Twistor and Polytope Interpretations for Subleading Color One-Loop Amplitudes

We use the relation of the one-loop subleading-color amplitudes to the one-loop $n$-point leading color amplitudes in ${\cal N}=4$ SYM, to derive a polytope interpretation for the former in the $MHV$ case, and a representation in momentum twistor space for the general $N^kMHV$ case. These techniques are explored in detail for the 5-point and 6-point amplitudes. We briefly discuss the implications for IR divergences.