Link-homotopy classes of 4-component links and claspers

TL;DR

This paper revises the classification of 4-component links under link-homotopy using Habiro's clasper theory, providing a more symmetrical perspective and practical algorithms for determining link-homotopy equivalence.

Contribution

It introduces a new classification approach for 4-component links via claspers, simplifying previous algebraic methods and enabling practical link-homotopy determination.

Findings

New classification scheme using claspers

Identification of subsets classified by invariants

Algorithm for link-homotopy equivalence

Abstract

Two links are link-homotopic if they are transformed into each other by a sequence of self-crossing changes and ambient isotopies. The link-homotopy classes of 4-component links were classified by Levine with enormous algebraic computations. We modify the results by using Habiro's clasper theory. The new classification gives more symmetrical and schematic points of view to the link-homotopy classes of 4-component links. As applications, we give several new subsets of the link-homotopy classes of 4-component links which are classified by comparable invariants and give an algorithm which determines whether given two links are link-homotopic or not.