A criterion for double sliceness (Q6614197)

Freedman famously proved that knots with trivial Alexander polynomial are topologically slice. Freedman's result was later extended in [\textit{S. Friedl} and \textit{P. Teichner}, Geom. Topol. 9, 2129--2158 (2005; Zbl 1120.57004)], where the authors give a homological condition guaranteeing the sliceness of certain knots with Alexander polynomial \((t-2)(t^{-1}-2)\). The starting remark for the paper under review is that knots with Alexander polynomial \(1\) are in fact doubly slice: doubling the disc obtained by Freedman's theorem gives the required unknotted \(2\)-sphere. The author then investigates the question of double sliceness for knots with Alexander polynomial \((t-2)(t^{-1}-2)\).\N\NThe main result gives a homological criterion guaranteeing double sliceness for knots with Alexander module \(\mathbb{Z}[t,t^{-1}]/(t-2)\oplus\mathbb{Z}[t,t^{-1}]/(t^{-1}-2)\). This criterion can be seen, in a certain precise sense, as a doubling of the aforementioned Friedl-Teichner sliceness criterion. In fact, the author gives a stronger result: for a knot with Alexander module \(\mathbb{Z}[t,t^{-1}]/(t-2)\oplus\mathbb{Z}[t,t^{-1}]/(t^{-1}-2)\), the criterion is satisfied if and only if the knot is realised as the equatorial cross-section of an unknotted \(2\)-sphere satisfying certain additional properties. On the other hand, an example is given of a knot with Alexander module \(\mathbb{Z}[t,t^{-1}]/(t-2)\oplus\mathbb{Z}[t,t^{-1}]/(t^{-1}-2)\) which is doubly slice but does not satisfy the criterion. It should also be mentioned that the author conjectures that, if a doubly slice knot has Alexander polynomial \((t-2)(t^{-1}-2)\), then its Alexander module is necessarily \(\mathbb{Z}[t,t^{-1}]/(t-2)\oplus\mathbb{Z}[t,t^{-1}]/(t^{-1}-2)\).\N\NThe author then applies its criterion to the study of satellite knots, showing that, under certain assumptions on a pattern \(R\), the satellite knot \(R(K)\) is doubly slice for any choice of companion \(K\). This is then used to construct an infinite family of doubly slice knots with Alexander module \(\mathbb{Z}[t,t^{-1}]/(t-2)\oplus\mathbb{Z}[t,t^{-1}]/(t^{-1}-2)\).\N\NThe proofs heavily rely on the results of Friedl-Teichner and of [\textit{A. Conway} and \textit{M. Powell}, Adv. Math. 391, Article ID 107960, 29 p. (2021; Zbl 1476.57007)]. In particular, for the double sliceness criterion, the homological condition ensures the existence of two (different!) slicing discs, and a fundamental group calculation shows that the sphere obtained by gluing these discs is indeed unknotted.