The classification of links up to link-homotopy

Abstract

Though the study of knots and links in dimension three has been with us for well over a century, progress towards the ultimate goal of their classification has been slow. Various methods have been used in their study, ranging from braid theory to the study of the link complement and its fundamental group.Braid theory succeeded in the classification of braids (i.e., in solving the word and conjugacy problems in the braid groups). An equivalence relation generated by Markov moves on braids was found yielding the set of isotopy classes of links (see [B]). However, the combinatorics of the Markov moves are very difficult, and braid theory has not yet led to tqe classification of links, although recent work of Birman and Menasco shows progress in that direction [BM]. On the other hand, it has led to polynomial invariants, via the work of Jones and others [J]. (Recently, Witten [W] has given a physical interpretation of these polynomials in terms of particle "transmission." Possible relations with the results in this work are yet to be explored.)The study of the fundamental group of the complement of a link has led to link invariants of many types. Milnor [M2] introduced higher order linking numbers, called JI-invariants, which have been shown (see [P] and [T]) to coincide with co homological invariants [Ma] of the link complement. These invariants have proven inadequate for the classification of links, in part due to a lack of understanding of their true indeterminancy. Several years prior to the publication of [M2], Milnor introduced the notion of link-homotopy (his terminology was simply "homotopy") in [M 1]. Some link-homotopy invariants of links were introduced there which turned out to be a subcollection of the JI-invariants in [M2]. It was hoped that, modulo a certain appropriate indeterminancy, these link-homotopy invariants would be able to classify links up to link-homotopy. Although this classification was achieved by Milnor for links with two and three components [M 1], it was only after more than thirty years that Levine accomplished such a classification for links with four components [Le2].The classification of links up to link-homotopy, presented here, represents a unification of the above two points of view. As such, it is a possible model for what the ultimate classification picture might look like. See [L] for a general-

Characterization of finite type string link invariants of degree <5

“…We refer the reader to [12] or [35] for a precise definition of Milnor invariants µ(I) of string links. The smallest length Milnor invariants µ σ (ij) of a string link σ coincide with the linking numbers lk(σ i ,σ j ).…”

“…For an element α of the symmetric group S 3 , denote by K α the knot obtained from the unknot U by surgery along the C 4 -tree k α represented in Figure 4.4. Note that K id , K (13) and K (12) are the three knots A, B and C illustrated in Figure 4.1. 4 By the AS and IHX relations, the abelian group SL 4 (1)/C 5 is generated by these sixelements K α , α ∈ S 3 .…”

On Braid Groups, Free Groups, and the Loop Space of the 2-Sphere

“…Then both Milnor, and Habegger-Lin prove that the link L n ∪ β is link homotopically trivial if and only if the element b projects to the identity in K n . [25,26,17,22]. Theorem 3.5.…”

“…The purpose of this article is to describe connections between the loop space of the 2-sphere, Artin's braid groups, a choice of simplicial group whose homotopy groups are given by modules called Lie(n), as well as work of Milnor [25], and Habegger-Lin [17,22] on "homotopy string links". The current article exploits Lie algebras associated to Vassiliev invariants in work of T. Kohno [19,20], and provides connections between these various topics.…”

“…Consider the "reduced free group" due to Milnor [25,26] used to analyze "homotopy string links", and studied in Habegger-Lin [17], [22]. The "reduced free group" K n is defined as the quotient of F n modulo the relations…”