Kirby Problem 4.24

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 $K\subset S^3=\partial B^4$ be a smooth knot with an integer framing $n$. Let $W(K,n)$ be its two-handle trace \[ W(K,n)=B^4\cup_{\nu K}(D^2\times D^2), \] where the attaching region is $\partial D^2\times D^2$ and the framing is measured relative to the Seifert longitude of $K$. The cocore is the disk $C=\{0\}\times D^2$ in the attached handle. Choose a meridian $U$ of $K$ in the attaching-boundary diagram and regard it as a curve in $\partial W(K,n)$. A carving disk is a smooth proper embedded disk $D\subset W(K,n)$ with $\partial D=U$. Arrange the interiors of $D$ and $C$ to be transverse and their boundaries disjoint. Their algebraic intersection is the signed count \[ D\cdot C=\sum_{p\in\operatorname{int}D\cap\operatorname{int}C}\epsilon_p, \qquad\epsilon_p\in\{1,-1\}, \] using chosen disk orientations and the orientation of the trace. Its vanishing does not require geometrically disjoint interiors. Remove an open normal disk-bundle neighborhood of $D$ and smooth corners; denote the resulting manifold by $B_D$.

Statement recorded in the supplied source.

Can this carving construction produce a nonstandard four-ball? In the framed-knot setting above, is there a carving disk $D$ such that \[ D\cdot C=0\qquad\text{and}\qquad B_D\not\cong_{\mathrm{diff}}B^4? \] The disk may enter the two-handle; it is not required to lie in the original zero-handle. The intersection condition is the geometric signed-count condition with the specified cocore. [S2]

Short English statement

Can carving a disk from a framed knot trace give a manifold not diffeomorphic to the four-ball when the disk has zero algebraic intersection with the handle cocore?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2900 (KP-4.24). 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 4.24 and Remarks, PDF page 210. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.