The Link Volume of 3-Manifolds

TL;DR

This paper introduces the link volume, an invariant measuring how efficiently hyperbolic 3-manifolds can be represented as branched covers of the 3-sphere, and explores its properties and bounds.

Contribution

It defines the link volume invariant, proves a structure theorem for it, and establishes bounds relating it to Dehn filling parameters.

Findings

Link volume is always greater than hyperbolic volume for hyperbolic manifolds.

A structure theorem similar to Jorgensen and Thurston's is proved for link volume.

Link volumes of Dehn filled manifolds are linearly bounded by continued fraction length.

Abstract

We view closed orientable 3-manifolds as covers of S^3 branched over hyperbolic links. For a p-fold cover M \to S^3, branched over a hyperbolic link L, we assign the complexity p Vol(S^3 minus L) (where Vol is the hyperbolic volume). We define an invariant of 3-manifolds, called the link volume and denoted LV, that assigns to a 3-manifold M the infimum of the complexities of all possible covers M \to S^3, where the only constraint is that the branch set is a hyperbolic link. Thus the link volume measures how efficiently M can be represented as a cover of S^3. We study the basic properties of the link volume and related invariants, in particular observing that for any hyperbolic manifold M, Vol(M) < LV(M). We prove a structure theorem that is similar to (and relies on) the celebrated theorem of Jorgensen and Thurston. This leads us to conjecture that, generically, the link volume of a hyperbolic 3-manifold is much bigger than its volume. Finally we prove that the link volumes of the manifolds obtained by Dehn filling a manifold with boundary tori are linearly bounded above in terms of the length of the continued fraction expansion of the filling curves.