Primary decomposition and the fractal nature of knot concordance

TL;DR

This paper introduces new filtrations of the knot concordance group based on derived series localized at polynomial sequences, revealing their fractal structure and infinite rank quotients, and distinguishes between certain classes of knots.

Contribution

It defines a new class of group filtrations using polynomial sequences, refining existing filtrations and providing insights into the structure of the knot concordance group.

Findings

Quotients of successive filtration terms have infinite rank.

The set of smooth concordance classes exhibits fractal properties.

No Cochran-Orr-Teichner knot is concordant to a Cochran-Harvey-Leidy knot.

Abstract

For each sequence of polynomials, P=(p_1(t),p_2(t),...), we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S^3, such a sequence of polynomials arises naturally as the orders of certain submodules of the sequence of higher-order Alexander modules of K. These group series yield new filtrations of the knot concordance group that refine the (n)-solvable filtration of Cochran-Orr-Teichner. We show that the quotients of successive terms of these refined filtrations have infinite rank. These results also suggest higher-order analogues of the p(t)-primary decomposition of the algebraic concordance group. We use these techniques to give evidence that the set of smooth concordance classes of knots is a fractal set. We also show that no Cochran-Orr-Teichner knot is concordant to any Cochran-Harvey-Leidy knot.