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We reveal new structures in the topological knot concordance group. As a key ingredient, we develop obstructions using  L  2 -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 &gt;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  L  2 -signature obstructions as well as Levine algebraic obstructions and Casson–Gordon invariants.

Amenable L 2 -Theoretic Methods and Knot Concordance