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.

arXiv comment25 pages, 10 figures; Version 3: minor corrections

References in corpus (16)

• Cluster algebras II: Finite type classification
• Cluster algebras and triangulated surfaces. Part II: Lambda lengths
• Cambrian Lattices
• $\tau$-tilting finite algebras, bricks and $g$-vectors
• Cambrian fans
• Sortable elements and Cambrian lattices
• Matroid polytopes, nested sets and Bergman fans
• The brick polytope of a sorting network
• Brick polytopes of spherical subword complexes and generalized associahedra
• Many non-equivalent realizations of the associahedron
• Non-kissing complexes and tau-tilting for gentle algebras
• Polytopal realizations of finite type $\mathbf{g}$-vector fans
• Polyhedral models for generalized associahedra via Coxeter elements
• Compatibility fans for graphical nested complexes
• Signed tree associahedra
• A $\tau$-Tilting Approach to Dissections of Polygons

Cited by in corpus (14)

• Accordiohedra as positive geometries for generic scalar field theories
• Non-kissing complexes and tau-tilting for gentle algebras
• On Positive Geometries of Quartic Interactions II : Stokes polytopes, Lower Forms on Associahedra and Worldsheet Forms
• Polytopal realizations of finite type $\mathbf{g}$-vector fans
• The positive geometry for ${\varphi }^p$ interactions
• Positive Geometries for all Scalar Theories from Twisted Intersection Theory
• Stokes Polytopes and Intersection Theory
• Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors
• Celebrating Loday's Associahedron
• Which nestohedra are removahedra?
• Towards Positive Geometry of Multi Scalar Field Amplitudes : Accordiohedron and Effective Field Theory
• On the Positive Geometry of Quartic Interactions III : One Loop Integrands from Polytopes
• Geometric Recursion from Polytope Triangulations and Twisted Homology
• A $\tau$-Tilting Approach to Dissections of Polygons