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  "body": "\n## Statement\n\nLet $K$ and $K'$ be tame [[knots|QYEAZTYGPT]] in $S^3$.\nSuppose that there is a [[homeomorphism|0QT00PB5G2]] $S^3 \\setminus K \\to S^3 \\setminus K'$.\nThen there is a homeomorphism $h \\colon S^3 \\to S^3$ with $h(K) = K'$.\n\n## Remarks\n\nThe proof reduces the theorem to a statement about [[Dehn surgery|ZA405B0Z1S]].\nA homeomorphism of complements that does not carry meridian to meridian would give a nontrivial surgery on $K'$ that yields $S^3$.\nGordon and Luecke showed that a nontrivial Dehn surgery on a nontrivial knot never gives $S^3$, using a careful study of the graphs formed by intersecting planar surfaces.\nThe homeomorphism $h$ may reverse orientation, so a knot and its mirror image have homeomorphic complements and count as equivalent here.\nThe analogue for links fails: twisting a link along an unknotted component changes the link but not the homeomorphism type of its complement.\nThe [[knot group|AQVHM3FP0D]] alone does not determine the knot: the square knot and the granny knot have isomorphic groups but are different knots.\n\n## Explanations\n\n### L1 human\n\nRemove a knot from space and keep the space that is left.\nThe theorem says that this leftover space remembers the knot completely.\nIf two knots leave behind the same space, up to stretching, then the knots themselves are the same, up to mirror image.\nSo questions about knots can be turned into questions about $3$-manifolds.\n\n### L2 human\n\nThe theorem justifies studying a knot through its complement, for instance through the hyperbolic structure of the complement of a [[hyperbolic knot|K47G2Q24DT]].\nCombined with Mostow–Prasad rigidity, it shows that a hyperbolic knot is determined by the isometry type of its complement.\nIt is close to the Property P question: both are about which surgeries on a knot can give simple manifolds.\nFor links, and for knots in other $3$-manifolds, the complement can lose information.\n",
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