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2020 Amenable signatures, algebraic solutions and filtrations of the knot concordance group
Taehee Kim
Algebr. Geom. Topol. 20(5): 2413-2450 (2020). 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 .

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Taehee Kim. "Amenable signatures, algebraic solutions and filtrations of the knot concordance group." Algebr. Geom. Topol. 20 (5) 2413 - 2450, 2020. https://doi.org/10.2140/agt.2020.20.2413

Information

Received: 15 June 2018; Revised: 11 September 2019; Accepted: 12 January 2020; Published: 2020
First available in Project Euclid: 10 November 2020

MathSciNet: MR4171570
Digital Object Identifier: 10.2140/agt.2020.20.2413

Subjects:
Primary: 57M25
Secondary: 57N70

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.20 • No. 5 • 2020
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