{
  "id": "1702.08739",
  "version": "v1",
  "published": "2017-02-28T11:02:44.000Z",
  "updated": "2017-02-28T11:02:44.000Z",
  "title": "On the reconstruction of polytopes",
  "authors": [
    "Eran Nevo",
    "Guillermo Pineda-Villavicencio",
    "Julien Ugon",
    "David Yost"
  ],
  "comment": "14",
  "categories": [
    "math.CO"
  ],
  "abstract": "Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its $1$-skeleton. Call a vertex of a $d$-polytope \\emph{nonsimple} if the number of edges incident to it is $>d$. We show that (1) the face lattice of any $d$-polytope with at most two nonsimple vertices is determined by its $1$-skeleton; (2) the face lattice of any $d$-polytope with at most $d-2$ nonsimple vertices is determined by its $2$-skeleton; and (3) for any $d>3$ there are two $d$-polytopes with $d-1$ nonsimple vertices, isomorphic $(d-3)$-skeleton and nonisomorphic face lattices. In particular, the result (1) is best possible for $4$-polytopes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-28T11:02:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B05",
      "52B12"
    ],
    "keywords": [
      "nonsimple vertices",
      "reconstruction",
      "nonisomorphic face lattices",
      "simple polytope",
      "edges incident"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}