1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Kyle Miller Nov 30 '18 at 7:53
1
$\begingroup$

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.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.