Classification of 3-component links up to link homotopy

Abstract

The concept of link-homotopy was introduced by Milnor’s paper in 1954, in which he achieved the classification of 3-component closed links up to link-homotopy. After more than thirty years Levine achieved such a classification for 4-component closed links. Then, in 1990, Habeggerr and Lin provided a new approach to a classification based on the closures of two different string links in the group of link homotopy classes. The closures of two string links are link-homotopic if and only if those two elements are related by a sequence of elements, such that two adjacent elements are either conjugate or partial conjugate, in the link group. We employed Habbeger-Lin’s approach to provide a full classification for 3-component string and closed links.