Tight contact structures on Seifert fibered 3-manifolds

I will start by introducing contact structures. They come in two flavors: tight and overtwisted. Classification of overtwisted  contact structures is well understood as opposed to tight contact structures. Tight contact structures have been classified on some 3  manifolds like S^3, R^3, Lens spaces, toric annuli, and almost all Seifert fibered manifolds with 3 exceptional fibers. We look at classification on one example of the Seifert fibered manifold with 4 exceptional fibers. I will explain the Legendrian surgery and convex surface theory, which help us calculate the lower bound and upper bound of a number of tight contact structures.  We will look at what more classification results can we hope to get using the same techniques and what is far-fetched.