Derivatives of links: Milnor’s concordance invariants and Massey’s products

Classification of string links up to self delta-moves and concordance

“…Since C 2n -equivalence for n-component (string) links implies self -equivalence [4, Lemma 1.2], we have that SL.n/=.s C c/ is a nilpotent group. Moreover, since the first nonvanishing -invariants are additive under the stacking product (for example see Cochran [2] and Habegger and Masbaum [8]), by Theorem 1.4, we have the following proposition. Proposition 1.6 The quotient SL.n/=.s C c/ forms a torsion-free nilpotent group under the stacking product.…”

Whitney towers and gropes in 4–manifolds

“…2 n + 2 (n−1) − 2) with certain additional restrictions on the type of allowable intersections (see Section 6). This definition of height is in rough correspondence with the usual definition of height for a symmetric (uncapped) grope, and in [5] it is shown that a knot in the 3-sphere S 3 = ∂B 4 is n-solvable if it bounds an embedded symmetric grope of height n + 2 or a Whitney tower of height n + 2 in B 4 . It is not known if these geometric conditions are equivalent to each other (or to being n-solvable) but we do have:…”

Covering Sato-Levine invariants

“…This geometric invariant was discovered independently by Sato [26] and Levine (unpublished) and has since been studied much more extensively by Cochran [5,6,7], Orr [25] and Stein [27]. The main result of this section is an observation that a stable version of the Sato-Levine invariant can be defined for semiboundary links with possibly disconnected components in the universal abelian cover of the exterior of any classical knot.…”

“…Kojima's η-function is a well-defined link concordance invariant, which satisfies both η L (x) = η L (x) and η L (1) = 0, and vanishes identically for boundary links by Kojima and Yamasaki [16]. This invariant is in fact completely determined by the 2-variable Alexander polynomial of L together with the 1-variable Alexander polynomials of its two components by Jin [13], and is equivalent to Cochran's derived invariants defined in [6], hence vanishes identically if and only if all Milnor invariants of the formμ(11...1122) vanish by Stein [27]; see also Theorem 6.10 of Cochran [7]. In a recent paper [19] of the author, Kojima's η-function is generalized to be definable for certain admissible links in higher dimensions.…”