{
  "id": "1410.5728",
  "version": "v1",
  "published": "2014-10-21T16:25:30.000Z",
  "updated": "2014-10-21T16:25:30.000Z",
  "title": "Spaces of Polynomial knots in low degree",
  "authors": [
    "Rama Mishra",
    "Hitesh Raundal"
  ],
  "comment": "31 pages, 11 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "{\\it We show that all knots up to $6$ crossings can be represented by polynomial knots of degree at most $7$; among which except for $5_2, 5_2^*, 6_1, 6_1^*, 6_2, 6_2^*$ and $6_3$ all are in their minimal degree representation. We provide concrete polynomial representation of all these knots. Durfee and O'shea had asked a question: Is there any $5$ crossing knot in degree $6$? In this paper we try to partially answer this question. We define the set $\\pd$ to be the set of all polynomial knots given by $t\\mapsto\\fght$ where $f, g$ and $h$ are real polynomials with $\\fghio$. This set can be identified with a subset of $\\rtd$ and thus it is equipped with the natural topology which comes from the usual topology $\\rtd$. In this paper we determine a lower bound on the number of path components of $\\pd$ for $d\\leq 7$. We define path equivalence between polynomial knots in the space $\\pd$ and show that path equivalence is stronger than the topological equivalence.}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-21T16:25:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57Q45"
    ],
    "keywords": [
      "polynomial knots",
      "low degree",
      "concrete polynomial representation",
      "minimal degree representation",
      "define path equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.5728M"
    }
  }
}