Kirby Problem 1.63

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.

In standard contact $\mathbb R^3$, $\xi_{\mathrm{std}}=\ker(dz-y\,dx)$. A Legendrian knot is tangent to $\xi_{\mathrm{std}}$. Its Thurston--Bennequin number $\operatorname{tb}(\Lambda)$ is the linking number with a contact-framing push-off. Positive and negative stabilization insert a zigzag in the front projection and decrease $\operatorname{tb}$ by one. A representative is non-destabilizable if it is not Legendrian isotopic to either stabilization of another representative. A smooth knot type $K$ is fixed throughout the quantifier below.

Statement recorded in the supplied source.

Does there exist a single smooth knot type $K$ such that for every $N\in\mathbb N$ there is a non-destabilizable Legendrian representative $\Lambda_N$ of $K$ satisfying $\operatorname{tb}(\Lambda_N)\le -N$? [S2]

Short English statement

Can one fixed knot type have non-destabilizable Legendrian representatives with arbitrarily negative Thurston--Bennequin number?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2722 (KP-1.63). 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 1.63 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.