Geometric Realizations of the Accordion Complex of a Dissection

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.