Kirby Problem 1.10

Definitions and mathematical statement

Let $L=K_1\cup\cdots\cup K_n\subset S^3$ be a smooth framed link, $n\geq1$. Zero framing on $K_i$ is its preferred longitude, characterized by linking number zero with $K_i$. Simultaneous zero-framed Dehn surgery replaces every tubular neighborhood with a solid torus whose meridian is glued to that longitude. Suppose that the resulting manifold is \[ S^3_0(L)\cong\mathop{\#}_{i=1}^n(S^1\times S^2). \] A framed handleslide replaces one component with a band sum with a parallel copy of another, using its specified framing and transporting the framing in the usual band sum. Framed isotopies and these slides preserve the surgery manifold.

The Generalized Property R Conjecture asks whether a finite sequence of such slides and isotopies always turns $L$ into the $n$-component zero-framed unlink. An unlink bounds $n$ pairwise disjoint embedded disks in $S^3$. The number of link components must stay $n$: adjoining auxiliary surgery components is not one of the permitted moves in this formulation. This specifies the unstabilized conjecture attached to the catalogue entry.

Short English statement

Can every zero-framed link giving a connected sum of $S^1\times S^2$ be slid to the zero-framed unlink?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2669 (KP-1.10), 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 1.10. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.