Timeline for Invariant knot for finite group actions on $S^3$
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| Feb 2, 2021 at 8:41 | history | edited | Sam Nead | CC BY-SA 4.0 |
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| Feb 1, 2021 at 22:51 | comment | added | Ryan Budney | Turning your question around a little, you are indirectly asking about the symmetry groups of knots in $S^3$. These are readily computed, by the techniques of Sakuma as mentioned. But you can also compute them for hyperbolic knots fairly rapidly with SnapPea. This was originally done by Jeff Weeks, at essentially the same time as Sakuma. I believe Sakuma's table of symmetry groups for "small" knots is in the back of Kawauchi's reference book on knot theory. A nice feature of the Snappea approach is you can algorithmically describe the fixed point sets. | |
| Feb 1, 2021 at 22:37 | answer | added | Strongly Negative Amphicheiral | timeline score: 2 | |
| Sep 11, 2020 at 9:43 | review | Close votes | |||
| Sep 20, 2020 at 3:07 | |||||
| Sep 11, 2020 at 9:18 | history | edited | YCor |
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| Sep 11, 2020 at 9:15 | comment | added | YCor | Probably you want a nontrivial action, and then the question reduces to faithful actions of cyclic groups of prime order. Any knot in Euclidean $\mathbf{R}^3$ or in the round sphere, with nontrivial isometry group, yields an example. For example, the usual representation of the trefoil knot has a isometry group of order $6$. | |
| Sep 10, 2020 at 22:54 | comment | added | zeraoulia rafik | Since invariant knot include knot Group, Other conventions consider knots to be embedded in the 3-sphere, in which case the knot group is the fundamental group of its complement in $S^3$ | |
| Sep 10, 2020 at 22:45 | comment | added | Dmitri Panov | Certainly. To get a simple example, take any toric knot, i.e. something given by equation $z_1^p=z_2^q$ in the unit sphere $|z_1|^2+|z_2|^2=1$. For any such knot there is a (linear) circle action on $S^3$ that preserves the knot, and you can take any finite subgroup of this circle. | |
| Sep 10, 2020 at 22:35 | history | edited | Ali Taghavi |
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| Sep 10, 2020 at 22:29 | history | asked | Ali Taghavi | CC BY-SA 4.0 |