Knot Concordance and Higher-Order Blanchfield Duality

TL;DR

This paper introduces new techniques to study the knot concordance group, proving that certain filtration quotients have infinite rank and resolving questions about the sliceness of specific knot families.

Contribution

It develops novel methods for analyzing knot concordance and demonstrates that the filtration quotients are infinitely generated, also resolving longstanding questions about slice knots.

Findings

F_n/F_{n.5} has infinite rank for all n

Established the same for smooth concordance group

Resolved whether certain knot families contain slice knots

Abstract

In 1997, T. Cochran, K. Orr, and P. Teichner defined a filtration {F_n} of the classical knot concordance group C. The filtration is important because of its strong connection to the classification of topological 4-manifolds. Here we introduce new techniques for studying C and use them to prove that, for each natural number n, the abelian group F_n/F_{n.5} has infinite rank. We establish the same result for the corresponding filtration of the smooth concordance group. We also resolve a long-standing question as to whether certain natural families of knots, first considered by Casson-Gordon and Gilmer, contain slice knots.