Extending knot isotopies to two-component links with linking number one

50 years ago D. Rolfsen posed the following problem: Is every knot isotopic to the unknot? Here a knot is an embedding (=injective continuous map) of $S^1$ in $S^3$; and an isotopy is a homotopy through embeddings. I have proved that Rolfsen's problem has a negative solution if the answer to the following question is positive. (For a sketch of the proof in Russian see https://www.youtube.com/watch?v=3myZM-mbw4M and https://www.youtube.com/watch?v=nsB24aa1hog.)

Question A. Does every isotopy of a knot extend to an isotopy of a 2-component link with linking number $1$?

Question A, which I asked explicitly at a 2005 conference in Siegen (Germany), was answered negatively by A. Zastrow (2022). He constructed an isotopy from the unknot to itself which does not extend to any link homotopy of a 2-component link with linking number 1. (A link homotopy is a homotopy which keeps distinct components disjoint.) This raises the question whether the link homotopy can be replaced by $n$-bordism for some $n$. An $n$-bordism is what one gets from a link homotopy $(K_t,Q_t)$ of a two-component link with linking number $1$, if the additional component $Q_t$ is allowed to be replaced by another one (at finitely many time instants $t$), as long as it represents the same conjugacy class in $G_t/G_t^{(n)}$. Here $G_t=\pi_1(S^3\setminus K_t)$, $G^{(0)}=G$, $G^{(n+1)}=[G^{(n)},G^{(n)}]$ and $G^{(n+0.5)}=[G^{(n-1)},G^{(n)}]$ for positive integer $n$. In this talk I plan to discuss Zastrow's example and the following two new results.

Theorem 1. Every isotopy of a knot extends to a $2$-bordism.

Theorem 2. A certain isotopy from the unknot to itself does not extend to any $2.5$-bordism.

Theorems 1 and 2 are interesting in view of the following result, whose proof will hopefully be discussed in subsequent talks.

Theorem 3. A certain knot (the so-called Bing sling) is not isotopic to the unknot by any isotopy which extends to a $2.5$-bordism.

In fact Theorems 2 and 3 can be improved by replacing $G_t/[G_t',G_t'']$ (which occurs in the definition of a $2.5$-bordism) with $G_t/[G_t',T]$, where $T$ is a certain subgroup intermediate between $G_t'$ and $G_t''$, namely, the preimage in $G_t'$ of the $\mathbb Z[\mathbb Z]$-torsion submodule of the knot module $G_t'/G_t''$. The proof of this addendum will be discussed if time permits.

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