{
  "id": "1407.3705",
  "version": "v2",
  "published": "2014-07-14T15:53:35.000Z",
  "updated": "2015-03-13T17:12:03.000Z",
  "title": "Representations of knot groups into $\\mathrm{SL}_n(\\mathbf{C})$ and twisted Alexander polynomials",
  "authors": [
    "Joan Porti",
    "Michael Heusener"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $\\Gamma$ be the fundamental group of the exterior of a knot in the three-sphere. We study deformations of representations of $\\Gamma$ into $\\mathrm{SL}_n(\\mathbf{C})$ which are the sum of two irreducible representations. For such representations we give a necessary condition, in terms of the twisted Alexander polynomial, for the existence of irreducible deformations. We also give a more restrictive sufficient condition for the existence of irreducible deformations. We also prove a duality theorem for twisted Alexander polynomials and we describe the local structure of the representation and character varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-14T15:53:35.000Z",
      "abstract": "Let $\\Gamma$ be the fundamental group of the exterior of a knot in the three-sphere. For a representation of $\\Gamma$ in $\\mathrm{SL}_n(\\mathbf{C})$ that is direct sum of two irreducible ones, we give necessary and sufficient conditions for the existence of deformations to irreducible representations, in terms of twisted Alexander polynomials. We also prove a duality theorem for twisted Alexander polynomials and we describe the local structure of the representation and character varieties.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-13T17:12:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M05",
      "57M27"
    ],
    "keywords": [
      "twisted alexander polynomials",
      "knot groups",
      "direct sum",
      "fundamental group",
      "sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.3705P"
    }
  }
}