Structure in the classical knot concordance group

By Tim D. Cochran, K. Orr, P. Teichner. 2002

131 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ce2a4375632a0157ba810a0a38dde6086a8ee0f2

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A Survey of Classical Knot Concordance (2003, result): “Recent work of Cochran, Orr and Teichner, [14] [15], has revealed a deeper structure to the knot concordance group.”
An infinite-rank summand of topologically slice knots (2013, unknown): “Further work (eg Jiang [21], Cochran, Orr and Teichner [6] and Cochran and”
Knot concordance and von Neumann $\rho$-invariants (2004, methodology): “The first part of the conjecture is true forn 2: n = 0 is verified using twist knots and the Seifert form obstructions, for n = 1 this can be established by using examples due to Casson-Gordon, and n = 2 is the main result of [ COT2 ].”
The structure of the rational concordance group of knots (2006, methodology): “For more details, see Theorem 4.13.”
Topologically slice knots with nontrivial Alexander polynomial (2010, background): “5 was proved in [9] and the infinite generation of Ci/Ci+.”
Knot concordance and higher-order Blanchfield duality (2007, methodology): “This family is simpler than the examples of Cochran and Teichner.”
Homology cobordism invariants and the Cochran-Orr-Teichner filtration of the link concordance group (2006, background): “For more about the grope filtration of a knot, see [11, 12, 13].”
Noncommutative knot theory. (2002, background): “= Z and it is known that (MK, φ0 : P−→ Z) is null-bordant, i.e. that MK bounds (V,ψ), and that σ(2) 0 (K) is the integral of the Levine signature function of K and σ(V) is the ordinary knot signature [COT2]. In this sense the ρngeneralize “ordinary” Levine-Tristram signatures associated to the Z–cov”
Primary decomposition and the fractal nature of knot concordance (2009, background): “Finally, recall that Cochran–Orr–Teichner exhibited the first knots with vanishing Casson–Gordon invariants that are not slice in the topological category and showed that Gn has infinite rank for n = 0, 1, 2 [18,19].”