# Characterization of the reduced peripheral system of links

## TL;DR

This paper explores the properties and characterizations of the reduced peripheral system in link theory, providing new diagrammatic and topological perspectives for links with multiple components.

## Contribution

It offers two new characterizations of links sharing the same reduced peripheral system, one diagrammatical and one topological, enhancing understanding of link invariants.

## Key findings

- Characterization via welded diagrams up to self-virtualization
- Topological characterization using ribbon solid tori in 4-space
- Insights into the limitations of the reduced peripheral system for links with four or more components

## Abstract

The reduced peripheral system was introduced by Milnor in the fifties for the study of links up to link-homotopy, i.e. up to isotopies and crossing changes within each link component. However, for four or more components, this invariant does not yield a complete link-homotopy invariant. This paper provides two characterizations of links having the same reduced peripheral system: a diagrammatical one, in terms of link diagrams, seen as welded diagrams up to self-virtualization, and a topological one, in terms of ribbon solid tori in 4--space up to ribbon link-homotopy.