GGT: Knots, links, and essential geodesic systole

Adams and Reid proved that if M is a closed 3-manifold that admits no hyperbolic metric, then for any hyperbolic link L in M, the systole (length of a shortest geodesic) is bounded above by 7.355... Lakeland and I extended this to the case when M admits a hyperbolic metric, with a bound on the systole that depends only on the volume of the hyperbolic metric on M (in particular, independent of L).  In both proofs, the systole is realized by a curve which is null-homotopic in M.  I will discuss joint work with Yandi Wu where we show that this is not an accident.  Specifically, we construct a sequence of hyperbolic links {L_n} in any closed, orientable 3-manifold M whose complements have arbitrarily large "essential systole", which is the length of the shortest closed geodesic in M-L_n that is non-null-homotopic in M.  More recently, we constructed a sequence of knots with the same property.  As an application of our construction, we find "filling" knots and links in any M, answering a question of Freedman and Krushkal.