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  "body": "\n## Statement\n\nLet $f \\colon S^3 \\to S^3$ be an orientation-preserving diffeomorphism of finite order $n \\ge 2$.\nSuppose that the fixed point set $\\set{x \\in S^3 : f(x) = x}$ is homeomorphic to a circle.\nThen this fixed circle is the [[unknot|1TQ06R4Q66]].\n\n## Remarks\n\nP. A. Smith showed that the fixed point set of such a map of prime order is empty or a circle, and asked whether the circle can be knotted.\nThe proof combined work of several authors, including Thurston on hyperbolic structures, Meeks and Yau on minimal surfaces, Bass on group theory, and Gordon and Litherland.\nOne case uses incompressible surfaces in the knot complement; the other uses a hyperbolic structure given by Thurston's work.\nThe orientation hypothesis matters: a reflection of $S^3$ has order $2$ and fixes a $2$-sphere.\nThe analogue in higher dimensions fails: in $S^4$, some knotted $2$-spheres are fixed sets of periodic diffeomorphisms.\nExample: the rotation by angle $2\\pi/n$ about the unknotted great circle $\\set{(z, 0)} \\subset S^3 \\subset \\C^2$, given by $(z, w) \\mapsto (z, e^{2\\pi i/n} w)$, fixes exactly that circle.\n\n## Explanations\n\n### L1 human\n\nImagine a rotation of three-dimensional space that, repeated a few times, comes back to where it started.\nThe points that do not move form an axis.\nIn ordinary space this axis is a straight line, and in the $3$-sphere it is an unknotted circle.\nThe theorem says that no smooth symmetry of finite order can have a knotted axis.\nSo a nontrivial [[knot|QYEAZTYGPT]] can never be the axis of such a rotation.\n\n### L2 human\n\nEquivalently, a nontrivial knot $K$ is never the branch set of a cyclic branched cover $S^3 \\to S^3$.\nThe theorem was a test case for Thurston's geometrization program: the decisive input came from hyperbolic geometry and minimal surfaces.\nIt is a precursor of the orbifold theorem, which handles more general finite group actions on $3$-manifolds.\n",
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