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  "body": "\n## Counterexample\n\nA [[knot|QYEAZTYGPT]] is a [[topological embedding|00ZG9Y9TR4]] $K \\colon S^1 \\to \\R^3$.\nIt is *tame* if there is a [[homeomorphism|0QT00PB5G2]] $h \\colon \\R^3 \\to \\R^3$ such that $h(K(S^1))$ is a closed polygon with finitely many sides; otherwise it is *wild*.\n\nLet $p$ be a point of $\\R^3$.\nTake a sequence of small trefoil knots, of sizes tending to $0$, placed one after the other along an arc that ends at $p$.\nTie them all into one closed curve that passes through $p$, so that every neighbourhood of $p$ contains infinitely many of the small trefoils.\nThe result is a knot, and it is wild.\nEvery point of a polygonal knot has a small ball around it that meets the knot in one unknotted arc.\nA homeomorphism of $\\R^3$ carries this local picture along.\nAt $p$ the knot has no such ball, because every small ball around $p$ contains whole knotted trefoils of the sequence.\nOne shows that this local knotting cannot be undone.\n\n## Remarks\n\nSo not every topological knot is tame, and knot theory usually restricts to tame knots, for example to smooth or polygonal ones.\nThe local flatness property at every point, which fails at $p$, characterizes tame knots among topological ones.\nWild knots and other wild embeddings, such as the Alexander horned sphere, are studied in the theory of wild embeddings.\n\n## Explanations\n\n### L1 human\n\nOrdinary knots are tied in a rope with a finite number of crossings.\nA wild knot is a closed curve with infinitely many knots tied in it, getting smaller and smaller and piling up at one point.\nNear that point the curve is knotted at every scale, so it cannot be straightened into a nice polygon.\n",
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  "front": {
    "kind": "counterexample",
    "msc": [
      "57M30",
      "57K10"
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    "name": "Wild knot",
    "same_as": {
      "wikidata": "Q2489408",
      "wikipedia": "Wild knot"
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    "sources": [
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        "url": "https://en.wikipedia.org/wiki/Wild_knot"
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    "tag": "QYDQ28X8MA",
    "uses": [
      "QYEAZTYGPT",
      "00ZG9Y9TR4",
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  "kind": "counterexample",
  "link": "https://math2.tamnd.com/node/QYDQ28X8MA/",
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      "field": "uses",
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      "target": "00ZG9Y9TR4"
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      "field": "uses",
      "target": "0QT00PB5G2"
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  "markdown": "https://math2.tamnd.com/node/QYDQ28X8MA.md",
  "msc": [
    "57M30",
    "57K10"
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  "name": "Wild knot",
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  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QYD/QYDQ28X8MA.md",
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  "tag": "QYDQ28X8MA",
  "uri": "m2://QYDQ28X8MA",
  "url": "https://math2.tamnd.com/wiki/wild-knot/",
  "writing": {
    "tag": "QYDQ28X8MA",
    "kind": "counterexample",
    "level": "W0",
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