Consider a $(p,q)$ torus knot $K$ in 3-dimensional Euclidean space $\mathbb R^3$ where $p,q \geq 2$ and $\operatorname{GCD}(p,q)=1$.

Let $\operatorname{Isom}(\mathbb R^3,K)$ be the isometries of $\mathbb R^3$ that preserve $K$.

It's a fairly standard argument using theorems about uniqueness of Seifert fiberings to prove that it's impossible for $\operatorname{Isom}(\mathbb R^3, K)$ to contain subgroups isomorphic to both $\mathbb Z_p$ and $\mathbb Z_q$. Of course, $\operatorname{Isom}(S^3,K)$ can and does for the standard embeddings of torus knots in $S^3$. In some sense the core issue is that when this does happen, the $\mathbb Z_p$ and $\mathbb Z_q$ subgroups of $\operatorname{Isom}(S^3,K)$ have disjoint fixed point sets.

My question: is there a reasonably elementary proof $\operatorname{Isom}(\mathbb R^3, K)$ does not contain subgroups isomorphic to both $\mathbb Z_p$ and $\mathbb Z_q$ that *avoid* the use of Seifert-fiber space techniques? I'm particularly interested if any "quantum topology" invariants can make this kind of symmetry argument. I thought a little about this, at least I'm not seeing how one could use the Alexander polynomial.

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