Hyperbolic four-manifolds, colourings and mutations

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