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  "body": "\n## Summary\n\nThe knot group of a [[knot|QYEAZTYGPT]] is the [[fundamental group|S1XK3HXHFX]] of its complement in $\\R^3$.\nIt is a [[knot invariant|GZQ5ZE3X6G]]: equivalent knots have isomorphic knot groups.\nIt is a powerful one, since it distinguishes the unknot from every other knot.\nA diagram of the knot gives an explicit presentation of it, the [[Wirtinger presentation|P6GR8ZQCK3]].\n\n## Explanations\n\n### L1 human\n\nImagine the knot as a fixed wire in space, and consider loops of string that start and end at a fixed point and may wind around the wire.\nTwo loops count as the same if one can be slid to the other without crossing the wire.\nFollowing one loop and then another gives a way to combine loops, and this makes them a group.\nFor the unknot, a loop is described by how many times it winds through the circle, so the group is the integers $\\Z$.\nFor the trefoil, the group is not commutative, which proves that the trefoil cannot be untied.\n",
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