  Recently, a new approach to computing the canonical forms of the momentum
amplituhedron in dual-momentum space was proposed by the authors. These are
relevant for the integrands of scattering amplitudes in planar N=4
super-Yang-Mills. At one-loop the idea was to view the set of all loop momenta,
which we refer to as the one-loop fiber geometry, as a fibration over the
tree-level kinematic data. This led to the notion of tree-level chambers,
subsets of the tree-level kinematic space for which the combinatorial structure
of the one-loop fiber remains unchanged, that allowed for a novel
representation of the one-loop integrand. The goal of this paper is to extend
these ideas to two loops for MHV integrands. Our approach will be to view the
geometry accessed by the second loop momentum, similarly referred to as the
two-loop fiber geometry, as a fibration over both the one-loop kinematic data
and the position of the first loop momentum in the one-loop fiber. This will
lead to the notion of one-loop chambers, subsets of the one-loop fibers for
which the combinatorial structure of the two-loop fiber remains unchanged. We
will characterise the full set of one-loop chambers and their corresponding
two-loop fibers and present formulae for their canonical forms. Ultimately,
this will result in a new formula for the two-loop MHV integrand written as a
fibration of fibration.
