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I believe many such links can be constructed by looking at a foliation similar to the hopf fibration, but the wrapping leaves replaced with $(p,q)$ torus knots. However, I'm interested in particular in whether there are classes of examples that don't fall under that construction.

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I think this paper exactly answers your question http://projecteuclid.org/euclid.dmj/1077378799 .

But you are basically right, the links you get are fibres (possibly degenerate) in some (possibly singular) Seifert fibration of $S^3$ and pretty much it can only be fibred in the ways you mentioned.

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The book of David Eisenbud and Walter Neumann on Three-dimensional link theory and invariants of plane curve singularities (Annals of Mathematics Studies, vol. 110, Princeton University Press, Princeton, NJ, 1985) contains a discussion of Seifert links (Chapter II, Section 7). In particular, yes, there are many more Seifert links than $(p,q)$ torus knots or links, for instance, the "key-chain" links. For more details, see also the paper of Ryan Budney, JSJ-decompositions of knot and link complements in $S^3$, Enseign. Math. 52 (2006), no. 3-4, 319–359.

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