On the detection of knots by their traces in high dimensions

Abstract

For every $n\ge 4$, we demonstrate the existence of non-isotopic, smooth $(n-2)$-knots in $S^n$ with diffeomorphic traces. We give two proofs: the first by generalising the RBG link construction to all dimensions, and the second as an application of work of Plotnick. Conversely, we prove that for every $n\ge 4$, the unknot in $S^n$ is detected by the diffeomorphism type of its surgery and hence of its trace.

V2: 49 pages, 6 figures. Referees' comments have been incorporated. Please note the slight change of title. Accepted version, to appear in Proc. LMS