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Associahedra via spines. (English) Zbl 1413.52017
The \(n\)-dimensional associahedron is a certain convex \(n\)-polytope whose vertices are in one-to-one correspondence with the triangulations of a convex \((n+3)\)-gon and whose edges represent pairs of triangulations related by a flip of a single diagonal. Various polytopal realizations of the associahedron are known. The present paper introduces the spine of a triangulation of the \((n+3)\)-gon as the dual graph of the triangulation together with a labeling and an orientation. The approach to associahedra via spines extends the classical description of associahedra in terms of binary trees [J.-L. Loday, Arch. Math. 83, No. 3, 267–278 (2004; Zbl 1059.52017)] and provides a new perspective on the realizations discovered in [C. Hohlweg and C. E. M. C. Lange, Discrete Comput. Geom. 37, No. 4, 517–543 (2007; Zbl 1125.52011)], leading in particular to new insights about their geometric and combinatorial properties.

52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
52B11 \(n\)-dimensional polytopes
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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