Doubly slice knots and metabelian obstructions

TL;DR

This paper introduces new $L^{(2)}$-signature obstructions to determine when certain high-dimensional knots with metabelian groups are doubly slice, providing examples where previous invariants fail.

Contribution

It develops novel $L^{(2)}$-signature obstructions for high-dimensional knots and constructs infinite families where these obstructions detect non-sliceness beyond prior methods.

Findings

Constructed infinite families of knots with non-zero obstructions

Obstructions detect non-sliceness not seen by previous invariants

Extended the understanding of doubly slice knots in higher dimensions

Abstract

For $\ell >1$, we develop $L^{(2)}$-signature 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 non-zero, but for which double sliceness is not obstructed by any previously known invariant.