Volume 6, issue 5 (2006)

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Geodesic knots in cusped hyperbolic 3–manifolds

Sally M Kuhlmann

Algebraic & Geometric Topology 6 (2006) 2151–2162

arXiv: 0906.5469

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.

Keywords
simple closed geodesic, knot, hyperbolic 3-manifold
Mathematical Subject Classification 2000
Primary: 57N10, 53C22
Secondary: 57M50, 30F40
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Publication
Received: 21 April 2006
Accepted: 5 September 2006
Published: 19 November 2006
Authors
Sally M Kuhlmann
Department of Mathematics and Statistics
University of Melbourne
Victoria, 3010
Australia