The amplituhedron crossing and winding numbers

TL;DR

This paper proves that points in the amplituhedron satisfy winding or crossing number descriptions depending on the parity of m, and for m=2, it also shows the converse, confirming conjectural descriptions of the amplituhedron.

Contribution

It establishes the validity of the winding and crossing number descriptions for points in the amplituhedron based on their origin from the positive Grassmannian, and proves the converse for m=2.

Findings

Points in the amplituhedron satisfy winding or crossing descriptions based on m's parity.

For m=2, points satisfying the winding description are inside the amplituhedron.

The conjectural descriptions are confirmed for points from the positive Grassmannian.

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.