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

TL;DR

This paper introduces new infinite rank subgroups in the knot concordance group's filtrations using algebraic solutions, expanding the understanding of knot invariants and obstructions.

Contribution

It constructs new infinite rank subgroups in the filtrations using algebraic $n$-solutions instead of iterated doubling operators, and generalizes algebraic solutions to $R$-algebraic $n$-solutions.

Findings

Constructed new subgroups with trivial intersection with known subgroups.

Used $L^2$-theoretic obstructions to analyze knot solvability.

Extended algebraic $n$-solutions to $R$-algebraic $n$-solutions.

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 each slice knot $K$ whose Alexander polynomial has degree greater than 2, we construct the generating knots such that they have the same derived quotients and higher-order Alexander invariants up to a certain order. 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 the notion of algebraic $n$-solutions to the notion of $R$-algebraic $n$-solutions where $R$ is either rationals or the field of $p$ elements for a prime $p$.