The Non-triviality of the Grope Filtrations of The Knot and Link Concordance Groups

TL;DR

This paper proves that the successive quotients in the Grope filtration of the knot and link concordance groups have infinite rank, revealing complex structure in these mathematical objects.

Contribution

It demonstrates the infinite rank of successive quotients in the Grope filtration for both knot and string link concordance groups, extending previous understanding.

Findings

Successive quotients in the Grope filtration have infinite rank.

The result applies to both knot and multi-component string link concordance groups.

The work extends the understanding of the algebraic structure of concordance groups.

Abstract

We consider the Grope filtration of the classical knot concordance group that was introduced in a paper of Cochran, Orr and Teichner. Our main result is that successive quotients at each stage in this filtration have infinite rank. We also establish the analogous result for the Grope filtration of the concordance group of string links consisting of more than one component.