{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $L \\subset S^3$ be an oriented tame [[link|1A5WRCNTM3]], for example an oriented [[knot|QYEAZTYGPT]].\nA *Seifert surface* for $L$ is a subset $\\Sigma \\subset S^3$ such that:\n\n1. $\\Sigma$ is a compact connected [[surface|VGJJVBZ1MQ]] with boundary, embedded in $S^3$ in a tame way;\n2. $\\Sigma$ is [[orientable|702SNFQG0W]];\n3. the boundary of $\\Sigma$ is $L$, and the orientation of $L$ is the boundary orientation induced by an orientation of $\\Sigma$.\n\n## Remarks\n\nThe *genus* $g(K)$ of a knot $K$ is the least [[genus|NJ787GZS1N]] of a Seifert surface for $K$.\nExample: the unknot $\\set{(x, y, 0) : x^2 + y^2 = 1}$ in $\\R^3 \\subset S^3$ bounds the flat disk $\\set{(x, y, 0) : x^2 + y^2 \\le 1}$, which is compact, connected, orientable and has the unknot as boundary; so the unknot has genus $0$.\nConversely, a knot of genus $0$ bounds a disk and is the unknot.\nNon-example: a band with one half-twist, a Möbius band, has the unknot as boundary, but it is not orientable, so it is not a Seifert surface.\nA band with three half-twists is a Möbius band whose boundary is the trefoil, so it fails for the same reason; the trefoil has genus $1$ instead.\nSeifert's algorithm builds a Seifert surface from any diagram.\nOne smooths each crossing according to the orientation, fills the resulting circles with disks, and joins the disks by twisted bands at the crossings.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "K866ND3WF7",
    "kind": "definition",
    "msc": [
      "57K10"
    ],
    "name": "Seifert surface",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Seifert_surface"
      }
    ],
    "tag": "QT4HP1H2YE",
    "uses": [
      "QYEAZTYGPT",
      "1A5WRCNTM3",
      "VGJJVBZ1MQ",
      "702SNFQG0W",
      "NJ787GZS1N"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/QT4HP1H2YE/",
  "links": [
    {
      "field": "defines",
      "target": "K866ND3WF7"
    },
    {
      "field": "uses",
      "target": "QYEAZTYGPT"
    },
    {
      "field": "uses",
      "target": "1A5WRCNTM3"
    },
    {
      "field": "uses",
      "target": "VGJJVBZ1MQ"
    },
    {
      "field": "uses",
      "target": "702SNFQG0W"
    },
    {
      "field": "uses",
      "target": "NJ787GZS1N"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/QT4HP1H2YE.md",
  "msc": [
    "57K10"
  ],
  "name": "Seifert surface",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QT4/QT4HP1H2YE.md",
  "status": null,
  "tag": "QT4HP1H2YE",
  "uri": "m2://QT4HP1H2YE",
  "url": "https://math2.tamnd.com/wiki/seifert-surface-definition/",
  "writing": {
    "tag": "QT4HP1H2YE",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}