Understanding noncyclic covers of knot complements

Question

Let $K$ be a knot in $S^3$. I'm familiar with the process of taking the $n$-fold cyclic cover $X_n(K)$ of the knot complement $X(K) = S^3 - K$ and I know that this cyclic cover can be completed to a manifold $\overline{X_n}(K)$ which is the $n$-fold cyclic branched covering of $S^3$ branched over $K$.

I am wondering why I have never seen this done for noncyclic covers of knot complements. What do these covers "look like"? Can they be completed to branched covers of $S^3$ branched over $K$ like in the cyclic case?

As a concrete example, I know the trefoil complements fundamental group surjects onto the symmetric group $S_3$, but I sure don't know how to picture the corresponding cover of the knot complement.

Answer 1

There are various things in the literature on branched coverings of $S^3$ over arbitrary knots and links.

Here are some examples:

• There is the theorem of Hilden and Montesinos saying that every closed, oriented 3-manifold is a branched cover over some link.
• There is a later theorem of Bill Thurston saying that there is a single link that does the job, i.e. there is a a universal link, having the property that every oriented 3-manifold is a branched cover over that link.