Smooth unknottedness of turned 1-twist-spun tori and the Cochran–Davis torus in the 4-sphere

Juhász and Powell (2024) proved that certain smoothly embedded genus-one tori in the 4-sphere with knot group $\mathbb{Z}$ are topologically unknotted: Boyle's turned 1-twist-spun tori $T_{K,1,1}$ for every classical knot $K$ , and the union of the Cochran–Davis genus-one Seifert surface (which admits no slice derivative) with a ribbon disc. Whether any of these tori is smoothly unknotted — that is, smoothly isotopic to the standard unknotted torus — is open. Hughes, Kim, and Miller (2025) posed the turned-torus case as an explicit open question and proved that the associated double branched covers are diffeomorphic to $S^2\times S^2$ , so no invariant of the cover can separate these surfaces from the standard torus. The problem is to decide the smooth isotopy class of exactly these topologically standard tori: either establish smooth unknottedness, or show that one of them is not smoothly isotopic to the standard torus.

Background

A knotted surface is a smoothly embedded closed surface in $S^4$ ; it is called smoothly (resp. topologically) unknotted if it is ambiently isotopic, in the smooth (resp. locally flat topological) category, to a standardly embedded surface of the same genus. A $\mathbb{Z}$ -surface is a connected orientable surface $\Sigma\subset S^4$ whose knot group $\pi_1(S^4\setminus\Sigma)$ is infinite cyclic, i.e. isomorphic to $\mathbb{Z}$ ; for example, an unknotted surface has this property. The topological category is classified by surgery theory: Conway and Powell (2023) showed that every $\mathbb{Z}$ -surface in $S^4$ of genus $g\notin\{1,2\}$ is topologically unknotted, via the topological classification of such surfaces in terms of the equivariant intersection form (the intersection pairing on the homology of the infinite cyclic cover of the surface exterior) of their exteriors. The genus-one and genus-two cases remain open — the topological unknotting conjecture — and have an algebraic formulation (Conjecture 2.6 of Juhász–Powell, originating in the Conway–Powell program): every stably surface-hyperbolic hermitian sesquilinear form over the group ring $\mathbb{Z}[\mathbb{Z}]$ of rank $2g$ with $g\in\{1,2\}$ is already surface hyperbolic, where a form is surface hyperbolic if it is isometric to $g$ copies of the hyperbolic form $\mathcal{H}_2$ , and stably surface hyperbolic if it becomes so after summing with copies of $\mathcal{H}_2$ . Juhász and Powell verified topological standardness for several significant families of genus-one $\mathbb{Z}$ -tori, giving evidence for this conjecture. Two families are central here. First, from twist-spinning: Artin associated to a knot $K\subset S^3$ a 2-knot — a 2-sphere smoothly embedded in $S^4$ — by spinning, Zeeman generalized this to $k$ -twist-spun knots, and Boyle's turned torus construction attaches a tube to a twist-spun knot, producing genus-one tori; the turned 1-twist-spun torus of $K$ is denoted $T_{K,1,1}$ (also written $\sigma_T^{\pm1}(K)$ ). Juhász–Powell proved $T_{K,1,1}$ is topologically unknotted for every $K$ , answering a question of Boyle in the topological category. Second, Cochran and Davis (2015) constructed a ribbon knot (a knot bounding an immersed disc in the 4-ball with only ribbon-type singularities) with a genus-one Seifert surface $F$ (an oriented surface in $S^3$ whose boundary is the knot) admitting no slice derivative (a notion from higher-order knot concordance obstructing certain ways that $F$ could relate to a slice disk); the union $\Sigma = F\cup D$ of $F$ with a ribbon disc $D$ (an embedded disc in the 4-ball bounded by the same knot, whose projection has only ribbon-type singularities) is a genus-one $\mathbb{Z}$ -torus, also proved topologically unknotted. In the smooth category the status of both families is unknown: no smoothly knotted but topologically unknotted orientable surface in $S^4$ is currently known, and these tori are the principal explicit candidates. The natural detection route through the double cover of $S^4$ branched along the surface is provably blind here: Juhász–Powell showed the branched cover of $T_{K,1,1}$ is $(S^2\times S^2)\#X$ for a homotopy 4-sphere $X$ (a smooth 4-manifold homotopy equivalent, but not known to be diffeomorphic, to $S^4$ ) whose Seiberg–Witten invariants (gauge-theoretic smooth invariants of 4-manifolds) agree with those of $S^2\times S^2$ , and Hughes–Kim–Miller (2025) strengthened this by proving $X$ is diffeomorphic to $S^4$ and that the double branched cover of every odd-twist turned torus (including $T_{K,1,1}$ ) is diffeomorphic to $S^2\times S^2$ ; for the Cochran–Davis torus the cover arises from $S^2\times S^2$ by knot surgery (the Fintushel–Stern operation replacing a tubular neighborhood of a torus with the exterior of a knot) along a 0-homologous torus. Thus any separation from the standard torus must come from surface-level smooth invariants or from an explicit smooth isotopy, not from invariants of the branched cover. Recently developed Real Seiberg–Witten invariants (an involution-equivariant refinement of Seiberg–Witten gauge theory for 4-manifolds with involution, in Atiyah's Real sense) and related Real Heegaard Floer invariants have produced exotic nonorientable surfaces and involutions in 4-manifolds, but have not been applied successfully to orientable $\mathbb{Z}$ -tori with standard branched covers; and Galvin–Orson–Powell (2026) showed that topologically isotopic surfaces that are trivial in $\mathbb{Z}/2$ -homology become smoothly isotopic after stabilizing the ambient manifold by connected sum with $S^2\times S^2$ , so these tori are smoothly standard after one stabilization — while the unstabilized question, which is the one at issue, remains open.

Problem Statement

Decide the smooth unknottedness of the topologically unknotted genus-one $\mathbb{Z}$ -tori of Juhász–Powell. Specifically: (1) for every knot $K\subset S^3$ , is the turned 1-twist-spun torus $T_{K,1,1}=\sigma_T^{\pm1}(K)\subset S^4$ smoothly isotopic to the standard unknotted torus? (2) is the Cochran–Davis torus $\Sigma=F\cup D$ , the union of the genus-one Seifert surface with no slice derivative with a ribbon disc, smoothly isotopic to the standard unknotted torus? An answer must either prove that these surfaces are smoothly unknotted, or exhibit a member of one of these families that is not smoothly isotopic to the standard torus, thereby refuting the smooth unknotting conjecture for genus-one $\mathbb{Z}$ -tori at these candidates. Results that hold only after ambient stabilization by connected sum with $S^2\times S^2$ , or only in the locally flat topological category, do not by themselves constitute an answer; the question concerns smooth isotopy in $S^4$ itself.

References

András Juhász and Mark Powell, Examples of topologically unknotted tori, Transactions of the American Mathematical Society, Series B 11 (2024) 1266–1293, DOI 10.1090/btran/202; arXiv:2312.06326, https://arxiv.org/abs/2312.06326
Anthony Conway and Mark Powell, Embedded surfaces with infinite cyclic knot group, Geometry & Topology 27 (2023) 739–821, DOI 10.2140/gt.2023.27.739, https://doi.org/10.2140/gt.2023.27.739; arXiv:2009.13461, https://arxiv.org/abs/2009.13461
Mark Hughes, Seungwon Kim, and Maggie Miller, Branched covers of twist-roll spun knots and turned twisted tori, arXiv:2402.11706 (2024; v3 2025), https://arxiv.org/abs/2402.11706
Tim D. Cochran and Christopher William Davis, Counterexamples to Kauffman's conjectures on slice knots, Advances in Mathematics 274 (2015) 263–284, DOI 10.1016/j.aim.2014.12.006, https://doi.org/10.1016/j.aim.2014.12.006
Jeffrey Boyle, The turned torus knot in $S^4$ , Journal of Knot Theory and Its Ramifications 2 (1993) 239–249, DOI 10.1142/S0218216593000155, https://doi.org/10.1142/S0218216593000155
Daniel Galvin, Patrick Orson, and Mark Powell, Smooth stable isotopy of topologically isotopic surfaces, arXiv:2606.06299 (2026), https://arxiv.org/abs/2606.06299
Anthony Conway and Allison N. Miller, Immersed surfaces with knot group $\mathbb{Z}$ , arXiv:2410.04635 (2024), https://arxiv.org/abs/2410.04635
David Baraglia, Exotic embedded surfaces and involutions from Real Seiberg–Witten theory, International Journal of Mathematics 37 (2026), article 2650028, DOI 10.1142/S0129167X2650028X, https://doi.org/10.1142/S0129167X2650028X; arXiv:2504.00281, https://arxiv.org/abs/2504.00281