The two-loop Amplituhedron

The loop-Amplituhedron ${\mathcal{A}}_n^{(L)}$ is a semialgebraic set in the product of Grassmannians ${\mathrm{G}\mathrm{r}}_{\mathbb{R}}(2,4{)}^L$. Recently, many aspects of this geometry for the case of $L=1$ have been elucidated, such as its algebraic and face stratification, its residual arrangement and the existence and uniqueness of the adjoint. This paper extends this analysis to the simplest higher loop case given by the two-loop four-point Amplituhedron ${\mathcal{A}}_4^{(2)}$.