Factorial growth rates for the number of hyperbolic 3-manifolds of a given volume

TL;DR

This paper demonstrates that the number of hyperbolic 3-manifolds with a fixed volume can grow factorially, providing explicit bounds and constructions that surpass previous exponential growth results.

Contribution

It introduces new constructions showing factorial growth in the count of hyperbolic 3-manifolds of a given volume, improving upon prior exponential growth bounds.

Findings

Number of hyperbolic knot complements grows at least factorially with volume

Explicit lower bounds for the count of hyperbolic 3-manifolds are provided

Constructs rely on volume-preserving mutations and Montesinos knots

Abstract

The work of J{\o}rgensen 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.