The minimal volume orientable hyperbolic 2-cusped 3-manifolds

TL;DR

This paper identifies the Whitehead link complement and the (-2, 3, 8) pretzel link complement as the minimal volume orientable hyperbolic 3-manifolds with two cusps, establishing their volume as 4 times Catalan's constant.

Contribution

It proves the minimal volume for 2-cusped orientable hyperbolic 3-manifolds and uses topological methods to derive lower bounds on volume.

Findings

Whitehead link complement has volume 3.66...

(-2, 3, 8) pretzel link complement has volume 3.66...

Topological arguments establish volume bounds and constraints.

Abstract

We prove that the Whitehead link complement and the (-2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... = 4 x Catalan's constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on volume and strong constraints on the manifolds that realize that lower bound.