Geodesic knots in the figure-eight knot complement

We address the problem of topologically characterising simple closed geodes ics in the figure-eight knot complement. We develop ways of finding these g eodesics up to isotopy in the manifold, and notice that many seem to have t he lowest-volume complement amongst all curves in their homotopy class. How ever, we find that this is not a property of geodesics that holds in genera l. 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 su spension flow on the figure-eight knot complement, many but not all of whic h are geodesic. We are led to conclude that geodesics of small tube radii m ay be difficult to distinguish topologically in their free homotopy class.