Abstract:
In 1974, D. Rolfsen asked: if two PL links $L_0$, $L_1$ in $S^3$ are isotopic, then are they PL isotopic? (Here "PL" means "piecewise linear" and "isotopy" means "homotopy through embeddings".) We show that they are PL isotopic to links $L_0'$, $L_1'$ 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 the first talk on this subject I plan to discuss a weaker result, which I proved 20 years ago [arXiv:math/0312007].
It easily implies (using the Kontsevich integral) that the links $L_0'$ and $L_1'$ are indistinguishable by rational finite type invariants.
A key lemma is that given a topological link $L$, for each $n$ there exists an $\epsilon>0$ such that if two PL links are $\epsilon$-close to $L$, then they are indistinguishable by those type $n$ invariants that are well-defined up to PL isotopy. This lemma (or rather its version for finite type invariants of colored links) is also used in my recent results on another problem of Rolfsen ("Is every knot isotopic to the unknot?"). The proof of this lemma is based on the notion of $n$-quasi-isotopy. This notion was originally motivated by the Isotopic Realization Problem for maps of a $1$-manifold in $S^3$ [arXiv:math/0103113]. This problem is still wide open and I also plan to discuss it briefly.
To complete the proof that the links $L_0'$ and $L_1'$ are indistinguishable by all (not only rational) finite type invariants I'm using the clasper theory of Gusarov and Habiro. This new argument is a little bit technical, and if it is to be discussed, it will take an additional seminar.
Connect to Zoom: https://zoom.us/j/92456590953 Access code: the Euler characteristic of the wedge of two circles
(the password is not the specified phrase but the number that it determines)
