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. 



Classifying links with essential annuli in the complement as torus links
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](https://projecteuclid.org/euclid.bams/1183548782)". 


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.


