Topological isotopy and finite type invariants

TL;DR

The paper tackles Rolfsen's question by linking PL and topological isotopy to the separation power of finite type invariants. It develops the framework of $n$-quasi-isotopy and colored finite type invariants, showing that PL isotopy-invariant quantities extend to isotopy for topological links via continuity, and establishes equivalences with several conjectures (including Habiro–Kirk–Livingston-type statements). A central outcome is that, if finite type invariants separate links in $S^3$, then topological isotopy implies PL isotopy for links, effectively solving Rolfsen's problem under this hypothesis; the paper also provides concrete colored-type invariants, such as coefficients of the reduced Conway polynomial, that extend continuously to topological links. These results provide a concrete program to approach long-standing questions about realizing topological knots/links by PL representatives and about the power of finite type invariants in distinguishing link types.

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.

Figures (7)

• Figure 1: The $n$th Milnor link for $n=4$.
• Figure 2: The only singular link in a generic homotopy from $M_n$ to the unlink is an $(n-1)$-quasi-embedding $(n=4)$. Moreover, it is a strong $(n-1)$-quasi-embedding, using that all the $P_i$ are collapsible, and hence their regular neighborhoods are balls.
• Figure 3: The $n$th Whitehead link for $n=3$.
• Figure 4: The only singular link in a generic homotopy from $W_n$ to the unlink is an $(n-1)$-quasi-embedding $(n=3)$.
• Figure 5: The $C_n$-move for $n=4$.

Theorems & Definitions (118)

• Theorem A
• Theorem 1.1
• Theorem B
• Corollary 1.2
• Example 1.3
• Remark 1.4
• Remark 1.5
• Example 1.6
• Theorem C
• Corollary 1.7