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