Non-triviality of the Cochran-orr-teichner Filtration of the Knot Concordance Group

We establish nontriviality results for certain filtrations of the smooth and topological knot concordance groups. First, as regards the n-solvable filtration of the topological knot concordance group, C, defined by K. Orr, P. Teichner and the first author [COT1]: 0 ⊂ · · · ⊂ F(n.5) ⊂ F(n) ⊂ · · · ⊂ F(1.5) ⊂ F(1.0) ⊂ F(0.5) ⊂ F(0) ⊂ C, we refine the recent nontriviality results of Cochran and Teichner [CT] and extend those of C. Livingston [Li] and the second author [K]. We exhibit similar nontriviality for the closely related symmetric Grope filtration of C considered in [CT]. We also show that the Grope filtration of the smooth concordance group is nontrivial using examples that cannot be distinguished by the Ozsváth-Szabó τ -invariant nor by J. Rasmussen’s s-invariant [OS][Ra]. Our broader contribution is to establish, in “the relative case”, the key homological results whose analogues Cochran-Orr-Teichner established in “the absolute case” in [COT1]. We say two knots K0 and K1 are concordant modulo n-solvability if K0#(−K1) ∈ F(n). Our main result is that, for any knot K whose classical Alexander polynomial has degree greater than 2, and for any positive integer n, there exist infinitely many knots Ki that are concordant to K modulo nsolvability, but are all distinct modulo n.5-solvability. Moreover, the Ki and K share the same classical Seifert matrix and Alexander module as well as sharing the same higher-order Alexander modules and Seifert presentations up to order n− 1.