Associahedra via spines

TL;DR

This paper introduces the concept of spines as dual trees with labels and orientations for triangulations of polygons, providing new insights and simplified proofs for associahedron realizations related to combinatorial and geometric properties.

Contribution

It extends the classical binary tree perspective of associahedra by defining spines, offering a new approach that simplifies existing constructions and enhances understanding of their properties.

Findings

Introduces the spine of a triangulation as a dual tree with labeling and orientation.

Provides a new perspective on associahedron realizations related to the symmetric group.

Simplifies and shortens previous proofs of associahedron properties.

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.