{
  "id": "1307.4391",
  "version": "v4",
  "published": "2013-07-16T19:45:32.000Z",
  "updated": "2015-06-22T14:22:35.000Z",
  "title": "Associahedra via spines",
  "authors": [
    "Carsten Lange",
    "Vincent Pilaud"
  ],
  "comment": "27 pages, 11 figures. Version 4: change to shorter title; added example with Figure 3 and a remark on Cambrian Hopf algebras in Section 6; updated references; minor stylistic changes",
  "categories": [
    "math.CO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-12-16T09:49:22.000Z",
      "title": "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.",
      "comment": "24 pages, 10 figures. Version 3: new Section 4 on Minkowski decompositions",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-06-22T14:22:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B05",
      "52B11",
      "20F55"
    ],
    "keywords": [
      "associahedron",
      "construction",
      "relevant combinatorial properties",
      "triangulations",
      "convex polygon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.4391L"
    }
  }
}