A tiling, also known as a triangulation, of the amplituhedron A(n,k,m) is defined by a collection of km-dimensional positroid cells in its preimage, the nonnegative Grassmannian in Gr(k,n), that map injectively to pairwise disjoint regions that cover a dense subset of the amplituhedron. For m=4, it has been conjectured and proven that the BCFW recursion gives rise to such tilings for all k and n. We explore the structure of the amplituhedron and the existence of tilings for higher values of m. The case m=6 is already far less understood, and new phenomena emerge. For m=6 and k=2, we propose various families of conjectural tilings for all values of n. In other cases, we provide strong evidence that no such tilings exists. Instead, we suggest refinements of the tiling notion, such as a collection of positroid cells that map onto a double cover of the amplituhedron.

 

Joint work with Nima Arkani-Hamed, Tsviqa Lakrec, Matteo Parisi, Melissa Sherman-Bennett, Ran Tessler, and Lauren Williams.