*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.