Doubly slice knots and metabelian obstructions

Abstract

An [Formula: see text]-dimensional knot [Formula: see text] is called doubly slice if it occurs as the cross section of some unknotted [Formula: see text]-dimensional knot. For every [Formula: see text] it is unknown which knots are doubly slice, and this remains one of the biggest unsolved problems in high-dimensional knot theory. For [Formula: see text], we use signatures coming from [Formula: see text]-cohomology to develop new obstructions for [Formula: see text]-dimensional knots with metabelian knot groups to be doubly slice. For each [Formula: see text], 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.