Link homotopic but not isotopic

TL;DR

This paper constructs examples of links in 3-sphere that are link homotopic but not link isotopic, with all proper sublinks isotopic to a given link, and explores properties related to Milnor invariants.

Contribution

It introduces a method to produce links that are link homotopic but not isotopic, extending understanding of link equivalence relations.

Findings

Existence of links link homotopic but not isotopic to a given link.

Proper sublinks are all isotopic to the original link.

Construction of links with zero Milnor invariants when starting from an unlink.

Abstract

Given an $m$-component link $L$ in $S^3$ ($m \ge 2$), we construct a family of links which are link homotopic, but not link isotopic, to $L$. Every proper sublink of such a link is link isotopic to the corresponding sublink of $L$. Moreover, if $L$ is an unlink then there exist links that in addition to the above properties have all Milnor invariants zero.