Kirby Problem 1.20

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.

Let $Y$ be a closed oriented connected $3$-manifold. 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 rationally null-homologous knot is Floer minimal if the two total ranks agree. A null-homologous knot has $[K]=0$ in $H_1(Y;\mathbb Z)$. An unknot bounds an embedded disk. The Borromean knot $B_g\subset\#^{2g}(S^1\times S^2)$ is the boundary of the page in the open book with once-punctured genus-$g$ page and identity monodromy. For an integral class in $H_2(Y;\mathbb Z)$, its Thurston norm is the minimum of the sum of $\max(0,-\chi)$ over components of oriented embedded representing surfaces. Extend this function by homogeneity to rational classes and continuously to $H_2(Y;\mathbb R)$. A simple knot in a lens space is represented by the standard doubly pointed genus-one Heegaard diagram with its minimal number of intersection points.

Statement recorded in the supplied source.

(a) Are there null-homologous Floer minimal knots with irreducible exteriors besides the knots $B_g$? (b) If $b_1(Y)<2$, is every null-homologous Floer minimal knot an unknot? (c) If the Thurston norm of $Y$ is not identically zero, must $Y$ contain no such knot with irreducible exterior? (d) Is every Floer minimal knot in a lens space simple? [S1][S2]

Short English statement

Classify knots attaining the minimal total knot Floer rank, with separate conjectures involving the first Betti number, Thurston norm and lens spaces.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2679 (KP-1.20). 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 1.20 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.