Geometric realizations of the accordion complex of a dissection

Overview & Synopsis

Consider $2n$ points on the unit circle and a reference dissection $D$ of the convex hull of the odd points. The accordion complex of $D$ is the simplicial complex of non-crossing subsets of the diagonals with even endpoints that cross a connected subset of diagonals of $D$. In particular, this complex is an associahedron when $D$ is a triangulation and a Stokes complex when $D$ is a quadrangulation. In this paper, we provide geometric realizations (by polytopes and fans) of the accordion complex of any reference dissection $D$, generalizing known constructions arising from cluster algebras.