Filling knots and essential geodesic systole

Abstract

We prove that every closed, orientable $3$--manifold contains hyperbolic knots with arbitrarily large essential geodesic systole. That is, for any $3$--manifold $M$, and $R> 0$ there is a hyperbolic knot $K \subset M$ such that every closed geodesic in the knot complement which is non-null-homotopic in $M$ has length at least $R$. The construction also produces filling knots, in the sense of Freedman-Krushkal [FreeKrus2023]. This answers two questions of Biringer.