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

Taehee Kim

Published 2016-06-22Version 1

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 to show that the successive quotients of the filtrations are nontrivial. In the proof, to show that the generating knots of the new infinite rank subgroups are linearly independent modulo (n.5)-solvability, we use Amenable Signature Theorem given by Cha. For this purpose, we generalize further 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.