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Doubly slice knots and metabelian obstructions

Patrick Orson

https://orcid.org/0000-0003-1707-0737

Department of Mathematics, Boston College, USA

E-mail Address: [email protected]

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and

Mark Powell

Department of Mathematical Sciences, Durham University, UK

E-mail Address: [email protected]

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https://doi.org/10.1142/S1793525321500229Cited by:0 (Source: Crossref)

Abstract

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 $ℓ > 1$ , we use signatures coming from $L^{(2)}$ -cohomology to develop new obstructions for $(4 ℓ - 3)$ -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.

Keywords:

AMSC: 57K45

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