{
  "id": "2102.01237",
  "version": "v1",
  "published": "2021-02-02T00:14:13.000Z",
  "updated": "2021-02-02T00:14:13.000Z",
  "title": "One-Dimensional Projections and Monotone Paths on Cross-Polytopes and Other Regular Polytopes",
  "authors": [
    "Alexander Black",
    "Jesús De Loera"
  ],
  "comment": "20 pages, 2 figures, 4 tables that contain figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Projections of regular convex polytopes have been studied for a long time (see work by Coxeter, Petrie, Schl\\\"afli, others). One-dimensional linear projections may not appear interesting at first sight, but a linear functional orders the vertices of a polytope and yields an orientation of the graph of the polytope in question. The monotone paths of a polytope are the directed paths on the oriented one-skeleton from the minimum to the maximum. Here we revisit projections for regular polytopes from the lens of monotone paths. Billera and Sturmfels introduced a construction in the early 1990s called the monotone path polytope, which is a polytope whose face lattice encodes the combinatorial structure of certain monotone paths on a polytope. They computed the monotone path polytopes of simplices and cubes, which yield a combinatorial cube and permutahedron respectively. However, the question remains: What are the monotone path polytopes of the remaining regular polytopes? We answer that question in this paper for cross-polytopes, the dodecahedron, the icosahedron, and partially for the 24-cell, 120-cell, and the 600-cell.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-02T00:14:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B05",
      "52B11",
      "52B12",
      "05C20",
      "06B99",
      "F.2.2",
      "G.2.2"
    ],
    "keywords": [
      "regular polytopes",
      "monotone path polytope",
      "one-dimensional projections",
      "cross-polytopes",
      "regular convex polytopes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}