{
  "id": "2204.13660",
  "version": "v1",
  "published": "2022-04-28T17:21:57.000Z",
  "updated": "2022-04-28T17:21:57.000Z",
  "title": "A novel connection between integral binary quadratic forms and knot polynomials",
  "authors": [
    "Amitesh Datta"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We establish a novel connection between algebraic number theory and knot theory. We show that the number of equivalence classes of integral binary quadratic forms of discriminant $t^2 - 4$ (for $t\\neq \\pm 2$) is equal to the number of isotopy classes of links in $\\mathbb{S}^3$ with prescribed values (depending on $t$) of three classical link invariants. The equality arises from a natural algebraic correspondence between integral binary quadratic forms (of discriminant $t^2 - 4$ for $t\\neq \\pm 2$) and isotopy classes of links of braid index at most three. In particular, the class numbers of certain quadratic number fields precisely measure the failure of the Alexander/Jones polynomial to distinguish non-isotopic links of braid index at most three.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-28T17:21:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E16",
      "57K14",
      "20F36"
    ],
    "keywords": [
      "integral binary quadratic forms",
      "novel connection",
      "knot polynomials",
      "quadratic number fields precisely measure",
      "braid index"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}