Associahedra via spines

Abstract

An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. Using labeled polygons, C. Hohlweg and C. Lange constructed various realizations of the associahedron with relevant properties related to the symmetric group and the classical permutahedron. We introduce the spine of a triangulation as its dual tree together with a labeling and an orientation. This notion extends the classical understanding of the associahedron via binary trees, introduces a new perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's original approach, and sheds light upon the combinatorial and geometric properties of the resulting realizations of the associahedron. It also leads to noteworthy proofs which shorten and simplify previous approaches.

27 pages, 11 figures. Version 5: minor corrections

References in corpus (9)

• The diameter of associahedra
• Brick polytopes of spherical subword complexes and generalized associahedra
• Cambrian Hopf Algebras
• Many non-equivalent realizations of the associahedron
• Polyhedral models for generalized associahedra via Coxeter elements
• Vertex barycenter of generalized associahedra
• Minkowski Decomposition of Associahedra and Related Combinatorics
• Which nestohedra are removahedra?
• Signed tree associahedra

Cited by in corpus (15)

• Cambrian Hopf Algebras
• Non-kissing complexes and tau-tilting for gentle algebras
• Polytopal realizations of finite type $\mathbf{g}$-vector fans
• Brick polytopes, lattice quotients, and Hopf algebras
• Quotientopes
• Compatibility fans for graphical nested complexes
• Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors
• The weak order on integer posets
• Removahedral congruences versus permutree congruences
• Hopf algebras on decorated noncrossing arc diagrams
• Signed tree associahedra
• Shard polytopes
• Acyclic reorientation lattices and their lattice quotients
• Cambrian triangulations and their tropical realizations
• The permuto-associahedron revisited