Geodesic knots in cusped hyperbolic 3-manifolds

Details Geodesic knots in cusped hyperbolic 3-manifolds Geodesic knots in cusped hyperbolic 3-manifolds

2009-06-30

arxiv.org/abs/0906.5469v1

Algebr. Geom. Topol. 6 (2006) 2151-2162

Mathematics

Geometric Topology

This is the version published by Algebraic & Geometric Topology on 19 November 2006

Scientific paper

10.2140/agt.2006.6.2151

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.

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