Mathematische Annalen

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

Primary decomposition and the fractal nature of knot concordance

  • Tim D. CochranEmail author
  • Shelly Harvey
  • Constance Leidy


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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  1. 1.
    Bartholdi, L., Grigorchuk, R., Nekrashevych, V.: From fractal groups to fractal sets. In: Fractals in Graz 2001. Trends Math., pp. 25–118. Birkhäuser, Basel (2003)Google Scholar
  2. 2.
    Casson, A.J., Gordon, C.McA.: On slice knots in dimension three. In: Algebraic and Geometric Topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, pp. 39–53. Amer. Math. Soc., Providence (1978)Google Scholar
  3. 3.
    Casson, A.J., Gordon, C.McA.: Cobordism of classical knots. In: À la recherche de la topologie perdue. Progr. Math., vol. 62, pp. 181–199. Birkhäuser Boston, Boston (1986). With an appendix by P. M. GilmerGoogle Scholar
  4. 4.
    Casson, A., Freedman, M.: Atomic surgery problems. In: Four-Manifold Theory (Durham, N.H., 1982). Contemp. Math., vol. 35, pp. 181–199. Amer. Math. Soc., Providence (1984)Google Scholar
  5. 5.
    Cha J.C.: The structure of the rational concordance group of knots. Mem. Am. Math. Soc. 189(885), x+95 (2007)MathSciNetGoogle Scholar
  6. 6.
    Cha J.C.: Injectivity theorems and algebraic closures of groups with coefficients. Proc. Lond. Math. Soc. (3) 96(1), 227–250 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cheeger J., Gromov M.: Bounds on the von Neumann dimension of L 2-cohomology and the Gauss-Bonnet theorem for open manifolds. J. Differ. Geom. 21(1), 1–34 (1985)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Cochran T., Harvey S.: Homology and derived series of groups. Geom. Topol. 9, 2159–2191 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cochran T., Harvey S., Leidy C.: Link concordance and generalized doubling operators. Algebr. Geom. Topol. 8, 1593–1646 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cochran T.D.: Noncommutative knot theory. Algebr. Geom. Topol. 4, 347–398 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cochran T.D., Friedl S., Teichner P.: New constructions of slice links. Comment. Math. Helv. 84, 617–638 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cochran T.D., Harvey S.: Homology and derived series of groups. Part II: Dwyer’s theorem. Geom. Topol. 12(1), 199–232 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cochran, T.D., Harvey, S., Leidy, C.: 2-Torsion in the n-solvable filtration of the knot concordance group. Proc. Lond. Math. Soc. (2010). doi: 10.1112/plms/pdq020
  14. 14.
    Cochran, T.D., Harvey, S., Leidy, C.: Knot concordance and Blanchfield duality. Preprint.
  15. 15.
    Cochran, T.D., Harvey, S., Leidy, C.: Knot concordance and Blanchfield duality. In: Oberwolfach Reports; European Mathematical Society Publishing House, vol. 3, issue 3 (2006)Google Scholar
  16. 16.
    Cochran T.D., Harvey S., Leidy C.: Knot concordance and higher-order Blanchfield duality. Geom. Topol. 13, 1419–1482 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cochran T.D., Kim T.: Higher-order Alexander invariants and filtrations of the knot concordance group. Trans. Am. Math. Soc. 360(3), 1407–1441 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cochran T.D., Orr K.E., Teichner P.: Knot concordance, Whitney towers and L 2-signatures. Ann. Math. (2) 157(2), 433–519 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cochran T.D., Orr K.E., Teichner P.: Structure in the classical knot concordance group. Comment. Math. Helv. 79(1), 105–123 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cochran T.D., Teichner P.: Knot concordance and von Neumann ρ-invariants. Duke Math. J. 137(2), 337–379 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fox R.H., Milnor J.W.: Singularities of 2-spheres in 4-space and cobordism of knots. Osaka J. Math. 3, 257–267 (1966)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Freedman, M.H., Quinn, F.: Topology of 4-manifolds. In: Princeton Mathematical Series, vol. 39. Princeton University Press, Princeton (1990)Google Scholar
  23. 23.
    Friedl S.: Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants. Algebr. Geom. Topol. 4, 893–934 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gilmer P., Livingston C.: Discriminants of Casson-Gordon invariants. Math. Proc. Camb. Philos. Soc. 112(1), 127–139 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gilmer P.M.: Slice knots in S 3. Quart. J. Math. Oxford Ser. (2) 34(135), 305–322 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gordon, C.McA.: Some aspects of classical knot theory. In: Knot Theory (Proc. Sem., Plans-sur-Bex, 1977). Lecture Notes in Math., vol. 685, pp. 1–60. Springer, Berlin (1978)Google Scholar
  27. 27.
    Harvey S.L.: Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm. Topology 44(5), 895–945 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Harvey S.L.: Monotonicity of degrees of generalized Alexander polynomials of groups and 3-manifolds. Math. Proc. Camb. Philos. Soc. 140(3), 431–450 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Harvey S.L.: Homology cobordism invariants and the Cochran-Orr-Teichner filtration of the link concordance group. Geom. Topol. 12(1), 387–430 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hillman, J.A.: Alexander ideals of links. Lecture Notes in Mathematics, vol. 895. Springer-Verlag, Berlin (1981)Google Scholar
  31. 31.
    Jiang B.J.: A simple proof that the concordance group of algebraically slice knots is infinitely generated. Proc. Am. Math. Soc. 83(1), 189–192 (1981)CrossRefzbMATHGoogle Scholar
  32. 32.
    Kervaire M.A., Milnor J.W.: On 2-spheres in 4-manifolds. Proc. Natl. Acad. Sci. USA 47, 1651–1657 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kim S.-G.: Polynomial splittings of Casson-Gordon invariants. Math. Proc. Camb. Philos. Soc. 138(1), 59–78 (2005)CrossRefzbMATHGoogle Scholar
  34. 34.
    Kim S.-G., Kim T.: Polynomial splittings of metabelian von Neumann rho-invariants of knots. Proc. Am. Math. Soc. 136(11), 4079–4087 (2008)CrossRefzbMATHGoogle Scholar
  35. 35.
    Kim T.: Filtration of the classical knot concordance group and Casson-Gordon invariants. Math. Proc. Camb. Philos. Soc. 137(2), 293–306 (2004)CrossRefzbMATHGoogle Scholar
  36. 36.
    Kim T.: New obstructions to doubly slicing knots. Topology 45(3), 543–566 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Leidy, C.: Higher-order linking forms for 3-manifolds. PreprintGoogle Scholar
  38. 38.
    Leidy C.: Higher-order linking forms for knots. Comment. Math. Helv. 81(4), 755–781 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Leidy, C., Maxim, L.: Higher-order Alexander invariants of plane algebraic curves. Int. Math. Res. Not., Art. ID 12976, 23 (2006)Google Scholar
  40. 40.
    Leidy, C., Maxim, L.: Obstructions on fundamental groups of plane curve complements. In: Contemporary Mathematics, vol. 459, pp. 117–130 (2008)Google Scholar
  41. 41.
    Levine, J.P.: Signature invariants of homology bordism with applications to links. In: Knots 90 (Osaka, 1990), pp. 395–406. de Gruyter, Berlin (1992)Google Scholar
  42. 42.
    Levine J.: Knot modules. I. Trans. Am. Math. Soc. 229, 1–50 (1977)CrossRefzbMATHGoogle Scholar
  43. 43.
    Livingston, C.: A survey of classical knot concordance. In: Handbook of Knot Theory, pp. 319–347. Elsevier, Amsterdam (2005)Google Scholar
  44. 44.
    Lück, W., Schick, T.: Various L 2-signatures and a topological L 2-signature theorem. In: High-Dimensional Manifold Topology, pp. 362–399. World Scientific, River Edge (2003)Google Scholar
  45. 45.
    Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory, 2nd edn. Dover, Mineola (2004). Presentations of groups in terms of generators and relationsGoogle Scholar
  46. 46.
    Milnor, J.W.: Infinite cyclic coverings. In: Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967), pp. 115–133. Prindle, Weber & Schmidt, Boston (1968)Google Scholar
  47. 47.
    Passman, D.S.: The algebraic structure of group rings. In: Pure and Applied Mathematics. Wiley-Interscience, New York (1977)Google Scholar
  48. 48.
    Rolfsen, D.: Knots and links. Mathematics Lecture Series, vol. 7. Publish or Perish Inc., Houston (1990). Corrected reprint of the 1976 originalGoogle Scholar
  49. 49.
    Stallings J.: Homology and central series of groups. J. Algebra 2, 170–181 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Stenström, B.: Rings of Quotients. Springer-Verlag, New York (1975). Die Grundlehren der Mathematischen Wissenschaften, Band 217. An introduction to methods of ring theoryGoogle Scholar
  51. 51.
    Stoltzfus N.W.: Unraveling the integral knot concordance group. Mem. Am. Math. Soc. 12(192), iv+91 (1977)MathSciNetGoogle Scholar
  52. 52.
    Terasaka H.: On null-equivalent knots. Osaka Math. J. 11, 95–113 (1959)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Tim D. Cochran
    • 1
    Email author
  • 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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