Kirby Problem 3.41

Definitions and mathematical statement

Let $K\subset S^3$ be a smooth knot and $X=S^3\setminus\operatorname{int}\nu K$ its exterior. Choose the meridian $\mu$ and the preferred longitude $\lambda$ on $\partial X$, with $\lambda$ null-homologous in $X$. For a reduced rational number $r=p/q$, $q>0$, write $S^3_r(K)$ for the oriented Dehn filling obtained by attaching a solid torus so that its meridian is identified with the slope $p\mu+q\lambda$. This fixes the surgery coefficient convention.

A positive cooriented contact structure on an oriented three-manifold is $\xi=\ker\alpha$ with $\alpha\wedge d\alpha>0$. It is tight when it has no embedded overtwisted disk, meaning a disk with Legendrian boundary whose contact framing agrees with the disk framing. The structure is allowed to depend on the surgery coefficient.

The high surgery conjecture asks whether \[ \forall K\subset S^3\quad\exists n_K\in\mathbb Z\quad \forall r\in\mathbb Q, \quad r\geq n_K\Longrightarrow S^3_r(K)\text{ admits a tight contact structure}. \] The threshold depends on $K$, but all sufficiently large rational slopes, not just integers or one subsequence, must qualify. No hyperbolicity or fiberedness assumption is imposed on the knot.

Short English statement

For every knot, do all sufficiently large positive rational surgeries admit tight contact structures?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2839 (KP-3.41), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.41. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.