Topological isotopy and finite type invariants

TL;DR

This paper investigates the relationship between finite type invariants and PL isotopy of links in $S^3$, establishing conditions under which these invariants can distinguish links and solving a longstanding question posed by Rolfsen.

Contribution

It proves that finite type invariants separate PL links if and only if Rolfsen's problem is affirmatively answered, linking invariants to topological isotopy.

Findings

Finite type invariants distinguish certain PL links.

Extension of invariants to topological links is continuous and isotopy-invariant.

Finite type invariants relate to several conjectures in link theory.

Abstract

In 1974, D. Rolfsen asked: If two PL links in $S^3$ 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 $S^3$, then Rolfsen's problem has an affirmative solution. In fact, we show that finite type invariants separate PL links in $S^3$ if and only if Rolfsen's problem has an affirmative solution and certain 5 other (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 $C^0$-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.