Geometric realizations of the accordion complex of a dissection

Abstract

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

25 pages, 10 figures; Version 3: minor corrections