We show that every orientation-preserving periodic diffeomorphism
f on a closed orientable 3-manifold M has a \textquotedblleft{}surgery
description\textquotedblright{}, that is, there is a framed link $\mathcal{L}\;\textrm{in}\; S^{3}$
which is invariant by a standard rotation $\varphi$ around a trivial
knot, such that M is obtained by surgery on $\varphi$ and that f
is conjugate to the periodic diff{}eomorphism induced by $\varphi$.
We will illustrate this result, by visualizing isometries of the complements
of 2-component hyperbolic links with $\leq$ 9 crossings which do
not extend to periodic maps of $S^{3}$.
