Hyperbolic 4-manifolds, colourings and mutations

We develop a way of seeing a complete orientable hyperbolic 4-manifold M as an orbifold cover of a Coxeter polytope P⊂H4 that has a facet colouring. We also develop a way of finding a totally geodesic sub-manifold N in M, and describing the result of mutations along N. As an application of our method, we construct an example of a complete orientable hyperbolic 4-manifold X with a single non-toric cusp and a complete orientable hyperbolic 4-manifold Y with a single toric cusp. Both X and Y have twice the minimal volume among all complete orientable hyperbolic 4-manifolds.