Key points of this research results
- Explicit classifications of link-homotopy classes of links were provided for 4- and 5-component links.
- The Habegger-Lin algorithm, which determines whether two links belong to the same link-homotopy class, was implemented for 4- and 5-component links.
- A table was constructed to facilitate the creation of invariants for link-homotopy.
- A conjecture was proposed regarding the number of generators for equivalence relations in link-homotopy classes.
Outline
The classification and determination of links are classical problems in knot theory, a branch of topology. A link is a collection of closed strings in three-dimensional space, and any link obtained by continuously moving the strings by hand is considered the same link. This study approaches the classification and determination of links under a weaker equivalence relation called link-homotopy. Under this relation, two links are considered the same if they are transformed from one to another by making strings pass through one another as long as they are connected.
Habegger and Lin demonstrated that link-homotopy classes of links are determined by a group action on string links. They also provided a classification and normal forms for link-homotopy classes of string links. However, explicit computations of this group action had not been given. In this study, explicit classifications of link-homotopy classes were obtained by computing the group action for 4- and 5-component links (links consisting of 4 or 5 strings). As an application, the Habegger-Lin algorithm, which determines whether two links belong to the same link-homotopy class, was made implementable for 4- and 5-component links, and specific examples were presented. Additionally, during the course of this research, a conjecture was proposed regarding the upper bound on the number of generators for equivalence relations in link-homotopy classes for general-component links. These results were achieved by applying local deformation theory of links to the classical theory of Habegger and Lin.

Paper Info
Yuka Kotorii, Atsuhiko Mizusawa, Clasper Presentations of Habegger-Lin’s Action on String Links, EXPERIMENTAL MATHEMATICS, 2024, VOL. 00, NO. 0, 1–45
https://doi.org/10.1080/10586458.2024.2398150