Linear Independence of Knots Arising from Iterated Infection Without the Use of Tristram Levine Signatures

TL;DR

This paper constructs explicit families of knots that are linearly independent within the (n)-solvable filtration of the knot concordance group, utilizing the ho^1-invariant and allowing for knots with vanishing Tristram-Levine signatures.

Contribution

It introduces a new method for constructing linearly independent knots deep in the filtration, without relying on Tristram-Levine signatures.

Findings

Explicit construction of linearly independent knots

Knots can have vanishing Tristram-Levine signatures

Deep in the (n)-solvable filtration

Abstract

We give an explicit construction of linearly independent families of knots arbitrarily deep in the (n)-solvable filtration of the knot concordance group using the \rho^1-invariant. A difference between previous constructions of infinite rank subgroups in the concordance group and ours is that the deepest infecting knots in the construction we present are allowed to have vanishing Tristram-Levine signatures.