On braids and links up to link-homotopy

TL;DR

This paper explores links and braids up to link-homotopy using Habiro's clasper calculus, providing a faithful linear representation of the homotopy braid group and classifying multi-component links up to link-homotopy.

Contribution

It introduces a new geometric approach to classify links and braids up to link-homotopy using clasper calculus, including a faithful representation and extended classifications.

Findings

Faithful linear representation of the homotopy braid group.

Geometric proof of Levine's classification for 4-component links.

Extended classification for 5-component links in the algebraically split case.

Abstract

This paper deals with links and braids up to link-homotopy, studied from the viewpoint of Habiro's clasper calculus. More precisely, we use clasper homotopy calculus in two main directions. First, we define and compute a faithful linear representation of the homotopy braid group, by using claspers as geometric commutators. Second, we give a geometric proof of Levine's classification of 4-component links up to link-homotopy, and go further with the classification of 5-component links in the algebraically split case.