W.
B. Raymond Lickorish: Every 3-manifold is the result of
surgery of the sphere along some link. Robion Kirby: It is well understood which pairs of links give the same 3 manifolds upon surgery. |

For example, appropriate surgery on "Hopf chains" yields Seifert Fibered Manifolds, and studying Hopf chains (in the appropriate manner) amounts to studying Seifert Fibered Manifolds.

A mathematical landscape by Zarko D. Mijajlovich