Kirby Problem 1.22

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Use the integral hat Heegaard Floer group $\widehat{\mathrm{HF}}(Y)$ and integral hat knot Floer group $\widehat{\mathrm{HFK}}(Y,K)$, with rank meaning the rank of the free abelian part. Knot Floer isomorphisms preserve Maslov and Alexander gradings. A rational homology sphere $Y$ is an L-space when $\operatorname{rank}\widehat{\mathrm{HF}}(Y)=|H_1(Y;\mathbb Z)|$. An L-space knot is a knot in $S^3$ having a positive integral surgery that is an L-space. A hyperbolic knot has a complete finite-volume hyperbolic complement. Write $\Sigma_2(K)$ for the two-fold cover of $S^3$ branched along $K$.

Statement recorded in the supplied source.

For every hyperbolic L-space knot $K$, is $\Sigma_2(K)$ not an L-space? [S1][S2]

Short English statement

The double branched cover of a hyperbolic L-space knot should fail to be an L-space.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2681 (KP-1.22). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, author preliminary edition (2026), Problem 1.22 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.