{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nThe *trefoil knot* is the image of the map $f \\colon \\R / 2\\pi\\Z \\to \\R^3$ given by\n$$f(t) = \\bigl((2 + \\cos 3t) \\cos 2t, \\ (2 + \\cos 3t) \\sin 2t, \\ \\sin 3t\\bigr).$$\nThe map is continuous, and it is injective.\nIndeed, if $f(t) = f(s)$, the last coordinate and the distance $2 + \\cos 3t$ from the $z$-axis give $3t \\equiv 3s \\pmod{2\\pi}$.\nThe angle around the $z$-axis gives $2t \\equiv 2s \\pmod{2\\pi}$.\nThen $t - s$ is a multiple of $2\\pi/3$ and of $\\pi$, so of $2\\pi$.\nHence the image is a [[knot|QYEAZTYGPT]].\nIt lies on a torus, and it winds $2$ times around the axis of the torus and $3$ times around its core.\nSo it is the [[torus knot|NS60VV786W]] of type $(2, 3)$.\n\n## Remarks\n\nIts [[crossing number|SWJZ4QRF9W]] is $3$, and it is the only knot with crossing number $3$ up to mirror image.\nIt is [[tricolorable|KNAGK0W65G]] while the unknot is not, so it is not the unknot.\nIts [[knot group|AQVHM3FP0D]] has the presentation $\\langle x, y \\mid xyx = yxy \\rangle$, which is isomorphic to $\\langle a, b \\mid a^2 = b^3 \\rangle$.\nIts Alexander polynomial is $t^{-1} - 1 + t$, and its genus is $1$.\nIt is not equivalent to its mirror image, a fact that the Jones polynomial detects.\n\n## Explanations\n\n### L1 human\n\nThe trefoil is the simplest real knot: tie an overhand knot in a string and glue the ends together.\nAny picture of it has at least three crossings, and the standard picture has three lobes like a clover leaf.\nIt comes in two mirror forms, left-handed and right-handed, and these cannot be moved into each other.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "instance_of": [
      "QYEAZTYGPT"
    ],
    "kind": "example",
    "msc": [
      "57K10"
    ],
    "name": "Trefoil knot",
    "same_as": {
      "wikidata": "Q168620",
      "wikipedia": "Trefoil knot"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Trefoil_knot"
      }
    ],
    "tag": "VYTYSVHN4A"
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/VYTYSVHN4A/",
  "links": [
    {
      "field": "instance_of",
      "target": "QYEAZTYGPT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/VYTYSVHN4A.md",
  "msc": [
    "57K10"
  ],
  "name": "Trefoil knot",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VYT/VYTYSVHN4A.md",
  "status": null,
  "tag": "VYTYSVHN4A",
  "uri": "m2://VYTYSVHN4A",
  "url": "https://math2.tamnd.com/wiki/trefoil-knot/",
  "writing": {
    "tag": "VYTYSVHN4A",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}