Geometry and Topology
The work of Jørgensen and Thurston shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. In this paper, we construct examples showing that the number of hyperbolic knot complements with a given volume v can grow at least factorially fast with v. A similar statement holds for closed hyperbolic 3-manifolds, obtained via Dehn surgery. Furthermore, we give explicit estimates for lower bounds of N(v) in terms of v for these examples. These results improve upon the work of Hodgson and Masai, which describes examples that grow exponentially fast with v. Our constructions rely on performing volume preserving mutations along Conway spheres and on the classification of Montesinos knots.
First published in Proceedings of the American Mathematical Society in volume 143, number 5, 2015, published by the American Mathematical Society.
Factorial growth rates for the number of hyperbolic 3-manifolds of a given volume.
Proceedings of the American Mathematical Society, 2015, volume 143, number 5, pages 2201-2214
Millichap, Christian, "Factorial Growth Rates for the Number of Hyperbolic 3-Manifolds of a Given Volume" (2015). Faculty Publications. Accepted Version. Submission 6.