Kirby Problem 4.8

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 closed simply connected smooth four-manifold, and let $T\subset X$ be a smoothly embedded torus with self-intersection $T\cdot T=0$ and simply connected complement $\pi_1(X\setminus T)=1$.

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 is prime if it is nontrivial and cannot be expressed as a connected sum of two nontrivial knots. A mirror is the image under an orientation-reversing diffeomorphism of $S^3$; isotopy of knots means ambient isotopy in $S^3$.

Statement recorded in the supplied source.

For prime knots $K_1,K_2$, does \[ X_{K_1}\cong_{\mathrm{diff}}X_{K_2} \quad\Longrightarrow\quad K_1\text{ is isotopic to }K_2\text{ or to its mirror} \] hold for the above knot-surgery construction? The target is recognition of the knot up to mirroring. The hypotheses are not restricted to cusp-embedded tori, nonzero gauge invariants, or a particular starting manifold. [S2]

Short English statement

If knot surgery with two prime knots produces diffeomorphic four-manifolds, must the knots agree up to isotopy and mirror image?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2884 (KP-4.8). 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 4.8 and Remarks, PDF page 197. 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.