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  "body": "\n## Example\n\nLet $C_1 = \\set{(x, y, 0) \\in \\R^3 : x^2 + y^2 = 1}$ and $C_2 = \\set{(x, 0, z) \\in \\R^3 : (x - 1)^2 + z^2 = 1}$.\nEach of them is a round [[circle|060W5CBZP6]].\nThey are disjoint: a common point would have $y = 0$ and $z = 0$, so $x = \\pm 1$ from $C_1$ and $x \\in \\set{0, 2}$ from $C_2$, which is impossible.\nSo $C_1 \\cup C_2$ is a [[link|1A5WRCNTM3]] with $2$ components, the *Hopf link*.\nThe circle $C_2$ passes once through the flat disk bounded by $C_1$, at the point $(0, 0, 0)$, and it meets that disk nowhere else.\nHence the [[linking number|SFJ329YB9F]] of the two components is $\\pm 1$, the sign depending on the orientations.\nThe unlink of two components has linking number $0$, so the Hopf link is not equivalent to it: the two rings cannot be pulled apart.\n\n## Remarks\n\nEach component is an [[unknot|1TQ06R4Q66]].\nThe Hopf link has a diagram with $2$ crossings, and it is the only nontrivial link with at most $2$ crossings.\nIts complement in $S^3$ is homeomorphic to $T^2 \\times (0, 1)$, the product of a torus and an open interval, so its link group is $\\Z \\times \\Z$.\nAny two distinct fibres of the [[Hopf fibration|M5AKNYC4QQ]] $S^3 \\to S^2$ form a Hopf link, which gives the name.\nIt is the torus link of type $(2, 2)$, a two-component relative of the [[torus knots|NS60VV786W]].\n\n## Explanations\n\n### L1 human\n\nThe Hopf link is two rings hooked together, like two neighbouring links of a chain.\nEach ring on its own is unknotted, but they cannot be separated without cutting one of them.\nIt is the simplest link that is really linked.\n",
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