3-manifold. Namely, instead of looking at the piece of string itself, we may put the string inside of the 3-sphere and study the surrounding space. In this viewpoint, we may now study the theory of knots using techniques of 3-manifolds.