A Second Order Algebraic Knot Concordance Group

TL;DR

This paper introduces an algebraic group structure using symmetric chain complexes to encapsulate the initial stages of the knot concordance filtration, providing a unified obstruction framework.

Contribution

It defines a second order algebraic knot concordance group with enhanced control over fundamental groups and their transformations during concordance.

Findings

Captures the first two stages of the Cochran-Orr-Teichner filtration

Provides a new algebraic obstruction for knot concordance

Enhances understanding of fundamental group behavior in concordance

Abstract

We define an algebraic group comprising symmetric chain complexes which captures the first two stages of the Cochran-Orr-Teichner solvable filtration of the knot concordance group in a single obstruction. To achieve this we impose additional structure on each chain complex which puts extra control on the fundamental groups, and in particular on the way in which they can change in a concordance.