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  "body": "\n## Definition\n\nLet $K \\subset \\R^3$ be a [[knot|QYEAZTYGPT]], and let $x_0$ be a point of $\\R^3 \\setminus K$.\nThe *knot group* of $K$ is the [[fundamental group|S1XK3HXHFX]]\n$$\\pi_1(\\R^3 \\setminus K, x_0).$$\n\n## Remarks\n\nThe complement $\\R^3 \\setminus K$ is path-connected, so the group does not depend on $x_0$ up to isomorphism.\nIf $K$ and $K'$ are equivalent, an ambient isotopy gives a homeomorphism of their complements, so their knot groups are isomorphic.\nRemoving a point at infinity does not change it: $\\pi_1(S^3 \\setminus K) \\cong \\pi_1(\\R^3 \\setminus K)$.\nFor a tame knot, the abelianization of the knot group is $\\Z$.\nExample: the complement of the [[unknot|1TQ06R4Q66]] deformation retracts onto the union of a $2$-sphere and a circle that meet in one point.\nSo its knot group is $\\Z$, generated by a loop that goes once around the unknot.\nFor the [[trefoil knot|VYTYSVHN4A]], the knot group has the presentation $\\langle x, y \\mid xyx = yxy \\rangle$; it is not abelian, so the trefoil is not equivalent to the unknot.\nNon-example: the fundamental group of the knot $K$ itself is not the knot group.\nIt is $\\Z$ for every knot, since $K$ is homeomorphic to a circle, so it distinguishes no knots.\n",
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