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In \cite{arkani2018unwinding}, Arkani-Hamed, Thomas and Trnka formulated twoconjectural descriptions of the tree amplituhedron $\ampli$ depending on theparity of $m$. When $m$ is even, the description involves the winding numberand when $m$ is odd the description involves the crossing number. In thispaper, we prove that if a point of the amplituhedron is in the image of thepositive Grassmannian by the amplituhedron map, then it satisfies the windingor crossing descriptions depending on the parity of $m$. When $m=2$, we alsoprove the other direction: a point satisfying the winding description is insidethe amplituhedron.

The amplituhedron crossing and winding numbers