Scattering rigidity, lens rigidity and knot theory

Scattering rigidity of a Riemannian manifold allows one to
tell the metric of a manifold with boundary by looking at the
directions of geodesics at the boundary. Lens rigidity allows one to
tell the metric of a manifold with boundary from the same information
plus the length of geodesics. There are a variety of results about
lens rigidity but very little is known for scattering rigidity. We
will discuss the subtle difference between these two types of
rigidities and prove that they are equivalent for two-dimensional
simple manifolds with boundaries. In particular, this implies that
two-dimensional simple manifolds (such as the flat disk) are
scattering rigid since they are lens/boundary rigid (Pestov--Uhlmann,
2005). Surprisingly, a key ingredient in the proof is knot theory.