Kirby Problem 1.21

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. Let $g(K)$ be the least genus of a Seifert surface. For coprime nonzero integers $p,q$ with $|p|,|q|>1$, $T_{p,q}$ is the torus knot of those parameters; changing one sign gives its mirror. No positivity restriction is imposed on the torus knot in part(c)(ii). An iterated torus knot is obtained from a torus knot by successive cablings. Normalize $\Delta_K(t)$ up to $\pm t^k$.

Statement recorded in the supplied source.

(a) For every fixed $g\geq1$, are there finitely many L-space knots of genus $g$? (b) Characterize all knots $L$ for which only finitely many isotopy classes $K$ have $\widehat{\mathrm{HFK}}(S^3,K)\cong\widehat{\mathrm{HFK}}(S^3,L)$. (c) Are all of the following true: (i) the knot Floer homology of $T_{2g+1,2}$ detects that knot; (ii) a knot with the knot Floer homology of any $T_{p,q}$ is an iterated torus knot; (iii) an L-space knot all of whose Alexander roots have modulus one is an iterated torus knot? [S1][S2]

Short English statement

Ask for finiteness at fixed genus or fixed knot Floer homology, and for the three specified detection statements involving torus knots and Alexander roots.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2680 (KP-1.21). 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 1.21 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.