Amenable signatures, algebraic solutions and filtrations of the knot concordance group

Abstract

It is known that each of the successive quotient groups of the grope and solvable filtrations of the knot concordance group has an infinite-rank subgroup. The generating knots of these subgroups are constructed using iterated doubling operators. In this paper, for each of the successive quotients of the filtrations we give a new infinite-rank subgroup which trivially intersects the previously known infinite-rank subgroups. Instead of iterated doubling operators, the generating knots of these new subgroups are constructed using the notion of algebraic $n$–solutions, which was introduced by Cochran and Teichner. Moreover, for any slice knot $K$ whose Alexander polynomial has degree greater than $2$, we construct the generating knots so that they have the same derived quotients and higher-order Alexander invariants up to a certain order as the knot $K$.

In the proof, we use an $L^2$–theoretic obstruction for a knot to being $n.5$–solvable given by Cha, which is based on $L^2$–theoretic techniques developed by Cha and Orr. We also generalize and use the notion of algebraic $n$–solutions to the notion of $R$–algebraic $n$–solutions, where $R$ is either the rationals or the field of $p$ elements for a prime $p$.