Structure in the classical knot concordance group

TL;DR

This paper investigates the algebraic structure of the knot concordance group, revealing a subgroup of infinite rank characterized by specific invariants and signatures, enhancing understanding of knot classification.

Contribution

It introduces a new subgroup within the knot concordance group, distinguished by vanishing Casson-Gordon invariants and detection via $L^{(2)}$ signatures, providing novel structural insights.

Findings

Existence of an infinite rank subgroup with specific invariants

Identification of knots with vanishing Casson-Gordon invariants

Detection of non-triviality through $L^{(2)}$ signatures

Abstract

We provide new information about the structure of the abelian group of topological concordance classes of knots in $S^3$. One consequence is that there is a subgroup of infinite rank consisting entirely of knots with vanishing Casson-Gordon invariants but whose non-triviality is detected by $L^{(2)}$ signatures.