Amenable L2-theoretic methods and knot concordance

TL;DR

This paper introduces new $L^2$-theoretic obstructions from amenable groups to study topological knot concordance, revealing structures undetectable by previous invariants, especially regarding $(h)$-solvable knots.

Contribution

It develops novel $L^2$-theoretic obstructions using amenable groups with torsion, providing deeper insights into the knot concordance group beyond existing invariants.

Findings

Existence of $(n)$-solvable knots not $(n.5)$-solvable with vanishing classical obstructions.

New obstructions detect structures in knot concordance not seen by Cochran-Orr-Teichner invariants.

Application of amenable group methods to knot theory advances understanding of topological knot concordance.

Abstract

We introduce new obstructions to topological knot concordance. These are obtained from amenable groups in Strebel's class, possibly with torsion, using a recently suggested $L^2$-theoretic method due to Orr and the author. Concerning $(h)$-solvable knots which are defined in terms of certain Whitney towers of height $h$ in bounding 4-manifolds, we use the obstructions to reveal new structure in the knot concordance group not detected by prior known invariants: for any $n>1$ there are $(n)$-solvable knots which are not $(n.5)$-solvable (and therefore not slice) but have vanishing Cochran-Orr-Teichner $L^2$-signature obstructions as well as Levine algebraic obstructions and Casson-Gordon invariants.