Kirby Problem 1.19

Definitions and mathematical statement

Let $Y=Y_1\#Y_2$ be a connected sum of closed oriented $3$-manifolds with $Y_i\not\cong S^3$. The separating sphere $S$ has a product neighborhood $S^2\times[-1,1]$. A sphere twist $\Phi:Y\to Y$ is the identity off this neighborhood and, within it, rotates $S^2$ along a loop representing the nonzero element of $\pi_1(\operatorname{SO}(3))\cong\mathbb Z/2$. The rotations equal the identity near the two ends, so this defines a diffeomorphism of $Y$. Assume that $\Phi$ is not isotopic to the identity.

Must there be a smooth embedded circle $K\subset Y$ such that $K$ and $\Phi(K)$ are not ambiently isotopic? Ambient isotopy means a continuous family of diffeomorphisms $h_t:Y\to Y$ with $h_0=\mathrm{id}$ and $h_1(K)=\Phi(K)$. The knot need not be null-homologous or null-homotopic.

The given diffeomorphism already identifies the two embedded pairs $(Y,K)$ and $(Y,\Phi(K))$. The issue is whether some knot detects that this particular mapping class is nontrivial, through failure of an identification isotopic to the identity.

Short English statement

Can a nontrivial twist about a connected-sum sphere always be detected by moving some knot to a different isotopy class?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2678 (KP-1.19), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.19. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.