The knot complement problem for Legendrian and transverse knots

The famous knot complement theorem by Gordon and Luecke states that two knots in the $3$-sphere are equivalent if and only if their complements are homeomorphic. In this talk I want to discuss the same question for Legendrian and transverse knots and links in contact $3$-manifolds. The main results are that Legendrian as well as transverse knots in the tight contact $3$-sphere are equivalent if and only if their exteriors are contactomorphic.