Higher-order Alexander invariants and filtrations of the knot concordance group

Abstract

We establish certain "non-triviality" results for several filtrations of the smooth and topological knot concordance groups. First, as regards the n-solvable filtration of the topological knot concordance group defined by K. Orr, P. Teichner and the first author, we refine the recent non-triviality results of Cochran and Teichner by including information on the Alexander modules. These results also extend those of C. Livingston and the second author. We exhibit similar structure in the closely related symmetric Grope filtration of the knot concordance group considered by the first author and Teichner. We also show that the Grope filtration of the smooth concordance group is non-trivial using examples that cannot be distinguished by the Ozsvath-Szabo tau invariant nor by J. Rasmussen's s-invariant. Our broader contribution is to establish, in "the relative case", the key homological results whose analogues Cochran-Orr-Teichner established in "the absolute case". We say two knots K and J are concordant modulo n-solvability if K#(-J) is n-solvable. 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 K_i that are concordant to K modulo n-solvability, but are all distinct modulo n.5-solvability. Moreover, the K_i 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.