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In 2005, Britto, Cachazo, Feng, and Witten gave a recurrence (now known as the BCFW recurrence) for computing scattering amplitudes inN= 4 super Yang–Mills theory. Arkani-Hamed and Trnka subsequently introduced the amplituhedron to give a geometric interpretation of the BCFW recurrence. Arkani-Hamed and Trnka conjectured that each way of iterating the BCFW recurrence gives a “triangulation” or “tiling” of the m=4 amplituhedron. In this article, we prove the BCFW tiling conjecture of Arkani-Hamed and Trnka. We also prove the cluster adjacency conjecture for BCFW tiles of the amplituhedron, which says that facets of tiles are cut out by collections of compatible cluster variables for the Grassmannian Gr4,n. Moreover we show that each BCFW tile is the subset of the Grassmannian where certain cluster variables have particular signs. 

Abstract The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in$$\mathcal {N}=4$$ $N=4$super Yang–Mills theory. It generalizescyclic polytopesand thepositive Grassmannianand 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$$m=4$$ $m=4$amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for$$\text{ Gr}_{4,n}$$ ${\text{Gr}}_{4,n}$. Secondly, we exhibit a tiling of the$$m=4$$ $m=4$amplituhedron which involves a tile which does not come from the BCFW recurrence—thespuriontile, 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$$\text{ Gr}_{4,n}$$ ${\text{Gr}}_{4,n}$. This paper is a companion to our previous paper “Cluster algebras and tilings for the$$m=4$$ $m=4$amplituhedron.” 

We study collections of planar curves that yield diagrams for all knots. In particular, we show that a very special class called potholder curves carries all knots. This has implications for realizing all knots and links as special types of meanders and braids. We also introduce and apply a method to compare the efficiency of various classes of curves that represent all knots.