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arXiv · 2206.03435

The amplituhedron crossing and winding numbers

Xavier Blot·Jian-Rong Li
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Abstract

In \cite{arkani2018unwinding}, Arkani-Hamed, Thomas and Trnka formulated two conjectural descriptions of the tree amplituhedron $\ampli$ depending on the parity of $m$. When $m$ is even, the description involves the winding number and when $m$ is odd the description involves the crossing number. In this paper, we prove that if a point of the amplituhedron is in the image of the positive Grassmannian by the amplituhedron map, then it satisfies the winding or crossing descriptions depending on the parity of $m$. When $m=2$, we also prove the other direction: a point satisfying the winding description is inside the amplituhedron.

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Xavier Blot, Jian-Rong Li. 2022-06-07. The amplituhedron crossing and winding numbers. https://arxiv.org/abs/2206.03435

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