The minimal volume orientable hyperbolic 3-manifold with 4 cusps

We prove that the 8 4 2 link complement is the minimal volume orientable hyperbolic manifold with 4 cusps. Its volume is twice the volume V 8 of the ideal regular octahedron; that is, 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 nonseparating boundary.