A criterion for double sliceness

Abstract

We describe a condition involving noncommutative Alexander modules which ensures that a knot with Alexander module $\mathbb{Z}[t^{\pm 1}]/(t-2)\oplus \mathbb{Z}[t^{\pm 1}]/(t^{-1}-2)$ is topologically doubly slice. As an application, we show that a satellite knot $R_{\hat{I}}\cdot (K)$ is doubly slice if the pattern $R$ has Alexander module $\mathbb{Z}[t^{\pm 1}]/(t-2)\oplus \mathbb{Z}[t^{\pm 1}]/(t^{-1}-2)$ and satisfies this condition, and if the infection curve $\hat{I}\cdot \subset S^3\setminus R$ lies in the second derived subgroup $\ddot{I}{€}_1(S^3\setminus R{)}^{(2)}.$

14 pages, 3 figures. v2. Minor changes following suggestions from an anonymous referee. To appear in NYJM