Kirby Problem 4.130

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 $X$ be a K3 surface with an elliptic fibration and let $T$ be a regular torus fiber, with the product framing $T\times D^2$. For a smooth knot $K\subset S^3$, let $E(K)=S^3\setminus\operatorname{int}\nu K$ be its exterior, with knot meridian $\mu_K$ and preferred (zero-framed) longitude $\lambda_K$ on its boundary. With fiber coordinate circles $a,b$ and normal meridian $\mu_T$ on $\partial(T\times D^2)$, longitudinal surgery uses the product gluing that identifies $\mu_K,\lambda_K$ with $a,b$ and the auxiliary circle $s$ in $S^1\times E(K)$ with $\mu_T$, with signs chosen for the oriented boundary gluing. Denote the resulting closed smooth manifold by \[ X_K^\lambda=(X\setminus\operatorname{int}(T\times D^2))\cup_\phi(S^1\times E(K)). \] This is the zero-framed knot surgery convention in the original construction. [S3] A closed oriented smooth four-manifold is reducible if it admits a smooth connected-sum decomposition $M_1\# M_2$ with neither $M_i$ a homotopy four-sphere, meaning neither is homotopy equivalent to $S^4$. [S4] It is completely decomposable if it is diffeomorphic to a connected sum of copies of $\mathbb{CP}^2$ and $\overline{\mathbb{CP}}^{2}$.

Statement recorded in the supplied source.

For every knot $K\subset S^3$, is the longitudinally surgered K3 manifold $X_K^\lambda$ reducible? Is it always completely decomposable? These are two separate conclusions about this gluing convention, which differs from ordinary Fintushel--Stern knot surgery. [S2]

Short English statement

Does longitudinal knot surgery on a regular fiber of a K3 surface always produce a connected sum with neither summand a homotopy four-sphere, and does it always split entirely into projective-plane summands?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3006 (KP-4.130). 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 4.130, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Clifford Henry Taubes, Some 4-manifold geometry from hyperbolic knots in $S^3$, arXiv:

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  1602.01687 (2016), §2(a), equation (2.1). https://arxiv.org/abs/1602.01687.
• S4 M. J. D. Hamilton and D. Kotschick, Minimality and irreducibility of symplectic four-manifolds, International Mathematics Research Notices (2006), Article ID 35032, §1.2, p. 2 (arXiv:math/0509204v2). https://arxiv.org/abs/math/0509204v2.