May I ask a question: Since a thickened torus is the completement of a tubular neighbourhood of a Hopf link in S^3, is an isotopy class of a link in thickened torus equivalent to the isotopy class of the disjoint union of this link with the Hopf link in S^3? Are they equivalent then, studying links in thickened torus and links in S^3 which has a hopf link as sublink. Thanks!

mapping torus