Implementation of the Habegger--Lin decision algorithm

TL;DR

This paper implements the Habegger--Lin decision algorithm for link-homotopy classification, enabling effective distinction of links, especially in 4- and 5-component cases, and reveals new link pairs indistinguishable by Milnor invariants.

Contribution

The paper provides an explicit implementation of the Habegger--Lin algorithm, facilitating practical link-homotopy classification and discovery of links indistinguishable by Milnor invariants.

Findings

Successfully distinguished new link pairs not separable by Milnor invariants.

Implemented the algorithm for 4- and 5-component links.

Made the implementation publicly available at the provided repository.

Abstract

Habegger and Lin gave a classification of link-homotopy classes of links in terms of that of string links modulo certain group actions. As an application, they constructed an algorithm for determining whether given two links are link-homotopic. In \cite{KM4}, we explicitly computed these group actions for the 4- and 5-component cases. Consequently, the Habegger--Lin algorithm can be effectively applied in these cases. In this paper, we present an implementation of this algorithm, which is available at \cite{KMcode}, and exhibit new pairs of links that are not link-homotopic yet cannot be distinguished by Milnor's link-homotopy invariants, called $\overline{\mu}$-invariants.