# 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 '17 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 '17 at 18:14