Geodesic Knots in the Figure-Eight Knot Complement

Abstract

We address the problem of topologically characterising simple closed geodesics in the figure-eight knot complement. We develop ways of finding these geodesics up to isotopy in the manifold, and notice that many seem to have the lowest-volume complement amongst all curves in their homotopy class. However, we find that this is not a property of geodesics that holds in general. The question arises whether under additional conditions a geodesic knot has least-volume complement over all curves in its free homotopy class. We also investigate the family of curves arising as closed orbits in the suspension flow on the figure-eight knot complement, many but not all of which are geodesic. We are led to conclude that geodesics of small tube radii may be difficult to distinguish topologically in their free homotopy class.