Abstract
The study of Legendrian and transverse knots in contact 3-manifolds, has been a key feature in our understanding of contact geometry in general. In this talk I will survey some of the ways in which Legendrian and transverse knot theory illuminates the nature of contact structures and some of subtle features in their classification. I will particularly focus on current work aimed at classifying Legendrian and transverse torus knots in overtwisted contact structures. This classification exhibits many features not previously seen before and is the start of a project to complete the decades old project to classify tight contact structures on small Seifert fibered spaces. This is joint work with Hyunki Min and Anubhav Mukherjee.
