Positive geometry for nonplanar $N=4$ super-Yang-Mills loop integrands
In plain words
The amplituhedron covers only the planar limit, where the number of colors is infinite. Finite numbers of colors require nonplanar diagrams, and no geometric shape for them is known.
Precise statement
Find a region in a space of kinematic data whose canonical form (the unique differential form with logarithmic singularities on the region's boundaries) equals the subleading-color (nonplanar) or the full color-dressed $L$-loop four-point integrand of $N=4$ super-Yang-Mills; momentum twistors are unavailable because there is no planar ordering. Nonplanar integrands with only logarithmic singularities and no poles at infinity have been constructed at low loop order.
What would settle it
An explicit geometry whose canonical form reproduces the known two- and three-loop nonplanar integrands and predicts higher loops.
Status in the literature
Unverified note
No geometry is known as of 2026.