In Chapter One of his notes (March 2002) Thurston says:
If $K$ is the trivial knot the cyclic branched covers are $S^3$. It seems intuitively obvious (but it is not known) that this is the only way $S^3$ can be obtained as a cyclic branched covering of itself over a knot.
This sentence sounds a little bit enigmatic to me. Is he saying that (we expect that) "if $K \subset S^3$ is such that all its cyclic branched covers are $S^3$ then $K$ is the unknot", or that "if for some $k\geq 2$ the $k$-fold cyclic branched cover over $K$ is $S^3$ then $K$ is the unknot"?
I would like to know which is the state of the art this conjecture. In particular if $S^3$ can appear as a branched cover over a non-trivial knot I would like to see some (families of) examples.