I have a question about whether Ryan Budney's question:

Torus knots in Euclidean space -- a symmetry argument

can be extended to links. He asks:

Suppose you have a $(p,q)$ torus knot $K$ in $\mathbb{R}^3$ fixed by a subgroup $G$ of $\operatorname{SO}(3)$. Budney asks (more or less) for a nicer proof that $G$ cannot contain $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{Z}/q\mathbb{Z}$ as subgroups.

Edit: I think I understand Charlie Frohman's point in the comments to that question now that this follows from knowing the periods of the torus knots AND the structure of the finite subgroups of SO(3), which are very different than the finite subgroups of SO(4).

I'd like to extend this same result to $(p,q)$ torus links: there should be no configuration in $\mathbb{R}^3$ which is both $p$-fold and $q$-fold symmetric. I feel like this sort of thing ``should be known'', but so far my literature search has not turned up anything.

EDIT: I think you could probably do this in a similar way if you knew the periods of the torus links (as opposed to the torus knots). But is \emph{this} known?