# On the geometrization of double branched covers

I recently got into Lickorish's paper Prime knots and tangles 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?

• In particular I would like to know a sufficient condition for hyperbolicity... – Antonio Alfieri Dec 16 '16 at 21:04
• Your question can be recast in terms of orbifolds. In particular, make the link into a $\pi$-orbifold. Then the analogue of the geometrization theorem is the orbifold theorem. For background on this and a statement, see ams.org/mathscinet-getitem?mr=1778789 – Ian Agol Dec 17 '16 at 8:20
• Thank you. I think you already gave this reference to me, I think this time I'm gonna read this! – Antonio Alfieri Jan 11 '17 at 19:58
• Another (much earlier) reference for this material is the Bonahon and Siebenmann paper: www-bcf.usc.edu/~fbonahon/Research/Preprints/BonSieb.pdf – Ryan Budney Jan 12 '17 at 3:50

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, 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, 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.

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 $M_L$.

• Thank you for your answer. I can't understand one thing: is condition 1) equivalent to say that the link $L$ is non-split and prime? – Antonio Alfieri Jan 11 '17 at 19:59
• Yes, you are correct. I didn't include a discussion split links. I will edit that section to correct for this omission. – Neil Hoffman Jan 11 '17 at 21:04
• Let me summarise what happens in the case of knots. According to Thurston a knot is either a satellite of a non-trivial knot, a torus knot or an hyperbolic knot. So if a knot is not a satellite (it is prime) and it is either a torus knot or an hyperbolic knot. In the first case its double cover $M_L$ is a Seifert manifold, while in the second one if there are no essential Conway spheres I can conclude that $M_L$ is hyperbolic. Am I right? – Antonio Alfieri Jan 11 '17 at 23:10
• No, the double branched cover is a closed manifold making the problem more subtle. There are hyperbolic knots and links with non-hyperbolic double branched covers. For example: a two bridge (knot or) link has double branched cover a lens space; and a Montesinos (knot or) link has double branched cover a Seifert fibered space. Except for (2,n) torus links a two bridge link complement is hyperbolic. For Montesinos links of length 3, there are no essential Conway spheres while only a small subset are not hyperbolic. – Neil Hoffman Jan 11 '17 at 23:22
• There are also links have double branched cover a Solv manifold, I believe these are generically hyperbolic as well. – Neil Hoffman Jan 11 '17 at 23:22