I'm starting to read Allen Hatcher's Space of Knots and at the end of the first paragraph he says that his work would apply to all knots if the following conjecture is true:
Every free action of a finite cyclic group on $S^3$ is equivalent to a standard linear action.
The paper is from 1999, so I don't know if this is still a conjecture. Where can I read about it and the progress on it?
I'm also interested in knowing how this conjecture applies to knot theory, in particular to the results Hatcher gives in the paper.
Take the quotient by the action. You find a 3-manifold with finite cyclic fundamental group. The geometrization theorem, proof due to Perelman, implies that necessarily your 3-manifold is equivalent to an elliptic 3-manifold: a lens space $L(n,q)$ for $1 \leq q < n$. Passing to the universal cover, we find an equivariant diffeomorphism $S^3 \cong S^3$, where the first $S^3$ is given your nonstandard action and the second $S^3$ has action $1 \cdot (z,w) = (e^{2\pi i/n} z, e^{2 \pi iq/n} w)$.
The assumption seems to appear in the first paragraph of page 2: "Assuming the action of $\Gamma_K$ is equivalent to an action by elements of $SO(4)$..." This is known to be true now by the above.