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Mathematische Annalen

, Volume 351, Issue 2, pp 443–508 | Cite as

Primary decomposition and the fractal nature of knot concordance

  • Tim D. Cochran
  • Shelly Harvey
  • Constance Leidy
Article

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.

Mathematics Subject Classification (2000)

Primary 57M25 Secondary 20J05 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Tim D. Cochran
    • 1
  • Shelly Harvey
    • 1
  • Constance Leidy
    • 2
  1. 1.Department of Mathematics, MS-136Rice UniversityHoustonUSA
  2. 2.Department of MathematicsWesleyan University, Wesleyan StationMiddletownUSA

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