A cluster of results on amplituhedron tiles.

The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in super Yang-Mills theory. It generalizes and the and has a very rich combinatorics with connections to cluster algebras. In this article, we provide a series of results about tiles and tilings of the amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for . Secondly, we exhibit a tiling of the amplituhedron which involves a tile which does not come from the BCFW recurrence-the tile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for . This paper is a companion to our previous paper "Cluster algebras and tilings for the amplituhedron."