{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *framed link* in $S^3$ is a [[link|1A5WRCNTM3]] $L = K_1 \\cup \\cdots \\cup K_r$ in $S^3$ with an integer $n_i$, its framing, attached to each component $K_i$.\nIt determines a closed oriented $3$-manifold $M_L$ by integral [[Dehn surgery|ZA405B0Z1S]] on $L$.\nIt also determines a compact $4$-manifold $W_L$, obtained from $D^4$ by attaching one $2$-handle along each $K_i$ with framing $n_i$.\nThis is a [[handle decomposition|9V11N8VVTP]] of $W_L$, and $\\partial W_L = M_L$.\n*Kirby calculus* is the system of the following moves on framed links, together with isotopy of links:\n\n1. Blow-up and blow-down: add to $L$, or remove from $L$, an unknot with framing $\\pm 1$ inside a ball disjoint from the rest of $L$.\n2. Handle slide: replace a component $K_i$ by a band sum of $K_i$ with a parallel copy of another component $K_j$.\n   The parallel copy is taken with respect to the framing $n_j$, and the framing of the new component changes accordingly.\n\n## Remarks\n\nBy the [[Lickorish–Wallace theorem|1YP4RF1QXV]], every closed connected oriented $3$-manifold is $M_L$ for some framed link $L$.\nKirby's theorem states that $M_L$ and $M_{L'}$ are orientation-preserving diffeomorphic if and only if $L$ and $L'$ are related by a finite sequence of the moves above.\nA handle slide does not change $W_L$ up to diffeomorphism, while a blow-up replaces $W_L$ by its connected sum with $\\pm \\C P^2$; both leave $M_L$ unchanged.\nExample: the unknot with framing $+1$ describes $S^3$ itself, since a blow-down turns it into the empty link, which describes $S^3$.\nNon-example: the unknot with framing $0$ cannot be blown down, because move 1 needs framing $\\pm 1$; indeed it describes $S^1 \\times S^2$, which is not $S^3$.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "AVQ9WN1MDV",
    "kind": "definition",
    "msc": [
      "57K40",
      "57K30",
      "57R65"
    ],
    "name": "Kirby calculus",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kirby_calculus"
      }
    ],
    "tag": "ZFNZ0M7X41",
    "uses": [
      "1A5WRCNTM3",
      "ZA405B0Z1S",
      "9V11N8VVTP",
      "1YP4RF1QXV"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/ZFNZ0M7X41/",
  "links": [
    {
      "field": "defines",
      "target": "AVQ9WN1MDV"
    },
    {
      "field": "uses",
      "target": "1A5WRCNTM3"
    },
    {
      "field": "uses",
      "target": "ZA405B0Z1S"
    },
    {
      "field": "uses",
      "target": "9V11N8VVTP"
    },
    {
      "field": "uses",
      "target": "1YP4RF1QXV"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/ZFNZ0M7X41.md",
  "msc": [
    "57K40",
    "57K30",
    "57R65"
  ],
  "name": "Kirby calculus",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZFN/ZFNZ0M7X41.md",
  "status": null,
  "tag": "ZFNZ0M7X41",
  "uri": "m2://ZFNZ0M7X41",
  "url": "https://math2.tamnd.com/wiki/kirby-calculus-definition/",
  "writing": {
    "tag": "ZFNZ0M7X41",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}