# On codimension two embeddings up to link-homotopy

## TL;DR

This paper classifies knotted annuli in 4-space, called 2-string-links, up to link-homotopy using 4-dimensional Milnor invariants, and extends the understanding of ribbon and higher-string links.

## Contribution

It introduces a classification of 2-string-links up to link-homotopy via 4D Milnor invariants and establishes that any 2-string link is link-homotopic to a ribbon one.

## Key findings

- Classification of 2-string-links up to link-homotopy using 4D Milnor invariants
- Any 2-string link is link-homotopic to a ribbon 2-string link
- Extension of results to ribbon k-string links for k ≥ 3

## Abstract

We consider knotted annuli in 4-space, called 2-string-links, which are knotted surfaces in codimension two that are naturally related, via closure operations, to both 2-links and 2-torus links. We classify 2-string-links up to link-homotopy by means of a 4-dimensional version of Milnor invariants. The key to our proof is that any 2-string link is link-homotopic to a ribbon one; this allows to use the homotopy classification obtained in the ribbon case by P. Bellingeri and the authors. Along the way, we give a Roseman-type result for immersed surfaces in 4-space. We also discuss the case of ribbon k-string links, for $k\geq 3$.