{
  "id": "1406.2042",
  "version": "v1",
  "published": "2014-06-09T00:01:04.000Z",
  "updated": "2014-06-09T00:01:04.000Z",
  "title": "On the Characterization Problem of Alexander Polynomials of Closed 3-Manifolds",
  "authors": [
    "Karin Alcaraz"
  ],
  "comment": "22 pages, 3 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We give a characterization for the Alexander Polynomials of closed orientable 3-manifolds M with first Betti number 1, as well as some partial results for the characterization problem for M having first Betti number > 1. We first prove an analogue of a theorem of Levine: that the product of an Alexander polynomial of M with a symmetric polynomial in the same number of variables having non 0 trace, is again an Alexander polynomial of a closed orientable 3-manifold. Using the fact that there exists M with Alexander polynomial = 1 for M with first Betti number 1, 2 or 3, we conclude that symmetric polynomials of non 0 trace in 1, 2 or 3 variables are Alexander polynomials of closed orientable 3-manifolds. When the first Betti number of M is 1 we prove that non 0 trace symmetric polynomials are the only ones that can arise. Finally, for M with first Betti number > 3 we prove that the Alexander polynomial can not be 1, implying that for such manifolds not all symmetric polynomials having non 0 trace will occur.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-09T00:01:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27"
    ],
    "keywords": [
      "alexander polynomial",
      "first betti number",
      "characterization problem",
      "trace symmetric polynomials",
      "closed orientable"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.2042A"
    }
  }
}