Positive Legendrian Torus Knots

Positive Legendrian torus knots in the standard contact 3-sphere were classified by J. Etnyre and K. Honda around 20 years ago via Giroux' Convex Surface Theory. Nowadays, we have tools to deal with convex surfaces “parametrically” so we can attempt to determine the homotopy type of some path-connected component of the space of Legendrian embeddings. In this talk I will explain how to do this for positive Legendrian (p,q)-torus knots with max tb number. This is joint work (in progress) with Hyunki Min.