Structure in the classical knot concordance group

Abstract

Abstract. We provide new information about the structure of the abelian group of topological concordance classes of knots in S 3 . One consequence is that there is a subgroup of infinite rank consisting entirely of knots with vanishing Casson-Gordon invariants but whose non-triviality is detected by von Neumann signatures. Mathematics Subject Classification (2000). 57M25.

Link concordance, homology cobordism, and Hirzebruch-type defects from iterated $p$-covers

“…In the case of knots, for n ≤ 2 there are infinitely many infinite order elements in F (n) which are independent (over Z) modulo F (n. 5) , that is, F (n) /F (n.5) is of infinite rank. (For n ≤ 1, this is a classical result, and for n = 2, it was shown in [20].) In the case of links, Harvey considered the solvable filtration {BF (n) } of the boundary string link concordance group instead of spherical links, so that the difficulty from disk basings is avoided, and proved that there are infinite order boundary string links which generate an infinite rank subgroup in (the abelianization of) BF (n) /BF (n+1) , using her invariant ρ n [26].…”

“…Roughly speaking, L 2 -signatures of the infected manifold or link associated to certain solvable coefficient systems reflect L 2 -signatures of the infection knot K associated to much simpler coefficient systems (e.g., abelian or metabelian ones). All recent works mentioned above [19,20,21,13,18,30,31,32,26,16,17,9] depend on results of this type. When the infection knot K is torsion, however, those L 2 -signatures have failed to detect anything.…”

“…Based on this property, in many papers infection has been used as a primary source of examples revealing the structure of knot and link concordance. In particular, in their landmark paper [19] and subsequent works [20,21], Cochran, Orr, and Teichner gave a new framework of a systematic study of concordance of knots and links in terms of a filtration related to the derived series, and used infection as a realization tool to illustrate the rich contents of their theory. (For definitions of concordance and the Cochran-Orr-Teichner filtration, see Sections 6 and 8.)…”

“…It is known that there exist (infinitely many) nontrivial elements in any depth of the filtration, i.e., in F (n) for any n. In case of knots, it was proved for n = 2 by CochranOrr-Teichner [19,20], and for all higher n by Cochran-Teichner [21]. (For n < 2 the result is classical; e.g., see [34,28].)…”

New constructions of slice links

“…The construction begins with a ribbon knot or link and modifies it by a procedure called a multi-infection (previously called infection by a string link [C04, p. 385] and a tangle sum [CO94, Section 1]) which generalizes the classical satellite construction. Special cases of this construction have been used extensively since the late 1970's to exhibit interesting examples of knots and links that are not slice [Gi83,L05,COT03,COT04,H06,Ci06]. Therefore the present paper complements these results, giving hope for an eventual complete resolution of the question of when this construction results in a slice knot or link.…”