{
  "id": "1106.1259",
  "version": "v2",
  "published": "2011-06-07T05:46:43.000Z",
  "updated": "2011-06-08T01:08:55.000Z",
  "title": "The canonical genus for Whitehead doubles of a family of alternating knots",
  "authors": [
    "Hee Jeong Jang",
    "Sang Youl Lee"
  ],
  "comment": "33 pages, 27 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "For any given integer $r \\geq 1$ and a quasitoric braid $\\beta_r=(\\sigma_r^{-\\epsilon} \\sigma_{r-1}^{\\epsilon}...$ $ \\sigma_{1}^{(-1)^{r}\\epsilon})^3$ with $\\epsilon=\\pm 1$, we prove that the maximum degree in $z$ of the HOMFLYPT polynomial $P_{W_2(\\hat\\beta_r)}(v,z)$ of the doubled link $W_2(\\hat\\beta_r)$ of the closure $\\hat\\beta_r$ is equal to $6r-1$. As an application, we give a family $\\mathcal K^3$ of alternating knots, including $(2,n)$ torus knots, 2-bridge knots and alternating pretzel knots as its subfamilies, such that the minimal crossing number of any alternating knot in $\\mathcal K^3$ coincides with the canonical genus of its Whitehead double. Consequently, we give a new family $\\mathcal K^3$ of alternating knots for which Tripp's conjecture holds.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-06-08T01:08:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M27"
    ],
    "keywords": [
      "alternating knot",
      "canonical genus",
      "whitehead double",
      "tripps conjecture holds",
      "minimal crossing number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.1259J"
    }
  }
}