Thurston claimed that almost all closed 3-manifolds are hyperbolic. To support this, he said that every closed 3-manifold is obtained by Dehn surgery along some link whose complement is hyperbolic. And he proved that most surgeries along this link have hyperbolic structure.
The Lickorish–Wallace theorem tells us that every closed orientable 3 manifold can be described via integral Dehn surgery along a link in $S^3$. So my Question is, given a link $L\subset S^3$, how to construct (modify) a link $L'$ whose complement is hyperbolic? Because we know that $\infty$-sugery along any knot does not change the manifold. Thus we can hope to prove Thurston's claim.
We know that if a knot $K\subset S^3$ is not satellite or torus, then it is hyperbolic. So let's say, if we start with a sattelite knot, then we can add an unknot which intersect the indecomposable torus non-trivially. Then in this process we can kill off the indecomposable torus. But I am unable to prove that in this process we have not constructed any new indecomposable torus. Also I don't know whether this complememnt is Seifert or hyperbolic.
EDIT: (Question 2) Can we construct $L'$ from $L$ by adding only trivial knots?
Thank you in advance.