{
  "id": "1809.01779",
  "version": "v1",
  "published": "2018-09-06T01:04:55.000Z",
  "updated": "2018-09-06T01:04:55.000Z",
  "title": "On a Nonorientable Analogue of the Milnor Conjecture",
  "authors": [
    "Stanislav Jabuka",
    "Cornelia A. Van Cott"
  ],
  "comment": "56 pages, 7 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "The nonorientable 4-genus $\\gamma_4(K)$ of a knot $K$ is the smallest first Betti number of any nonorientable surface properly embedded in the 4-ball, and bounding the knot $K$. We study a conjecture proposed by Batson about the value of $\\gamma_4$ for torus knots, which can be seen as a nonorientable analogue of Milnor's Conjecture for the orientable 4-genus of torus knots. We prove the conjecture for many infinite families of torus knots, by relying on a lower bound for $\\gamma_4$ formulated by Ozsv\\'ath, Stipsicz, and Szabo. As a side product we obtain new closed formulas for the signature of torus knots. We provide a comparison between $\\gamma_4$ and $\\gamma_3$ for torus knots, where $\\gamma_3(K)$ is the cross-cap number of $K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-06T01:04:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "57M25"
    ],
    "keywords": [
      "torus knots",
      "nonorientable analogue",
      "milnor conjecture",
      "smallest first betti number",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 56,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}