Legendrian realizations of double twist knots

Legendrian knot theory is a rich field of contact topology which, although widely investigated, has still many fundamental open questions. For example, we hardly know any results on the natural problem of classifying Legendrian knots up to Legendrian isotopy. In this talk, I am going to present a lower and an upper bound on the number of different Legendrian realizations of a family of double twist knots with prescribed tb invariant. The lower bound provides an infinite family of Legendrian non-simple knot types, while the upper bound will hopefully help the future study of Legendrian simple ones.