Geodesic knots in cusped hyperbolic 3–manifolds
Copyright policy )Geodesic knots in cusped hyperbolic 3–manifolds
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.
This is the version published by Algebraic & Geometric Topology on 19 November 2006
- University of Melbourne Australia
Topology of general \(3\)-manifolds, simple closed geodesic, Geometric Topology (math.GT), 53C22, 57N10, 30F40, 57M50, Geodesics in global differential geometry, 53C22, 57M50, 30F40, 57N10, Mathematics - Geometric Topology, knot, Kleinian groups (aspects of compact Riemann surfaces and uniformization), General geometric structures on low-dimensional manifolds, FOS: Mathematics, hyperbolic 3-manifold
Topology of general \(3\)-manifolds, simple closed geodesic, Geometric Topology (math.GT), 53C22, 57N10, 30F40, 57M50, Geodesics in global differential geometry, 53C22, 57M50, 30F40, 57N10, Mathematics - Geometric Topology, knot, Kleinian groups (aspects of compact Riemann surfaces and uniformization), General geometric structures on low-dimensional manifolds, FOS: Mathematics, hyperbolic 3-manifold
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