Geodesic knots in closed hyperbolic 3-manifolds

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Every such manifold contains at least one geodesic knot by results of Adams, Hass and Scott in (Adams et al. Bull. London Math. Soc. 31: 81–86, 1999). In (Kuhlmann Algebr. Geom. Topol. 6: 2151–2162, 2006) we showed that every cusped orientable hyperbolic 3-manifold in fact contains infinitely many geodesic knots. In this paper we consider the closed manifold case, and show that if a closed orientable hyperbolic 3-manifold satisfies certain geometric and arithmetic conditions, then it contains infinitely many geodesic knots. The conditions on the manifold can be checked computationally, and have been verified for many manifolds in the Hodgson-Weeks census of closed hyperbolic 3-manifolds. Our proof is constructive, and the infinite family of geodesic knots spiral around a short simple closed geodesic in the manifold.