I recently got into Lickorish's paper [Prime knots and tangles](http://www.ams.org/journals/tran/1981-267-01/S0002-9947-1981-0621991-2/S0002-9947-1981-0621991-2.pdf) and a question, which I didn't have the first time I read it, naturally emerged. 


The Thurston-Perelman Geometrization Theorem asserts that given a compact, closed, orientable three-manifold $M$, either:


1) $M$ contains an essentrial sphere, or


2) $M$ contains an incompressible torus, or


3) $M$ is Seifert fibered, or 


4) $M$ is hyperbolic. 


Given a link $L \subset S^3$ we can form a closed three-manifold $M\_L$ by considering the double-branched cover of $S^3$ over $L$. 



> 
> **QUESTION.** How do conditions 1-4, regarding the manifold $M\_L$, translate to the topology of the link $L$? 
> 
> 
> 


Following Lickorish's paper it is easy to find some sufficient conditions based on the possibility to recognize certain tangle configurations in a diagram of $L$. What about "if and only if" conditions? 



On the geometrization of double branched covers
As Ian Agol mentioned in his comment, the OP's question can be thought of in terms of the Orbifold Theorem. There are two (contemporaneous and) independent proofs of the Orbifold Theorem: 


*Daryl Cooper, Craig D. Hodgson, and Steven P. Kerckhoff*, MR 1778789 [**Three-dimensional orbifolds and cone-manifolds**](http://www.ams.org/mathscinet-getitem?mr=1778789), ISBN: 4-931469-05-1. (The relevant background chapter is available here <https://projecteuclid.org/download/pdf_1/euclid.msjm/1389985818>) 


*Michel Boileau, Bernhard Leeb, and Joan Porti*, MR 2178962 [**Geometrization of 3-dimensional orbifolds**](http://dx.doi.org/10.4007/annals.2005.162.195), *Ann. of Math. (2)* **162** (2005), no. 1, 195--290. 


The background section of either paper seems to answer the OP's question. However, here is my attempt to present the necessary details:


Consider the warm up case, if the double branched cover $M\_L$ of a link (embedded in $S^3$) $L$ is geometric (in the sense of admitting one the eight Thurston geometries), then $M$ double covers a geometric orbifold $Q$ which has underlying space $S^3$ and the singular locus has cone angle $\pi$ and is isotopic to $L$ (as an embedded link in $S^3$). This covers cases 3) and 4) of the OP's question. Good background for Case 3) is the work of Montesinos, especially his book [**Classical Tessellations and Three-Manifolds**](http://rads.stackoverflow.com/amzn/click/3540152911).


Otherwise, there is an obstruction to geometrization. For closed, orientable 3-manifolds, the obstructions to geometrization are embedded $S^2$ which do not bound $B^3$'s and incompressible $T^2$. As a result of the two papers cited above, the obstructions to (closed) orbifold geometrization are more or less analogous. 


First, we need to discuss *bad* 2-orbifolds of $Q$, i.e. 2-orbifolds which have underlying space $S^2$ transversely intersect the singular locus $Q$ in exactly one point. This is equivalent to such a 2-orbifold not having a manifold cover (see for example Chapter 13 of Thurston's notes). 


For the rest of the argument we will restrict to the case of an orbifold $Q$ with underlying space $S^3$ and having a singular locus a link $L$ labeled only by cone angle $\pi$. 


**EDIT: the paragraph below originally forgot to consider split links. Thanks to the OP for pointing this out.**


1) In this context $Q$ is *reducible* if it contains a) an embedded bad sub-2-orbifold, b) an embedded 2-sphere which does not bound a 3-ball, or c) an embedded 2-fold orientable quotient of 2-sphere which does not bound the quotient obtained by rotating a 3-ball along an unknotted arc. This perspective more naturally generalizes to arguments of the above two papers. In the case of the question, we see that $Q$ will contain an embedded 2-sphere which does not bound a 3-ball if and only if $L$ is split and $Q$ will contain an embedded 2-fold orientable quotient of 2-sphere as in case c) if and only if $L$ is not prime. Furthermore, if $L$ is split or non-prime link, then each piece of a prime decomposition of the link $L$ will correspond to a prime decomposition of the orbifold $Q$ and a prime decomposition of $M\_L$. (The general case involves consideration of more 2-orbifold quotients of $S^2$.) So the orbifold version of reducible involves both contains an essential 2-sphere quotient and bad 2-orbifolds. 


2) The final case to consider is if $Q$ is irreducible but not geometric. In this case, $Q$ is the orbifold equivalent of being toroidal. The second reference uses the term toric for this case, but we will follow the first reference an say that an orbifold $Q$ is *orbifold-atoroidal*. Before the precise definition, notice that a torus $T^2$ has a unique orientable 2-fold quotient $S^2(2,2,2,2)$ (a 2-sphere with four cone points of order 2, sometimes called a pillow case). A *rational tangle* in a 3-ball can be obtained by taking two embedded unknotted arcs in 3-ball and then twisting along their boundaries. Assuming $Q$ irreducible, an embedded $T^2$ is *incompressible* if it does not bound solid torus to one side. In an irreducible 3-orbifold, an embedded $S^2(2,2,2,2)$ is *incompressible* if it does not bound a rational tangle to one side. If $Q$ contains an incompressible $T^2$ or $S^2(2,2,2,2)$, then we say $Q$ is *orbifold-toroidal* and otherwise $Q$ is *orbifold-atoroidal*. (The more general definition of orbifold-atoroidal involves considering more general quotients of $T^2$.) For the specific instance above, $Q$ is *orbifold-toroidal* if $L$ is a satellite link or if $L$ contains an essential Conway sphere, which is really just a concise restatement of the above paragraph. The decomposition of $L$ along satellite tori and essential Conway spheres leads to a JSJ-decomposition of $M\_L$. 


