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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, those are almost the only examples! 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, an annulus with one orbifold point, or a pair of pants.

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.

The correct statement is a classification of knots. See Corollary 2.5 of Thurston's article "Three dimensional manifolds, kleinian groups, and hyperbolic geometry".

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, those are almost the only examples! 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, an annulus with one orbifold point, or a pair of pants.