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

Kim, Taehee

doi:10.2140/agt.2020.20.2413

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

Taehee Kim

Algebraic & Geometric Topology 20 (2020) 2413–2450

DOI: 10.2140/agt.2020.20.2413

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

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Keywords

knot, concordance, grope, $n$–solution, algebraic $n$–solution, amenable signature

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57N70

References

Publication

Received: 15 June 2018

Revised: 11 September 2019

Accepted: 12 January 2020

Published: 4 November 2020

Authors

Taehee Kim
Department of Mathematics Konkuk University Seoul South Korea

Volume 20, issue 5 (2020)