*This is a follow-up question to [Can I wrap a suitcase with hair ties](https://mathoverflow.net/questions/424215/can-i-wrap-a-suitcase-with-hair-ties).*


Now we know that it is possible to wrap a suitcase with hair ties without tying them together,



> 
> but can you do it with large rotational symmetry?
> 
> 
> 


In other words, I am looking for a nontrivial link $L$ in a solid torus made by short circles that is trivial in the ambient euclidean space and such that $L$ is invariant with respect to a large cyclic group of rotations of the solid torus.



Wrapping a suitcase with large rotational symmetry
**Edit: my previous answer was incorrect**


No, this one you cannot do. If one had such an arrangement of circles in the solid torus inside $R^3$, and quotiented by a rotation, then by the [equivariant Dehn's lemma](https://link.springer.com/article/10.1007/BF02566211) there would be a collection of essential disks in the complement of the split link that are invariant under the symmetry.


Since the link components get permuted, no disk can have a fixed point of the cyclic action. So they are disjoint from the axis of symmetry of the rotation.
Hence the components are unknotted in the solid torus, a contradiction


