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For each sequence of polynomials, P=(p_1(t),p_2(t),...), we define acharacteristic 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 theorders of certain submodules of the sequence of higher-order Alexander modulesof K. These group series yield new filtrations of the knot concordance groupthat refine the (n)-solvable filtration of Cochran-Orr-Teichner. We show thatthe quotients of successive terms of these refined filtrations have infiniterank. These results also suggest higher-order analogues of the p(t)-primarydecomposition of the algebraic concordance group. We use these techniques togive evidence that the set of smooth concordance classes of knots is a fractalset. We also show that no Cochran-Orr-Teichner knot is concordant to anyCochran-Harvey-Leidy knot.

Primary decomposition and the fractal nature of knot concordance