Stabilization and satellite construction of doubly slice links

TL;DR

This paper introduces new concepts and examples related to doubly slice links in 4-dimensional topology, including an invariant that measures how far a link is from being strongly doubly slice.

Contribution

It provides the first examples of boundary links that are weakly but not strongly doubly slice and introduces a new invariant $g_{st}$ for homotopically trivial links.

Findings

Constructed examples with arbitrarily large doubly slice genus but $g_{st}=1

Established $g_{st}$ as a lower bound for the doubly slice genus

Connected the Conway-Orson signature bound to $g_{st}

Abstract

A 2-component oriented link in $S^3$ is called weakly doubly slice if it is a cross-section of an unknotted sphere in $S^4$, and strongly doubly slice if it is a cross-section of a 2-component trivial spherical link in $S^4$. We give the first example of 2-component boundary links which are weakly doubly slice but not strongly doubly slice. We also introduce a new invariant $g_{st}$ of homotopically trivial links that measures the failure of a link from being strongly doubly slice and that bounds the doubly slice genus $g_{ds}$ from below. Our examples have arbitrarily large doubly slice genus but satisfy $g_{st}=1$. We also prove that the Conway-Orson signature lower bound on $g_{ds}$ is actually a lower bound on $g_{st}$.