Kirby Problem 1.66

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 $\Lambda_-=\varnothing$.

Statement recorded in the supplied source.

On Legendrian isotopy classes of links, set $[\Lambda_1]\preceq[\Lambda_2]$ when there exists an exact Lagrangian cobordism from $\Lambda_1$ to $\Lambda_2$. Is this relation a partial order? In particular, if exact cobordisms exist in both directions, must $\Lambda_1$ and $\Lambda_2$ be Legendrian isotopic? [S2] [S3]

Short English statement

Does existence of an exact Lagrangian cobordism define a partial order on Legendrian link types?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2725 (KP-1.66). 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 1.66 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.