# Understanding noncyclic covers of knot complements

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.

• I have been told that Fox's work on 'spreads' and Montesinos "Branched coverings after Fox" have to do with the noncyclic version of branched covers. – Kyle Miller Nov 30 '18 at 7:53

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