On the detection of knotted spheres by their traces in high dimensions

TL;DR

This paper extends the understanding of high-dimensional knots by constructing non-isotopic knots with identical traces and proving that the unknot can be uniquely identified by its trace in dimensions four and higher.

Contribution

It generalizes the RBG link construction to all dimensions and establishes that the unknot is detectable via its trace in high dimensions.

Findings

Existence of non-isotopic knots with diffeomorphic traces in all dimensions n ≥ 4

Unknot detection by the diffeomorphism type of its trace in dimensions n ≥ 4

Generalization of RBG link construction to all dimensions

Abstract

For every $n \geq 4$, we demonstrate the existence of non-isotopic smooth $(n-2)$-knots in $S^n$ with diffeomorphic traces by generalising the RBG link construction to all dimensions. Conversely, we prove that for every $n \geq 4$, the unknot in $S^n$ is detected by the diffeomorphism type of its surgery and hence by its trace.