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Doubly slice knots and metabelian obstructions
Journal of Topology and Analysis  (IF0.457),  Pub Date : 2021-02-06, DOI: 10.1142/s1793525321500229
Patrick Orson, Mark Powell

An $n$-dimensional knot $Sn⊂Sn+2$ 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 $ℓ>1$, we use signatures coming from $L(2)$-cohomology to develop new obstructions for $(4ℓ−3)$-dimensional knots with metabelian knot groups to be doubly slice. For each $ℓ>1$, 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.