Kirby Problem 4.99

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.

Let $(X,\omega)$ be a closed symplectic four-manifold and let $T\subset X$ be a smoothly embedded symplectic torus with $T\cdot T=0$. Symplectic means that $\omega$ is a closed nondegenerate two-form and $\omega|_T$ is an area form.

For a smooth knot $K\subset S^3$, let $M_K$ be zero-framed Dehn surgery on $K$, using the longitude bounding a Seifert surface in its exterior. Let $m$ be a meridian of $K$, regarded as a framed circle in $M_K$, and set $T_m=m\times S^1\subset M_K\times S^1$. If $T\subset X$ is a torus with trivial normal bundle, Fintushel--Stern knot surgery is \[ X_K=\left(X\setminus\operatorname{int}(T\times D^2)\right) \cup_\varphi\left((M_K\times S^1)\setminus\operatorname{int}(T_m\times D^2)\right). \] Here $\varphi$ is the usual orientation-reversing identification of the boundary three-tori, preserving the normal-circle class $[\{\mathrm{pt}\}\times\partial D^2]$ in the fiber-sum convention. Choose the framing and boundary identification as part of the knot-surgery data. Equivalently, the replacement piece is $S^1$ times the knot exterior, with its Seifert-longitude class identified with the normal circle of $T$. [S3]

A knot $K$ is fibered if its exterior $S^3\setminus\operatorname{int}\nu K$ is a smooth fiber bundle over $S^1$ with fiber a compact connected Seifert surface, and the fibration sends a meridian to a degree-one loop in the base.

Statement recorded in the supplied source.

For such $(X,T)$ and a knot $K$, if the knot-surgery manifold $X_K$ admits a symplectic structure, must $K$ be fibered? \[ X_K\text{ admits a symplectic form with its inherited orientation}\quad\Longrightarrow\quad K\text{ is fibered}. \] The orientation of $X_K$ is inherited from the orientation of $(X,\omega)$ and the fiber-sum gluing; the resulting symplectic form must induce that orientation. It need not agree with $\omega$ on the unchanged region. No simple-connectivity, complement-group, or cusp-embedding hypothesis is imposed. [S2]

Short English statement

If knot surgery on a symplectic torus yields a manifold that admits any symplectic structure, must the surgery knot be fibered?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2975 (KP-4.99). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3—A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), author version, Problem 4.99 and Remarks, PDF page 272. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Ronald Fintushel and Ronald J. Stern, Knots, Links, and 4-Manifolds, Inventiones Mathematicae 134 (1998), 363--400; arXiv:dg-ga/9612014v2, Introduction, pages 3--4. https://arxiv.org/pdf/dg-ga/9612014v2.