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.



Is it known which links have Seifert fibered complements?
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.


