Geodesic knots in cusped hyperbolic 3–manifolds

Abstract

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Previous results show that a least one geodesic knot always exists [Bull. London Math. Soc. 31(1) (1999) 81–86], and that certain arithmetic manifolds contain infinitely many geodesic knots [J. Diff. Geom. 38 (1993) 545–558], [Experimental Mathematics 10(3) (2001) 419–436]. In this paper we show that all cusped orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. Our proof is constructive, and the infinite family of geodesic knots produced approach a limiting infinite simple geodesic in the manifold.