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

Abstract

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.

Mutations and short geodesics in hyperbolic 3-manifolds

“…In Section 5, we construct and describe our class of hyperbolic pretzel knots which are mutants of one another. We also highlight a theorem from our past work [27] that describes how many of these mutant pretzel knot complements are non-isometric and have the same volume. In Section 6, we analyze the geometry of our pretzel knots by realizing them as Dehn fillings of untwisted augmented links, whose complements have a very simple polyhedral decomposition.…”

“…Now, M σ 2n+1 (p, q) and M σ ′ 2n+1 (p, q) have the same volume by Ruberman's work. In [27,Theorem 3], we show that M σ 2n+1 (p, q) and M σ ′ 2n+1 (p, q) are non-isometric by choosing (p, q) sufficiently large so that the core geodesics resulting from this Dehn filling are the systoles of their respective manifolds. This comes from the work of Neumann-Zagier [31].…”

“…Let M = S 3 \ K and let M(p, q) denote the closed manifold obtained by performing a (p, q)-Dehn surgery along the knot K. In [27,Theorem 3], we show that for each n ∈ N, n ≥ 2, and for (p, q) sufficiently large, M σ 2n+1 (p, q) and M σ ′ 2n+1 (p, q) have the same volume and are non-isometric closed hyperbolic 3-manifolds, whenever M σ 2n+1 and M σ ′ 2n+1 are nonisometric. This proof relies on another result of Ruberman's [38,Theorem 5.5] which shows that corresponding Dehn surgeries on a hyperbolic knot K and its fellow mutant K µ will often result in manifolds with the same volume.…”

“…Proof. In [27], we constructed our K 2n+1 so that all Conway spheres in {(S a , σ a )} 2n a=1 are unlinked. However, here we have slightly modified this construction of each K 2n+1 .…”

Counting and effective rigidity in algebra and geometry

“…Millichap [61,62] constructed roughly (2n)! incommensurable hyperbolic 3-manifolds that have the same first 2n + 1 (complex) geodesic lengths.…”