A Tourist's Account of Characteristic Foliations on Convex Surfaces in 3-D Contact Geometry

Abstract

We begin with a rapid introduction to the theory of contact topology, first spending more time than you would probably want on developing the notion of contact manifold before launching right into the thick of the theory. The tools of characteristic foliations and convex surfaces are introduced next, concluding with an overview of Legendrian knots in contact 3-manifolds. Next, we develop a number of lemmas as tools for dealing with characteristic foliations, concluding with some sightseeing with regards to the theory of so-called "movies", allowing a glimpse into the workings of a theorem due to Colin: Two smoothly isotopic embeddings of S^2 into a tight contact 3-manifold inducing the same characteristic foliation are necessarily contact isotopic. We finish with an original observation that Colin’s theorem can be used to replace a key step in Eliashberg and Fraser’s classification of topologically trivial knots, thus providing an alternate proof of that result and thereby highlighting the power of the aforementioned theorem. We provide a simplification of this proof using intermediate results we encountered along the way.