{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $K$ be a tame [[knot|QYEAZTYGPT]] in $\\R^3$, and let $D$ be a diagram of $K$.\nA *crossing change* at a crossing of $D$ exchanges the over-strand and the under-strand at that crossing.\nLet $u(D)$ be the least number of crossings of $D$ whose simultaneous change gives a diagram of the [[unknot|1TQ06R4Q66]].\nThe *unknotting number* of $K$ is\n$$u(K) = \\min \\set{u(D) : D \\text{ is a diagram of } K}.$$\n\n## Remarks\n\nThe number $u(D)$ is finite: changing suitable crossings of any diagram gives a diagram of the unknot.\nExample: the standard three-crossing diagram of the [[trefoil knot|VYTYSVHN4A]] becomes a diagram of the unknot after one crossing change.\nThe trefoil is not the unknot, so its unknotting number is $1$.\nNon-example of the minimum: a single diagram may need more changes than $u(K)$, so $u(D)$ for one fixed diagram $D$ is only an upper bound for $u(K)$.\nA knot has $u(K) = 0$ if and only if it is the unknot.\nThe unknotting number of the [[torus knot|NS60VV786W]] $T(p, q)$ with coprime $p, q \\ge 2$ is $(p-1)(q-1)/2$; this is the [[Milnor conjecture|5SDA7ZXT95]].\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "PW9D9QKAD4",
    "kind": "definition",
    "msc": [
      "57K10"
    ],
    "name": "Unknotting number",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Unknotting_number"
      }
    ],
    "tag": "MCPFA4DC9Z",
    "uses": [
      "QYEAZTYGPT",
      "1TQ06R4Q66"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/MCPFA4DC9Z/",
  "links": [
    {
      "field": "defines",
      "target": "PW9D9QKAD4"
    },
    {
      "field": "uses",
      "target": "QYEAZTYGPT"
    },
    {
      "field": "uses",
      "target": "1TQ06R4Q66"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/MCPFA4DC9Z.md",
  "msc": [
    "57K10"
  ],
  "name": "Unknotting number",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MCP/MCPFA4DC9Z.md",
  "status": null,
  "tag": "MCPFA4DC9Z",
  "uri": "m2://MCPFA4DC9Z",
  "url": "https://math2.tamnd.com/wiki/unknotting-number-definition/",
  "writing": {
    "tag": "MCPFA4DC9Z",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}