Kirby Problem 4.66

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.

For a compact oriented smooth four-manifold $X$ and a framed oriented link $L\subset\partial X$, let $\mathcal S_0^2(X;L)$ be the Khovanov--Rozansky $\mathfrak{gl}_2$ skein lasagna module with integral coefficients. Its generators are framed oriented surfaces in $X$ with finitely many interior four-balls removed, having outer boundary $L$, together with a label in the corresponding $\mathfrak{gl}_2$ link homology for the link on each input sphere. Impose multilinearity in those labels and the local relations given by the functorial link-homology maps of surface cobordisms inside the input balls. The standard module records homological grading, quantum grading, and the relative surface homology class in $H_2(X,L;\mathbb Z)$. This is the construction of Section 2.1 in Source S3.

When $X$ is closed, $L=\varnothing$ and write $\mathcal S_0^2(X)$. A closed exotic pair means closed oriented smooth four-manifolds $X_0,X_1$ that are homeomorphic but not diffeomorphic. Compare homology-class gradings using the isomorphism induced by a homeomorphism; the other two gradings remain fixed.

Statement recorded in the supplied source.

Can this skein lasagna module distinguish a closed exotic pair? Does there exist a pair $X_0,X_1$ as above for which the graded modules $\mathcal S_0^2(X_0)$ and $\mathcal S_0^2(X_1)$ cannot be identified under the homology correspondence of any homeomorphism between $X_0$ and $X_1$? The four-manifolds must be closed, rather than the knot traces with boundary already treated in the source. [S2][S3]

Short English statement

Can the Khovanov skein lasagna module distinguish two homeomorphic closed smooth four-manifolds with different smooth structures?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2942 (KP-4.66). 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 4.66 and Remarks, PDF page 243. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Qiuyu Ren and Michael Willis, Khovanov homology and exotic 4-manifolds, arXiv:2402.10452v3 (2025), Introduction and Section 2.1, pages 9--10. https://arxiv.org/pdf/2402.10452v3.