Kirby Problem 1.9

Definitions and mathematical statement

For a knot $K\subset S^3$, Dehn surgery removes its open tubular neighborhood and glues in a solid torus, with the meridian of the added torus identified with a specified essential simple curve on the boundary. A slope $p/q$ means the curve $p\mu+q\lambda$ for coprime integers $p,q$, where $\mu$ is the meridian and $\lambda$ the preferred longitude. Surgery is nontrivial when its slope is not the meridian. A lens space is a closed $3$-manifold obtained by gluing two solid tori; write $L(p,q)$ for those with finite cyclic fundamental group of order $|p|$.

A knot is doubly primitive, or a Berge knot, if it can be placed on a genus-two Heegaard surface $S^3=H_1\cup_\Sigma H_2$ and there are compressing disks $D_i\subset H_i$ such that $|\partial D_i\cap K|=1$ for $i=1,2$. The conjecture is that every knot admitting a nontrivial surgery to a lens space is doubly primitive. The conclusion concerns the knot itself, including its embedding in $S^3$, rather than merely the homeomorphism type of the resulting lens space.

Short English statement

Must every knot with a lens-space surgery lie primitively on both sides of a genus-two Heegaard surface?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2668 (KP-1.9), 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 1.9. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.