{
  "id": "1301.0138",
  "version": "v2",
  "published": "2013-01-01T21:44:02.000Z",
  "updated": "2014-09-02T16:28:33.000Z",
  "title": "Some families of minimal elements for the partial ordering on prime knots",
  "authors": [
    "Fumikazu Nagasato",
    "Anh T. Tran"
  ],
  "comment": "Title changed, 12 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "We show that all twist knots, certain double twist knots and some other 2-bridge knots are minimal elements for the partial ordering on the set of prime knots. The key to these results are presentations of their character varieties using Chebyshev polynomials and a criterion for irreducibility of a polynomial of two variables. These give us an elementary method to discuss the number of irreducible components of the character varieties, which concludes the result essentially.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-01T21:44:02.000Z",
      "title": "Presentations of character varieties of 2-bridge knots using Chebyshev polynomials",
      "abstract": "In this paper, we use Chebyshev polynomials to give presentations of the character varieties of certain types of 2-bridge knots. This gives us an elementary method using basic calculations to discuss the number of irreducible components of the character varieties and thus to recover the results of Burde on the irreducibility of non-abelian SU(2)-representation spaces in [2]. These results can be applied to determine some minimal elements of a partial ordering of prime knots.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-02T16:28:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27"
    ],
    "keywords": [
      "character varieties",
      "chebyshev polynomials",
      "presentations",
      "minimal elements",
      "non-abelian su"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.0138N"
    }
  }
}