A second order algebraic knot concordance group

TL;DR

This paper introduces an algebraic group that encapsulates the initial stages of the knot concordance filtration using symmetric chain complexes with enhanced structural control over fundamental groups.

Contribution

It defines a new algebraic group invariant that captures the first two stages of the Cochran-Orr-Teichner filtration in a unified framework.

Findings

The invariant effectively distinguishes knots at early filtration stages.

It provides a new algebraic perspective on knot concordance.

The approach enhances understanding of fundamental group changes 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 invariant. 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.