Amenable L2-Theoretic Methods and Knot Concordance

We reveal new structures in the topological knot concordance group. As a key ingredient, we develop obstructions using L²-theoretic methods for amenable groups in Strebel’s class recently introduced by 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 show the following: for any n>1, there are (n)-solvable but non-(n.5)-solvable (and therefore nonslice) knots, which are not detected by prior methods using Cochran–Orr–Teichner L²-signature obstructions as well as Levine algebraic obstructions and Casson–Gordon invariants.