Kirby Problem 1.70

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.

An oriented link $L\subset(S^3,\xi_{\mathrm{std}})$ is positively transverse when its oriented tangent is positively transverse to the standard contact planes. For a Seifert surface $\Sigma\subset S^3$, the self-linking $\operatorname{sl}_\Sigma(L)$ is the linking with a contact push-off defined by a nonzero contact-plane section over $\Sigma$. A Seifert surface is a compact oriented embedded spanning surface without closed components. A quasipositive braid is a product of conjugates of positive braid generators. A strongly quasipositive braid is a product of positive embedded band generators \[\sigma_{i,j}=(\sigma_i\cdots\sigma_{j-2})\sigma_{j-1}(\sigma_i\cdots\sigma_{j-2})^{-1},\qquad i<j.\] Their closures define quasipositive and strongly quasipositive link types respectively. A quasipositive Seifert surface is the corresponding disk-and-positive-band surface for a strongly quasipositive braid, up to isotopy. For a smooth compact oriented properly embedded surface $F\subset B^4$ with boundary $L$ and no closed components, the boundary self-linking is denoted $\operatorname{sl}_F(L)$ using the boundary contact framing and the surface framing. Let $g_3(L)$ and $g_4(L)$ denote the least genus of connected oriented spanning surfaces in $S^3$ and of connected smooth oriented spanning surfaces in $B^4$, respectively. A ribbon surface in $B^4$ has a radial Morse function without index-2 critical points.

Statement recorded in the supplied source.

Ask both:

1. If a transverse $L\subset(S^3,\xi_{\mathrm{std}})$ bounds a smooth oriented properly embedded surface $F\subset B^4$ without closed components with $\operatorname{sl}_F(L)=-\chi(F)$, must it be quasipositive?
2. Is a link strongly quasipositive exactly when it is quasipositive and $g_3(L)=g_4(L)$?

The source further asks whether a ribbon surface realizing the equality in the first question must be properly isotopic to the intersection of a nonsingular complex curve in $\mathbb C^2$ with $B^4$. [S2]

Short English statement

Does sharpness of the slice-Bennequin bound imply quasipositivity? Does equality of Seifert and slice genera distinguish strong quasipositivity among quasipositive links? Also ask whether a ribbon surface attaining the bound is isotopic to a complex-curve piece.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2729 (KP-1.70). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, eds., K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 1.70 and its remarks; Chapters 1 and 2 for the stated conventions. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.