# What are the order 5 symmetries of the the torus knot T(3,4)?

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:

and

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?

• 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. Aug 6 at 20:19
• Are you asking if all element of order $5$ in $Diff(S^3, T_{3,4})$, are conjugate? Aug 6 at 20:20
• 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. Aug 6 at 20:25
• 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 $S^1 \to SO_4$. Aug 6 at 21:51
• 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. Aug 6 at 22:03