Clasper presentations of Habegger-Lin's action on string links

TL;DR

This paper explicitly calculates the actions of conjugations and partial conjugations on 4- and 5-component string links, enabling classification and comparison of links up to link-homotopy.

Contribution

It provides explicit calculations of these actions for 4- and 5-component links, facilitating practical link-homotopy classification.

Findings

Explicit formulas for conjugation actions on 4- and 5-component string links

Classification of link-homotopy classes for 4- and 5-component links

Algorithm implementation for link-homotopy determination

Abstract

Habegger and Lin gave a classification of the link-hmotopy classes of links as the link-homotopy classes of string links modulo the actions of conjugations and partial conjugations for string links. In this paper, we calculated the actions of the partial conjugations and the conjugations explicitly for 4- and 5-component string links which gave classifications (presentations) of the link-homotopy classes of 4- and 5-component links. As an application, we can run Habegger and Lin's algorithm which determines whether given two links are link-homotopoic or not for 4- and 5-component links.