Using spines to revisit a construction of the associahedron

Abstract. An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. C. Hohlweg and C. Lange constructed various realizations of the associahedron, with relevant combinatorial properties in connection to the symmetric group and to the classical permutahedron. We revisit this construction focussing on the spines of the triangulations, i.e. on their (oriented and labeled) dual trees. This new perspective leads to a noteworthy proof that these polytopes indeed realize the associahedron, and to new insights on various combinatorial properties of these realizations