2‐torsion in the n‐solvable filtration of the knot concordance group

Cochran–Orr–Teichner introduced in [‘Knot concordance, Whitney towers and L2‐signatures’, Ann. of Math. (2) 157 (2003) 433–519] a natural filtration of the smooth knot concordance group C …⊂Fn+1⊂Fn.5⊂Fn⊂…⊂F1⊂F0.5⊂F0⊂C, called the (n)‐solvable filtration. We show that each associated graded abelian group { Gn=Fn/Fn.5|n∈N },n⩾2, n ⩾ 2, contains infinite linearly independent sets of elements of order 2 (this was known previously for n = 0, 1). Each of the representative knots is negative amphichiral, with vanishing s‐invariant, τ‐invariant, δ‐invariants and Casson–Gordon invariants. Moreover, each is slice in a rational homology 4‐ball. In fact we show that there are many distinct such classes in Gn, distinguished by their Alexander polynomials and, more generally, by the torsion in their higher‐order Alexander modules.