{
  "id": "1302.1631",
  "version": "v2",
  "published": "2013-02-07T02:39:39.000Z",
  "updated": "2013-09-04T00:25:38.000Z",
  "title": "On the twisted Alexander polynomial for representations into SL_2(C)",
  "authors": [
    "Anh T. Tran"
  ],
  "comment": "Minor changes. To appear in Journal of Knot Theory and Its Ramifications",
  "categories": [
    "math.GT"
  ],
  "abstract": "We study the twisted Alexander polynomial $\\Delta_{K,\\rho}$ of a knot $K$ associated to a non-abelian representation $\\rho$ of the knot group into $SL_2(\\BC)$. It is known for every knot $K$ that if $K$ is fibered, then for every non-abelian representation, $\\Delta_{K,\\rho}$ is monic and has degree $4g(K)-2$ where $g(K)$ is the genus of $K$. Kim and Morifuji recently proved the converse for 2-bridge knots. In fact they proved a stronger result: if a 2-bridge knot $K$ is non-fibered, then all but finitely many non-abelian representations on some component have $\\Delta_{K,\\rho}$ non-monic and degree $4g(K)-2$. In this paper, we consider two special families of non-fibered 2-bridge knots including twist knots. For these families, we calculate the number of non-abelian representations where $\\Delta_{K,\\rho}$ is monic and calculate the number of non-abelian representations where the degree of $\\Delta_{K,\\rho}$ is less than $4g(K)-2$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-09-04T00:25:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27"
    ],
    "keywords": [
      "twisted alexander polynomial",
      "non-abelian representation",
      "twist knots",
      "knot group",
      "special families"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.1631T"
    }
  }
}