Geometric realizations of the accordion complex of a dissection

TL;DR

This paper introduces geometric realizations of the accordion complex associated with any dissection of a convex polygon, extending known constructions from cluster algebra theory to more general cases.

Contribution

It provides new polytope and fan realizations of the accordion complex for arbitrary dissections, broadening the scope of geometric models beyond triangulations and quadrangulations.

Findings

Polytope and fan realizations for the accordion complex are constructed.

Generalization of known cluster algebra related constructions to arbitrary dissections.

Connections established between accordion complexes and geometric combinatorics.

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.