Primary decomposition and the fractal nature of knot concordance

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.

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

References in corpus (4)

• 2-torsion in the n-solvable filtration of the knot concordance group
• Homology and Derived p-Series of Groups
• Knot concordance and Blanchfield duality
• Higher-order linking forms for 3-manifolds

Cited by in corpus (29)

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• Injectivity of satellite operators in knot concordance
• 2-torsion in the n-solvable filtration of the knot concordance group
• Satellites of Infinite Rank in the Smooth Concordance Group
• Satellite operators as group actions on knot concordance
• Satellite operators with distinct iterates in smooth concordance
• The geometry of the knot concordance space
• On the Upsilon invariant and satellite knots
• Shake slice and shake concordant knots
• Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders
• Grope metrics on the knot concordance set
• Splittings of von Neumann rho-invariants of knots
• A second order algebraic knot concordance group
• Higher-order linking forms for 3-manifolds
• Topological concordance of knots in homology spheres and the solvable filtration
• 2-torsion in the grope and solvable filtrations of knots
• A colored operad for string link infection
• Cut open null-bordisms and derivatives of slice knots
• Whitney towers, Gropes and Casson-Gordon style invariants of links
• Slice knots which bound punctured Klein bottles
• Polynomial splittings of correction terms and doubly slice knots
• Legendrian Satellites and Decomposable Concordances
• Linear independence of cables in the knot concordance group
• Knots having the same Seifert form and primary decomposition of knot concordance
• Linear Independence of Knots Arising from Iterated Infection Without the Use of Tristram Levine Signatures
• The Role Of Link Concordance In Knot Concordance
• Strong coprimality and strong irreducibility of Alexander polynomials
• Amenable signatures, algebraic solutions, and filtrations of the knot concordance group
• Action of the Mazur pattern up to topological concordance