An $n$-dimensional knot $S^n\subset S^{n+2}$ is called doubly slice if it occurs as the cross section of some unknotted $(n+1)$-dimensional knot. For every $n$ it is unknown which knots are doubly slice, and this remains one of the biggest unsolved problems in high-dimensional knot theory. For $\ell >1$, we use signatures coming from $L^{(2)}$-cohomology to develop new obstructions for $(4\ell -3)$-dimensional knots with metabelian knot groups to be doubly slice. For each $\ell >1$, we construct an infinite family of knots on which our obstructions are nonzero, but for which double sliceness is not obstructed by any previously known invariant.

中文翻译：

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双切片结和代谢障碍

一个$n$维结${\mathrm{小号}}^n\subset {\mathrm{小号}}^{n+\mathrm{2个}}$如果它作为某些未打结的横截面出现，则称为双切片$(n+\mathrm{1个})$维结。对于每一个$n$不知道哪些结是双片的，这仍然是高维结理论中最大的未解决问题之一。为了$\ell >\mathrm{1个}$，我们使用来自${\mathrm{大号}}^{(\mathrm{2个})}$- 上同调开发新的障碍$(\mathrm{4个}\ell -\mathrm{3个})$维结与 metabelian 结组被双切片。对于每个$\ell >\mathrm{1个}$，我们构建了一个无限的结族，在这些结上我们的障碍物是非零的，但是双切片不会被任何先前已知的不变量所阻碍。