  In this letter we compute a canonical set of cuts of the integrand for MHV
amplitudes in planar ${\cal N}=4$ SYM, where all internal propagators are put
on-shell. These "deepest cuts" probe the most complicated Feynman diagrams and
on-shell processes that can possibly contribute to the amplitude, but are also
naturally associated with remarkably simple geometric facets of the
amplituhedron. The recent reformulation of the amplituhedron in terms of
combinatorial geometry directly in the kinematic (momentum-twistor) space plays
a crucial role in understanding this geometry and determining the cut. This
provides us with the first non-trivial results on scattering amplitudes in the
theory valid for arbitrarily many loops and external particle multiplicities.
