# Classifying links with essential annuli in the complement as torus links

I've seen it claimed in several places, though never with a detailed proof, that every non-split link is either a hyperbolic, satellite, or torus link (see for example pg. 95 of Cromwell's "Knots and Links").

I understand from Thurston's work that if the complement of a non-split link is atoroidal and anannular, then it is hyperbolic. Because the presence of an essential torus implies the link is a satellite, I see how Thurston's result implies the first two thirds of the above classification. What I'm having trouble understanding is how the presence of an essential annulus in the complement of a non-satellite link $L$ implies that $L$ is a torus link.

Consider, for example, the standard torus $T$ embedded in $S^3$, which separates $S^3$ into two solid tori $V_1$ and $V_2$. Let $L_1$ be a nontrivial $(p,q)$-torus knot embedded on $T$, let $L_2$ be the core of $V_2$, and let $L = L_1 \cup L_2$

Then $L$ is non-split, its complement is atoirodal, and hence $L$ is not a satellite link (I don't have a proof of this statement however, so if I am wrong please let me know). Its complement does however contain an essential annulus coming from the torus knot $L_1$, and hence is not hyperbolic. Though it cannot be a torus link, because $L_1$ is knotted, while $L_2$ is unknotted. I'm thus having a hard time seeing where this example falls in the above classification.

Can the above classification of links (as hyperbolic, satellite, or torus) be made precise in a way that handles such situations? Are there any references which classify the types of essential annuli that can show up in the complement of a link?

Thank you in advance for any help.

• There may be some confusion in the terminology `satellite' as applied to links. For instance, I would have thought that the link you describe is in fact a satellite of the Hopf link. Dec 27, 2017 at 21:21
• I hadn't thought of it that way. I was taking the definition of a satellite link to be one with an incompressible, non-boundary parallel torus in its complement (which seemed to me to be the natural generalization from the knot situation). I guess that is an additional point that needs clarifying when dealing with this classification. Dec 28, 2017 at 18:14

## 1 Answer

Thurston only claims a classification of knots, not of links. See Corollary 2.5 of Thurston's article "Three dimensional manifolds, kleinian groups, and hyperbolic geometry".

Cromwell's statement is incorrect, as your examples show. However, your examples are almost the only ones that he does not cover. The correct statement for links is as follows: Every link complement in the three-sphere is either hyperbolic, toroidal (that is, satellite), or Seifert fibered. In the last case, we can obtain a lot of control over the base orbifold; it is either a disk with two orbifold points (giving a torus knot), an annulus with one orbifold point (your examples), or a pair of pants.

• Do you happen to know where I can find a reference or proof for the correct statement for links here? Oct 21, 2021 at 14:48
• You can find a nice exposition of Seifert fibered spaces in Hatcher's notes on three-manifolds: pi.math.cornell.edu/~hatcher/3M/3Mdownloads.html . I spent some time searching for an exact reference for the claim (for links) but failed to find one. It follows from Thurston's hyperbolisation theorem for Haken manifolds, Scott's version of the torus theorem, the Seifert fibered space conjecture, and the classification of Seifert fibered spaces. Oct 22, 2021 at 13:57
• Oh man, I have a lot to learn. Thank you very much! Oct 22, 2021 at 13:58