I'm considering finite index universal abelian (regular) covering of link complement:

$ X \rightarrow S^3\backslash L$

where $L$ is a minimally twisted chain link.

I'm interested in covering space. Can we compute its cohomology (or maybe fundamental group) in terms of $S^3\backslash L$? Is it true that $X$ is link complement? I would appreciate any helpful tip on this