Abelian invariants of doubly slice links

TL;DR

This paper introduces new algebraic invariants derived from multivariable signatures, Blanchfield forms, and Seifert matrices to obstruct links in 3-spheres from being cross sections of unlinked spheres in 4-spheres, and provides bounds on doubly slice genus.

Contribution

It develops novel obstructions based on algebraic invariants for determining when links are doubly slice in 4-spheres, extending to higher genus surfaces.

Findings

Obstructions from multivariable signature and Blanchfield form for doubly slice links

Lower bounds on doubly slice genus for links with higher genus surfaces

New algebraic criteria to distinguish doubly slice links from non-doubly slice ones

Abstract

We provide obstructions to a link in $S^3$ arising as the cross section of any number of unlinked spheres in $S^4$. Our obstructions arise from the multivariable signature, the Blanchfield form and generalised Seifert matrices. We also obtain obstructions in the case of surfaces of higher genera, leading to a lower bound on the doubly slice genus of links.