{
  "id": "1301.1987",
  "version": "v3",
  "published": "2013-01-09T21:09:43.000Z",
  "updated": "2017-08-24T16:48:46.000Z",
  "title": "Extending Tutte and Bollobás-Riordan Polynomials to Rank 3 Weakly-Colored Stranded Graphs",
  "authors": [
    "Remi C. Avohou",
    "Joseph Ben Geloun",
    "Mahouton N. Hounkonnou"
  ],
  "comment": "42 pages, 32 figures; new title, improved version, some statements corrected",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "abstract": "The Bollob\\'as-Riordan polynomial [Math. Ann. 323, 81 (2002)] is a universal polynomial invariant for ribbon graphs. We find an extension of this polynomial for a particular family of combinatorial objects called rank 3 weakly-colored stranded graphs. Stranded graphs arise in the study of tensor models for quantum gravity in physics, and generalize graphs and ribbon graphs. We present a seven-variable polynomial invariant of these graphs which obeys a contraction/deletion recursion relation similar to that of the Tutte and Bollob\\'as-Riordan polynomials. However, it is defined on a much broader class of objects, and furthermore captures properties that are not encoded by the Tutte or Bollob\\'as-Riordan polynomials of the underlying graphs.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-04-01T12:08:21.000Z",
      "title": "A Polynomial Invariant for Rank 3 Weakly-Colored Stranded Graphs",
      "abstract": "The Bollobas-Riordan polynomial [Math. Ann. 323, 81 (2002)] is a universal polynomial invariant for ribbon graphs. We find an extension of this polynomial for a particular family of graphs called rank 3 weakly-colored stranded graphs. These graphs live in a 3D space and appear as the gluing stranded vertices with stranded edges according to a definite rule (ordinary graphs and ribbon graphs can be understood in terms of stranded graphs as well). They also possess a color structure in a specific sense [Gurau, Commun. Math. Phys. 304, 69 (2011)]. The polynomial constructed is a seven indeterminate polynomial invariant of these graphs which responds to a similar contraction/deletion recurrence relation as obeyed by the Tutte and Bollobas-Riordan polynomials. It is however new due to the particular cellular structure of the graphs on which it relies. The present polynomial encodes therefore additional data that neither the Tutte nor the Bollobas-Riordan polynomials can capture for the type of graphs described in the present work.",
      "comment": "47 pages, 33 figures; Improved version, some statements corrected",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2017-08-24T16:48:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C10",
      "57M15"
    ],
    "keywords": [
      "weakly-colored stranded graphs",
      "bollobas-riordan polynomial",
      "similar contraction/deletion recurrence relation",
      "seven indeterminate polynomial invariant",
      "ribbon graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.1987A"
    }
  }
}