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  "body": "\n## Problem\n\nThe input is a finite diagram of a tame [[knot|QYEAZTYGPT]] in $\\R^3$, for example a polygon with finitely many vertices given by rational coordinates.\nThe question is whether the knot is the [[unknot|1TQ06R4Q66]], that is, whether it is ambient isotopic to a round circle.\nThe problem asks for an algorithm that answers this question for every input, and for the cost of such an algorithm.\n\n## Remarks\n\nDehn's work of 1910 began the search for such an algorithm, through the fundamental group of the knot complement.\nHaken proved with his theory of normal surfaces, published in 1961, that the problem is decidable.\nHis method searches for a disk bounded by the knot among finitely many normal surfaces in the knot complement.\nHass, Lagarias and Pippenger proved in 1999 that the problem lies in the complexity class NP.\nLackenby proved without extra hypotheses that it also lies in co-NP.\nIt is not known whether the problem can be solved in polynomial time.\n[[Reidemeister moves|T3HAGWYKK7]] give another approach: a diagram of the unknot can be reduced to a circle by such moves, and there are explicit bounds on the number of moves needed.\n\n## Explanations\n\n### L1 human\n\nTake a tangled loop of string whose ends are glued together.\nThe question is whether you can untangle it into a plain circle without cutting it.\nA messy picture of the plain circle can look very knotted, so the answer is not obvious.\nMathematicians know that a computer can always decide this, but they do not know whether it can be done quickly for large diagrams.\n",
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      "first_stated": "1910",
      "posed_by": "Max Dehn"
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    "kind": "problem",
    "msc": [
      "57K10",
      "68Q17"
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    "name": "Unknotting problem",
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      "wikidata": "Q7896950",
      "wikipedia": "Unknotting problem"
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      {
        "url": "https://en.wikipedia.org/wiki/Unknotting_problem"
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      {
        "locator": "Section 1",
        "url": "https://arxiv.org/abs/math/9807016"
      }
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    "tag": "VXSD46NRX0",
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      "T3HAGWYKK7"
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  "msc": [
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