The volume of hyperbolic alternating link complements

TL;DR

This paper establishes bounds on the volume of hyperbolic alternating link complements based on a simple diagram invariant, the twist number, and explores the topological properties of these link complements.

Contribution

It introduces a straightforward method to estimate hyperbolic link volumes from diagram twist numbers and proves the set of such link complements is topologically closed.

Findings

Volume bounds are proportional to the twist number t(D).

Examples asymptotically achieve the volume bounds.

The set of hyperbolic alternating link complements is closed in the geometric topology.

Abstract

If a hyperbolic link has a prime alternating diagram D, then we show that the link complement's volume can be estimated directly from D. We define a very elementary invariant of the diagram D, its twist number t(D), and show that the volume lies between v_3(t(D) - 2)/2 and v_3(16t(D) - 16), where v_3 is the volume of a regular hyperbolic ideal 3-simplex. As a consequence, the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite volume hyperbolic 3-manifolds, in the geometric topology. The appendix by Ian Agol and Dylan Thurston, which was written after the first version of this paper was distributed, improves the upper bound on volume to v_3(10t(D) - 10). In addition, examples of alternating links are given which asymptotically achieve this bound.