Kirby Problem 1.69

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. In a general contact 3-manifold $(Y,\xi)$ the relative self-linking is defined with the same spanning-surface section convention. A Legendrian ribbon is an oriented surface retracting onto a Legendrian graph, tangent to $\xi$ along the graph and transverse to $\xi$ elsewhere, with positively transverse boundary.

Statement recorded in the supplied source.

If a transverse link $L\subset(S^3,\xi_{\mathrm{std}})$ admits a Seifert surface $\Sigma$ with $\operatorname{sl}_\Sigma(L)=-\chi(\Sigma)$, must $L$ be strongly quasipositive? Must this particular surface $\Sigma$ be isotopic to a quasipositive Seifert surface? More generally, for a transverse link in a contact 3-manifold $(Y,\xi)$ satisfying $\operatorname{sl}_{\xi,\Sigma}(L)=-\chi(\Sigma)$, must $\Sigma$ be isotopic to a Legendrian ribbon? [S2]

Short English statement

Does sharpness of the Seifert-surface Bennequin bound imply strong quasipositivity, including quasipositivity of the surface itself? Ask the analogous Legendrian-ribbon question in general contact 3-manifolds.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2728 (KP-1.69). 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 1.69 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.