If $M$ is a complete Riemannian manifold and $N$ be any submanifold of it. Then the cut locus of $N$ is the collection of points $q\in M$ such that minimal geodesic joining $q$ to $N$ will no longer be minimum beyond $q$. For example, in $S^2$ the cut locus of the equator is the north pole and the south pole.
I am looking for a reference (or maybe closely related to reference): Cut locus of a knot. Maybe trefoil knot (as it is one of the simplest ones).