Homotopy string links and the $\kappa$-invariant

TL;DR

This paper constructs a new map for string links that distinguishes homotopy classes and shows it factors through Koschorke's original link map, advancing the understanding of link invariants.

Contribution

It introduces an analogous map for string links, proves it separates homotopy classes, and demonstrates its relation to Koschorke's original map.

Findings

The new map separates homotopy string links.

Koschorke's map factors through the new map and a closure map.

The work advances link homotopy classification methods.

Abstract

Koschorke introduced a map from the space of closed $n$-component links to the ordered configuration space of $n$-tuples of points in $\mathbb{R}^3$, and conjectured that this map separates homotopy links. The purpose of this paper is to construct an analogous map for string links, and to prove (1) this map in fact separates homotopy string links, and (2) Koschorke's original map factors through the map constructed here together with an analogue of Markov's closure map defined on the level of certain function spaces.