Atoroidal, irreducible 3-manifolds and 3-fold branched coverings of \(S^ 3\) (Q796867)

In the midst of 70's it was shown independently by U. Hirsch, H. Hilden, and J. Montesinos that every closed, connected, orientable 3-manifold M is a 3-fold irregular branched cover of the \(S^ 3\) branched over a knot. Moreover, the knot always may be chosen to be fibered. The author refines these results in the case of irreducible and atoroidal M. Namely, in this case at least one of the following is true: (a) M is a 3-fold (cyclic or irregular) branched cover of \(S^ 3\) branched over a simple, fibered knot; (b) M is a 2-fold (cyclic) branched cover of \(S^ 3\) or \(S^ 1\times S^ 2\) or a lens space branched over a simple link.