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Generally, I am interested in understanding the free actions of finite cyclic groups on $S^3$ which leave invariant an oriented torus knot $T(p,q)$. For a specific example, consider the knot $T(3,4)$ and the group $\mathbb{Z}/5\mathbb{Z}$. My understanding is that there is a unique (up to equivariant homoemorphism of pairs $(S^3,K)$) order 5 symmetry of each torus knot, see for example the top of page 2 here, since there is a unique order 5 subgroup of $S^1$. However, here are two free order 5 symmetries of $T(3,4)$, where $H$ is a right-handed half twist, $F^2$ is two right-handed full twists, $I$ is the trivial tangle, $-F$ is a left-handed full twist, and $T(3,4)$ is the closure of the shown braid:

A free order 5 symmetry given by sliding <span class=$1/5$th of the way to the right, and then doing $1/5$th of a left-handed full twist." />,

and

A free order 5 symmetry given by sliding <span class=$1/5$th of the way to the right, and then doing $2/5$ths of a right-handed full twist." />.

It seems to me that in the first example the quotient of $S^3$ is the lens space $L(5,-1)$ and in the second example the quotient is the lens space $L(5,2)$. Since these 3-manifolds are not homeomorphic, there then could not be an equivariant homemomorphism of pairs $(S^3,T(3,4))$ between these two symmetries. This seems to contradict the uniqueness of the order 5 symmetry on $T(3,4)$.

Is there indeed a unique order 5 symmetry of $T(3,4)$? If so, up to what equivalence of $\mathbb{Z}/5\mathbb{Z}$ symmetries? Can I see directly that these two symmetries are the same under that equivalence?

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  • $\begingroup$ Your first paragraph seems a little confused to me. You are citing a Hatcher paper on the homotopy-type of the space of embeddings $Emb(S^1, S^3)$ but you are trying to come to some conclusion about uniqueness of an order 5 symmetry. Hatcher does not talk about this issue. He does discuss the subgroup of $Isom(S^3) \simeq O_4$ that preserves a torus knot. But that is quite different from studying finite subgroups of $Diff(S^4, K)$. I think it would help to state your question clearly. $\endgroup$ Aug 6 at 20:19
  • $\begingroup$ Are you asking if all element of order $5$ in $Diff(S^3, T_{3,4})$, are conjugate? $\endgroup$ Aug 6 at 20:20
  • $\begingroup$ Thanks, I think this is exactly my confustion - do you know of a reference discussing finite order subgroups of $Diff(S^3,K)$ for torus knots? Indeed I am interested to know how many conjugacy classes of order 5 elements there are. $\endgroup$ Aug 6 at 20:25
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    $\begingroup$ Those don't look like symmetries of $T_{4,3}$ to me. For the torus knot in one of its standard embeddings the symmetries are $(z,w) \longmapsto (t^4 z, t^3 w)$ where $t$ is a unit complex number, i.e. I am describing the homomorphism from $S^1 \to SO_4$. $\endgroup$ Aug 6 at 21:51
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    $\begingroup$ Thanks for your help! I'll do some more reading and see if I can explain the symmetries I have in mind clearly / post a specific question later. $\endgroup$ Aug 6 at 22:03

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