Topological isotopy and finite type invariants

In 1974, D. Rolfsen asked: If two PL links in S³ are isotopic (=homotopic through embeddings), then are they PL isotopic? We prove that they are PL isotopic to another pair of links which are indistinguishable from each other by finite type invariants. Thus if finite type invariants separate PL links in S³, then Rolfsen's problem has an affirmative solution. In fact, we show that finite type invariants separate PL links in S³ if and only if certain 6 rather diverse conjectures hold simultaneously. We also show that if v is a finite type invariant (or more generally a colored finite type invariant) of PL links, and v is invariant under PL isotopy, then v assumes the same value on all sufficiently close C⁰-approximations of any given topological link; moreover, the extension of v by continuity to topological links is an invariant of isotopy. Some specific invariants of this kind are discussed.