Is there a convenient place in the literature where the geometric decompositions of cyclic branched covers of $S^3$ branched over "small" knots is recorded?
By small knots, I'm referring to things like torus knots, 2-bridge knots and the Rolfsen knot table.
For 2-sheeted coverings the Bonahon-Siebenman paper is a canonical reference, and for 2-bridge knots there's the standard connection with lens spaces. But I'd like to know about $n$-sheeted coverings for larger $n$ (3,4,5,6 is a little larger) and with standard small knots.
I imagine there are a few papers out there where much of this is worked out, perhaps a Montesinos paper? But I'm not having much luck finding this kind of information.
In principle I suppose I could generate tables using SnapPy and Regina but I imagine there must be work of this kind that predates such technology.
Thanks for any insights.
Best Answer
Such a classification follows from the orbifold theorem. The point is that the $n$-fold cyclic branched cover over a knot is an $n$-fold orbifold cover over the orbifold whose singular locus is the knot with a cone angle of $2\pi/n$ (in fact, one only needs the theorem for orbifolds of "cyclic type"). The orbifold theorem states that a good orbifold admits a geometric decomposition. An orbifold whose underlying singular locus is a knot in $S^3$ is automatically good, so satisfies the hypothesis of the orbifold theorem (there can be no teardrop or bad football suborbifolds for straightforward reasons).
There is a version of the prime decomposition for good 3-orbifolds. In the case at hand, the only spherical orbifold that may appear is a football, which corresponds to a connect sum. So the connect sum decomposition of the knot will correspond to the prime decomposition of the cyclic orbifolds.
Thus, assume the knot is prime. Now, the only toroidal orbifolds with the same order singular points are the $2222$ pillowcase and the $333$ turnover. Clearly the turnover cannot occur for a knot orbifold, and thus one only sees the $2222$ pillowcases for the cyclic 2-orbifold. This toroidal decomposition then corresponds to the decomposition along Conway spheres discussed by Bonahon-Siebenmann. Thus, one may reduce to considering tangles with no Conway spheres. The Seifert-fibered pieces will correspond to Montesinos tangles, where one inserts rational tangles in a standard picture.
There are a few exceptional cases, corresponding to the other geometric structures, which were classified by Dunbar. One can browse his tables and discover that the only other case is a Euclidean orbifold which is the 3-fold cover of the figure 8 knot complement.