I work in the field of geometric topology, and focus on 3-dimensional manifolds and knot theory. I have written papers on Heegaard splittings, sutured manifolds, and bridge surfaces for graphs in 3-dimensional manifolds.
How do the properties of 3-dimensional spaces affect the properties of the 1-dimensional objects (knots, links, graphs) in them? How do the properties of a knot affect the sort of 3-dimensional spaces which can be obtained from it? How is 2-handle addition like Dehn surgery? What do sutured manifold hierarchies tell us about a 3-manifold? What good is being thin?
