Some properties of ordinary sense slice 1-links: Some answers to the problem (26) of Fox

TL;DR

This paper investigates properties of ordinary sense slice 1-links, establishing the Arf invariant as zero for such links and demonstrating the existence of larger links containing any given link as a sublink.

Contribution

It introduces new results on the Arf invariant for ordinary sense slice 1-links and proves the existence of larger links containing arbitrary links as sublinks.

Findings

Arf(L)=0 for any ordinary sense slice 1-link L

Any m-component 1-link can be embedded as a sublink in a 3m-component ordinary sense slice 1-link

Provides answers to problem (26) of Fox regarding slice links

Abstract

We prove that, for any ordinary sense slice 1-link $L$, we can define the Arf invariant and Arf(L)=0. We prove that, for any m-component 1-link L_1, there exists a 3m-component ordinary sense slaice 1-link L_2 of which L_1 is a sublink.