#### Volume 20, issue 5 (2020)

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

### Taehee Kim

Algebraic & Geometric Topology 20 (2020) 2413–2450
##### 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 $p$.

 Dedicated to the memory of Tim D Cochran
##### Keywords
knot, concordance, grope, $n$–solution, algebraic $n$–solution, amenable signature
Primary: 57M25
Secondary: 57N70