Primary decomposition and the fractal nature of knot concordance

Details Primary decomposition and the fractal nature of knot concordance Primary decomposition and the fractal nature of knot concordance

2009-06-07

arxiv.org/abs/0906.1373v2

Math. Annalen 351, no. 2 (2011) 443-508

Mathematics

Geometric Topology

60 pages, added 4 pages to introduction, minor corrections otherwise; Math. Annalen 2010

Scientific paper

10.1007/s00208-010-0604-5

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.

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