Geodesic
Geodesic knots in cusped hyperbolic 3–manifolds
Kuhlmann, Sally M1
1Department of Mathematics and Statistics, University of Melbourne, Victoria, 3010, Australia
Algebraic and Geometric Topology, Tome 6 (2006) no. 5, pp. 2151-2162
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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.

DOI : 10.2140/agt.2006.6.2151
Keywords: simple closed geodesic, knot, hyperbolic 3-manifold

Kuhlmann, Sally M  1

1 Department of Mathematics and Statistics, University of Melbourne, Victoria, 3010, Australia 
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Kuhlmann, Sally M. Geodesic knots in cusped hyperbolic 3–manifolds. Algebraic and Geometric Topology, Tome 6 (2006) no. 5, pp. 2151-2162. doi: 10.2140/agt.2006.6.2151

[1] C Adams, J Hass, P Scott, Simple closed geodesics in hyperbolic $3$-manifolds, Bull. London Math. Soc. 31 (1999) 81

[2] T Chinburg, A W Reid, Closed hyperbolic $3$-manifolds whose closed geodesics all are simple, J. Differential Geom. 38 (1993) 545

[3] S M Kuhlmann, Geodesic knots in closed hyperbolic 3-manifolds, in preparation

[4] S M Kuhlmann, Geodesic knots in hyperbolic 3-manifolds, PhD thesis, University of Melbourne (2005)

[5] S M Miller, Geodesic knots in the figure-eight knot complement, Experiment. Math. 10 (2001) 419

[6] T Sakai, Geodesic knots in a hyperbolic $3$-manifold, Kobe J. Math. 8 (1991) 81

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