{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $p$ and $q$ be coprime integers.\nThe *$(p, q)$-torus knot* $T(p, q)$ is the image of the map $f \\colon \\R / 2\\pi\\Z \\to \\R^3$,\n$$f(t) = \\big( (2 + \\cos qt) \\cos pt, \\ (2 + \\cos qt) \\sin pt, \\ -\\sin qt \\big).$$\nA [[knot|QYEAZTYGPT]] is a *torus knot* if it is isotopic to $T(p, q)$ for some coprime $p, q$ with $\\abs{p} \\ge 2$ and $\\abs{q} \\ge 2$.\n\n## Remarks\n\nThe image lies on the standard [[torus|03NM7DZXXC]] obtained by revolving the circle $(x - 2)^2 + z^2 = 1$ about the $z$-axis.\nThe map $f$ is injective because $p$ and $q$ are coprime.\nExample: $T(2, 3)$ is the [[trefoil knot|VYTYSVHN4A]]; here $\\abs{p}, \\abs{q} \\ge 2$, so it is a torus knot in the sense above.\nNon-example: $T(1, q)$ is isotopic to the [[unknot|1TQ06R4Q66]], so the bound $\\abs{p} \\ge 2$ fails and it is not counted as a torus knot.\n$T(p, q)$ is isotopic to $T(q, p)$, and $T(p, -q)$ is the mirror image of $T(p, q)$.\nFor $p, q \\ge 2$ the Seifert genus of $T(p, q)$ is $(p-1)(q-1)/2$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "NS60VV786W",
    "kind": "definition",
    "msc": [
      "57K10"
    ],
    "name": "Torus knot",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Torus_knot"
      }
    ],
    "tag": "MD1QFAEFF0",
    "uses": [
      "QYEAZTYGPT",
      "03NM7DZXXC"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/MD1QFAEFF0/",
  "links": [
    {
      "field": "defines",
      "target": "NS60VV786W"
    },
    {
      "field": "uses",
      "target": "QYEAZTYGPT"
    },
    {
      "field": "uses",
      "target": "03NM7DZXXC"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/MD1QFAEFF0.md",
  "msc": [
    "57K10"
  ],
  "name": "Torus knot",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MD1/MD1QFAEFF0.md",
  "status": null,
  "tag": "MD1QFAEFF0",
  "uri": "m2://MD1QFAEFF0",
  "url": "https://math2.tamnd.com/wiki/torus-knot-definition/",
  "writing": {
    "tag": "MD1QFAEFF0",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}