On braids and links up to link-homotopy
Emmanuel Graff

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.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology · Algebraic structures and combinatorial models
