The minimal volume orientable hyperbolic 3-manifold with 4 cusps

TL;DR

This paper proves that the 8^4_2 link complement is the smallest volume orientable hyperbolic 3-manifold with four cusps, establishing its volume as twice that of the ideal regular octahedron.

Contribution

It identifies the minimal volume hyperbolic 3-manifold with four cusps and provides a volume calculation based on Agol's method.

Findings

8^4_2 link complement has minimal volume among 4-cusped hyperbolic manifolds.

Its volume is exactly twice the volume of the ideal regular octahedron.

The proof involves estimating volumes of manifolds with geodesic boundaries containing essential surfaces.

Abstract

We prove that the 8^4_2 link complement is the minimal volume orientable hyperbolic manifold with 4 cusps. Its volume is twice of the volume V_8 of the ideal regular octahedron, i.e. 7.32... = 2V_8. The proof relies on Agol's argument used to determine the minimal volume hyperbolic 3-manifolds with 2 cusps. We also need to estimate the volume of a hyperbolic 3-manifold with totally geodesic boundary which contains an essential surface with non-separating boundary.