Dihedral sign patterns on M_{0,n}

The connected components of M_{0,n}(R), the configuration space of n distinct points on the real projective line, are in bijection with the (n-1)!/2 dihedral orderings of [n]. We give an alternative characterization of these connected components in terms of sign patterns for the dihedral embedding, proving a conjecture of Arkani-Hamed, He, and Lam in the case of M_{0,n}. More precisely, we show that the connected components of M_{0,n} are in one-to-one correspondence with the sign patterns that are consistent with the extended u-relations. If I have time, I’ll present our implementation of the algorithm which produces the dihedral ordering corresponding to a given consistent sign pattern. This is joint work with Veronica Calvo Cortes (arXiv:2508.14714).