Uniqueness of branched covers of knots

Question

Let $K \subset S^3$ be a knot and $q$ be a positive integer. I just read in Rolfsen's Knots and Links the construction for a 3-manifold $M$ together with a $q$-fold branched cover $f: M \to S^3$ where the branching occurs over $K$. I am curious to what extent $M$ is uniquely characterized by this. Namely, is it true that given any other 3-manifold $N$ with a $q$-fold branched cover $g : N \to S^3$ branched over $K$ there is a homeomorphism $\phi : M \to N$ with $g \circ \phi = f$.

Answer 1

You have to decide what's your definition of a $q$-fold branched covering over $S^3$ branched over a knot $K$. The standard definition is that you take $E(K)=S^3 - N(K)$, observe that $H_1(E(K))\cong {\mathbb Z}$, compose the (Hurewicz) homomorphism with the quotient ${\mathbb Z}\to {\mathbb Z}_q$ (reduction mod $q$): $$ \pi_1(E(K))\to H_1(E(K))\to {\mathbb Z}\to {\mathbb Z}_q. $$
(Everything is canonical so far, at least the kernel of the composition, which is all what matters.) This composition corresponds (by the covering theory) to a unique $q$-fold (necessarily regular) covering $p: M'\to E(K)$ with the covering group $\cong {\mathbb Z}_q$. There is a unique (up to isotopy) way to attach the solid torus $D\times S^1$ to the boundary of $M'$, so that $p$ has an extension to a branched covering $p: M=M_q\to S^3$. Uniqueness here is a nice exercise in understanding covering maps $T^2\to T^2$ and their extensions to branched covers $D\times S^1\to D\times S^1$. This $M_q$ is unique up to a homeomorphism (and even the map $M_q\to S^3$ is unique in in the sense of isomorphic branched coverings).

Now, you can ask what else one can do along these lines if a finite $q$-fold covering $L'\to E(K)$ is not required to be regular or is not required to be cyclic. With some work, one can get examples which are not of the type as described above and which still extend to $q$-fold branched coverings $L\to S^3$ (branched over $K$), where $L$ is some closed 3-manifold, not homeomorphic to $M_q$ as above. For instance, I can do this for torus knots and, probably for hyperbolic knots (for some choice of $q$). I am not sure, however, that this is what you want.