Link-homotopy classes of 4-component links, claspers and the Habegger–Lin algorithm

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 links with up to three components were classified by the Milnor homotopy invariants. Levine investigated the indeterminacy of Milnor homotopy invariants and gave a classification of the link-homotopy classes of 4-component links. In this paper, we give another classification of the link-homotopy classes of 4-component links by using the clasper theory. This classification is obtained through a tetrahedron standard form of the link-homotopy classes of 4-component links. As an application, we give several new subsets of the link-homotopy classes of 4-component links which are classified by computable invariants. Moreover, the classification allows us to run the Habegger–Lin algorithm which determines whether two given links are link-homotopic or not.