Topologically but not smoothly H-slice knots in punctured connected sums of K3 surfaces for n ≥ 4 (Problem 4.26 of Konno–Miyazawa–Taniguchi)

For a knot $K \subset S^3$ and a 4-manifold $X$ with boundary $S^3$, call $K$ H-slice in $X$ if it bounds a null-homologous embedded disk in $X$, smoothly or locally flat (topologically). Manolescu–Marengon–Piccirillo produced the first knot that is topologically H-slice but not smoothly H-slice in the punctured K3 surface, and Iida–Mukherjee–Taniguchi extended this to the punctured $\#^n K3$ for $n \leq 3$; all known gauge-theoretic engines (Bauer–Furuta-type invariants, relative 10/8 inequalities, and the doubled Seiberg–Witten Floer K-theory invariant $\kappa$ of Konno–Miyazawa–Taniguchi) stall at $b_2^+ = 12$, i.e. at $\#^4 K3$. This record states, in the source-faithful form of Problem 4.26 of Konno–Miyazawa–Taniguchi (Involutions, knots, and Floer K-theory), the still-open question: for an integer $n \geq 4$, does there exist a knot in $S^3$ that is topologically H-slice but not smoothly H-slice in the punctured $\#^n K3$?

Background

A knot $K \subset S^3$ is called homologically slice (H-slice) in a smooth compact 4-manifold $X$ with $\partial X = S^3$ if $K$, viewed in $\partial X = S^3$, bounds a properly embedded 2-disk $D \subset X$ whose homology class vanishes in $H_2(X;\mathbb{Z})$ (such a disk is called null-homologous); 'smoothly H-slice' means $D$ is smooth, while 'topologically H-slice' means $D$ is only locally flat, i.e. every point of $D$ has a neighborhood pair homeomorphic to $(\mathbb{R}^4,\mathbb{R}^2)$. The K3 surface is the closed, simply connected, spin 4-manifold — spin means the second Stiefel–Whitney class vanishes, so the intersection form on second homology is even — with intersection form $2(-E_8) \oplus 3H$, where $E_8$ is the even negative-definite rank-8 lattice and $H$ the hyperbolic plane (the rank-2 indefinite unimodular lattice); thus $b_2(K3) = 22$, $b_2^+(K3) = 3$ — throughout, $b_2^+$ denotes the maximal dimension of a subspace of $H_2(X;\mathbb{R})$ on which the intersection form is positive definite — and $\sigma(K3) = -16$, where $\sigma$ is the signature of the intersection form. $\#^n K3$ denotes the $n$-fold connected sum, and 'punctured $\#^n K3$' means $\#^n K3$ with the interior of a 4-ball removed, a spin manifold with boundary $S^3$ and $b_2^+ = 3n$. By Freedman's disk-embedding theorem, any knot with trivial Alexander polynomial — the classical polynomial invariant $\Delta_K(t) \in \mathbb{Z}[t^{\pm1}]$ computed from the homology of the infinite cyclic cover of the knot complement — is topologically slice in $B^4$, hence automatically topologically H-slice in every such $X$; so the topological half of the question below can be arranged for free. The content is entirely on the smooth side: obstructing smooth H-sliceness requires gauge theory, specifically invariants defined from the Seiberg–Witten equations, a system of partial differential equations for a connection and a spinor field on a smooth 4-manifold. Furuta's 10/8-inequality bounds $-\sigma(W)/8 \leq b_2^+(W)$ for closed spin 4-manifolds $W$; doubling a purported slice-disk exterior along the boundary produces exactly such a $W$, which is how 10/8-type constraints on slice disks arise. Manolescu's relative 10/8-inequality (a version of the 10/8 constraint for 4-manifolds with boundary, applied to properly embedded surfaces) and the 10/8+4 theorem let Manolescu–Marengon–Piccirillo (arXiv:2012.12270; published as Mathematische Annalen 390 (2024), 1481–1506) produce the first topologically-but-not-smoothly H-slice knot in the punctured K3 surface ($n = 1$). Iida–Mukherjee–Taniguchi (arXiv:2102.02076; Advances in Mathematics 466 (2025)), via an adjunction inequality for Bauer–Furuta-type invariants (stable-homotopy refinements of the Seiberg–Witten invariants introduced by Bauer and Furuta), extended this to infinite families, linearly independent in the smooth concordance group (knots modulo smooth cobordism by embedded annuli in $S^3 \times [0,1]$), in the punctured $\#^n K3$ for $n = 1, 2, 3$ (their Theorem 1.10). However, the Bauer–Furuta-type invariant they use vanishes on $\#^n K3$ once $n \geq 4$. Konno–Miyazawa–Taniguchi then developed, in the source paper of this candidate, a version of Seiberg–Witten Floer K-theory for knots: to a knot $K$, viewed through the natural involution on its double branched cover (the 2-fold cover of $S^3$ branched along $K$), they attach a smooth-concordance invariant $\kappa(K)$, and their 10/8-type inequality yields genus bounds for null-homologous surfaces in punctured $\#^n K3$: their Theorem 4.23 states that for a connected sum $K$ of two-bridge knots (knots admitting a projection with exactly two local maxima) with Arf invariant zero (the Arf invariant being a $\mathbb{Z}/2$-valued concordance invariant), $-\tfrac{1}{2}\sigma(K) - n \leq g(K)$, where $g(K)$ is the minimal genus of a null-homologous properly embedded surface in the punctured $\#^n K3$ with boundary $K$; for $K = \#^m 5_2$, the $m$-fold connected sum of the two-bridge knot $5_2$ from the standard knot tables, this obstructs smooth H-sliceness whenever $m > n$. But knots with such nontrivial signature have nontrivial Alexander polynomial, and whether they are topologically H-slice in $\#^n K3$ is itself unresolved: the best classical topological obstruction, from the Levine–Tristram signatures $\sigma_K(e^{2\pi i r/m})$ (generalizations of the knot signature depending on a unit complex number, computed as signatures of the Seifert form twisted by that number), yields only $|\sigma_K(e^{2\pi i r/m}) - 16n| \leq 22n$ for topologically H-slice knots (the source paper's Theorem 6.7 and its consequence (70)), which permits but does not establish topological H-sliceness of these knots. Conversely, for topologically slice knots ($\sigma = 0$) no smooth obstruction is currently known in $\#^n K3$ for $n \geq 4$. This is the structural wall at $b_2^+ = 12$, and it is exactly against this wall that the source paper poses its Problem 4.26.

Problem Statement

As posed in Problem 4.26 of Konno–Miyazawa–Taniguchi, Involutions, knots, and Floer K-theory (arXiv:2110.09258; Compositio Mathematica 161 (2025), no. 11, 2852–2910): let $n$ be a positive integer with $n \geq 4$; does there exist a knot $K \subset S^3 = \partial(\#^n K3 \setminus \mathrm{int}\,D^4)$ that is topologically H-slice but not smoothly H-slice in the punctured $\#^n K3$ — i.e. that bounds a locally flat, null-homologous embedded disk in $\#^n K3 \setminus \mathrm{int}\,D^4$ but bounds no smoothly embedded null-homologous disk there? A resolution would either produce such a knot with proof, or prove that every topologically H-slice knot in the punctured $\#^n K3$ is smoothly H-slice. The target is the standard smooth punctured $\#^n K3$, considered up to diffeomorphism fixing the boundary identification. (The source paper's genus bound Theorem 4.23 is stated for any homotopy $\#^n K3$, i.e. any smooth 4-manifold homotopy-equivalent to $\#^n K3$; but homotopy $\#^n K3$'s need not be diffeomorphic to $\#^n K3$, so an answer relative to a non-diffeomorphic homotopy $\#^n K3$ would not by itself resolve Problem 4.26 as posed.)

References

1. Hokuto Konno, Jin Miyazawa, Masaki Taniguchi, Involutions, knots, and Floer K-theory, arXiv:2110.09258 (2021), DOI 10.48550/arXiv.2110.09258, https://arxiv.org/abs/2110.09258; published as Compositio Mathematica 161 (2025), no. 11, 2852–2910, DOI 10.1112/S0010437X25102820, https://doi.org/10.1112/S0010437X25102820
2. Nobuo Iida, Anubhav Mukherjee, Masaki Taniguchi, An adjunction inequality for the Bauer–Furuta type invariants, with applications to sliceness and 4-manifold topology, arXiv:2102.02076 (2021), DOI 10.48550/arXiv.2102.02076, https://arxiv.org/abs/2102.02076; published as Advances in Mathematics 466 (2025), article 110134, DOI 10.1016/j.aim.2025.110134, https://doi.org/10.1016/j.aim.2025.110134
3. Ciprian Manolescu, Marco Marengon, Lisa Piccirillo, Relative genus bounds in indefinite four-manifolds, arXiv:2012.12270 (2020), DOI 10.48550/arXiv.2012.12270, https://arxiv.org/abs/2012.12270; published as Mathematische Annalen 390 (2024), no. 1, 1481–1506, DOI 10.1007/s00208-023-02787-4, https://doi.org/10.1007/s00208-023-02787-4
4. Hokuto Konno, Jin Miyazawa, Masaki Taniguchi, Involutions, links, and Floer cohomologies, arXiv:2304.01115 (2023), DOI 10.48550/arXiv.2304.01115, https://arxiv.org/abs/2304.01115; published as Journal of Topology 17 (2024), no. 2, DOI 10.1112/topo.12340, https://doi.org/10.1112/topo.12340
5. Imogen Montague, Non-smoothable $\mathbb{Z}/p$-actions on nuclei, arXiv:2401.00244 (2023), DOI 10.48550/arXiv.2401.00244, https://arxiv.org/abs/2401.00244
6. Anthony Conway, Patrick Orson, Mark Pencovitch, Simply slicing knots, arXiv:2507.00431 (2025), DOI 10.48550/arXiv.2507.00431, https://arxiv.org/abs/2507.00431