Kirby Problem 1.65

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.

Put $\alpha=dz-y\,dx$ on $\mathbb R^3$, $\xi_{\mathrm{std}}=\ker\alpha$, and equip the symplectization $\mathbb R_t\times\mathbb R^3$ with $\omega=d\lambda$, where $\lambda=e^t\alpha$. A Legendrian link is tangent to $\xi_{\mathrm{std}}$. An exact Lagrangian cobordism $F$ from $\Lambda_-$ to $\Lambda_+$ is a properly embedded surface with $\omega|_F=0$ and $\lambda|_F=df$, cylindrical over these links outside a compact region; there are constants $c_-,c_+\in\mathbb R$ such that the primitive equals $c_-$ on the entire negative cylindrical end and $c_+$ on the entire positive cylindrical end. Each constant is common to all link components at its corresponding end. A filling has $\Lambda_-=\varnothing$. The Morse index below is the index of a critical point of the height $t|_F$. A decomposable cobordism is a composition of traces of Legendrian isotopies, births of standard Legendrian unknots, and elementary Lagrangian saddle (pinch) attachments. Lagrangian isotopy is an isotopy through Lagrangian surfaces, with the prescribed cylindrical ends. A Legendrian link is non-destabilizable if it is not the positive or negative stabilization of another Legendrian link.

Statement recorded in the supplied source.

Is every exact Lagrangian cobordism between Legendrian knots or links in $(\mathbb R^3,\xi_{\mathrm{std}})$ whose height has no index-2 critical points Lagrangian isotopic to a decomposable one? The source also asks whether every exact filling is so isotopic, and the broader variant with negative end any non-destabilizable Legendrian link, including the empty link. [S2] [S3]

Short English statement

Are exact Lagrangian cobordisms without maxima decomposable up to Lagrangian isotopy? Also ask the corresponding filling and non-destabilizable-negative-end variants.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2724 (KP-1.65). 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 1.65 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.
• S3 Tobias Ekholm, Ko Honda and Tamás Kálmán, Legendrian knots and exact Lagrangian cobordisms, arXiv:1212.1519v3 (2012), Definition 1.1 and footnote 1, p. 2; published in J. Eur. Math. Soc. 18 (2016), pp. 2627--2689. https://arxiv.org/pdf/1212.1519v3.