Geodesic knots in cusped hyperbolic 3-manifolds

TL;DR

This paper proves that every cusped orientable finite volume hyperbolic 3-manifold contains infinitely many simple closed geodesics, called geodesic knots, and provides a constructive method to generate them approaching a limiting geodesic.

Contribution

It establishes the existence of infinitely many geodesic knots in all cusped hyperbolic 3-manifolds with a constructive proof.

Findings

All cusped orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots.

Constructive method to produce geodesic knots approaching a limiting geodesic.

Extension of previous results showing existence in specific cases.

Abstract

We consider the existence of simple closed geodesics or "geodesic knots" in finite volume orientable hyperbolic 3-manifolds. Previous results show that at 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.