Primary decomposition and the fractal nature of knot concordance

Abstract

For each sequence $${\mathcal{P}=(p_1(t),p_2(t),\dots)}$$ of polynomials we define a characteristic series of groups, called the derived series localized at $${\mathcal{P}}$$ . These group series yield filtrations of the knot concordance group that refine the (n)-solvable filtration. We show that the quotients of successive terms of these refined filtrations have infinite rank. The new filtrations allow us to distinguish between knots whose classical Alexander polynomials are coprime and even to distinguish between knots with coprime higher-order Alexander polynomials. This provides evidence of higher-order analogues of the classical 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.