Algorithmic unsolvability of the triviality problem for multidimensional knots


We prove that for any fixedn≥3 there is no algorithm deciding whether or not a given knotf: S n→ℝn+2 is trivial. Some related results, are also presented.

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  1. [AC]S. Aanderaa andD. E. Cohen,Modular machines, the word problem for finitely presented groups and Collins' theorem, inWord problems II, eds. S. I. Adian, W. W. Boone, G. Higman, North-Holland, 1980, 1–16.

  2. [ABB]F. Acquistapace, R. Benedetti andF. Broglia,Effectiveness-non-effectiveness in semi-algebraic and PL geometry, Inv. Math.102 (1) (1990), 141–156.

    Article  MathSciNet  MATH  Google Scholar 

  3. [BCR]J. Bochnak, M. Coste andM.-F. Roy,Géometrie algébrique réelle, Springer, 1987.

  4. [BJ]G. Boolos andR. Jeffrey,Computability and Logic, Third Edition, Cambridge University Press, 1989.

  5. [BHP]W. Boone, W. Haken andV. Poenaru,On recursively unsolvable problems in topology in topology and their classification, inContributions to Mathematical Logic, eds. H. Arnold Schmidt, K. Schutte and H.-J. Thiele, North-Holland, 1968.

  6. [G]M. Gromov,Asymptotic invariants of infinite groups, inGeometric Group Theory, vol. 2, eds. G. Niblo and M. Roller, Cambridge University Press, 1993.

  7. [Hf]A. Haefliger,Differentiable embeddings of S n inS n+q forq>2, Ann. Math.83 (1966), 402–436.

    Article  MathSciNet  Google Scholar 

  8. [H]W. Haken,Theorie der Normalflächen, Acta Math.105 (1961), 245–375.

    Article  MathSciNet  MATH  Google Scholar 

  9. [K]M. Kervaire,On higher dimensional knots, inDifferential and Combinatorial Topology, ed. S. Cairns, Princeton Univ. Press, 1965, 105–120.

  10. [K2]M. Kervaire,Smoth homology spheres and their fundamental groups, Trans. Amer. Math. Soc.144 (1969), 67–72.

    Article  MathSciNet  MATH  Google Scholar 

  11. [K3]M., Kervaire,Multiplicateurs de Schur et K-theorie, inEssays on Topology and Related Topics, ed. A. Haefliger and R. Narasimhan, Springer, 1970, pp. 212–225.

  12. [L]J. Levine,A classification of differnetiable knots, Ann. Math. 82 (1965), 15–50.

    Article  Google Scholar 

  13. [M]C. F. Miller,Decision problems for groups-survey and reflections, inAlgorithms and Classification in Combinatorial Group Theory, eds. G. Baumslag and C. F. Miller, Springer, 1989.

  14. [Mi]J. Milnor,Introduction to algebraic K-theory, Ann. of Math. Studies, Princeton University Press, 1971.

  15. [N0]A. Nabutovsky,Non-recursive functions in real algebraic geometry, Bull. Amer. Math. Soc.20 (1989), 61–65.

    MathSciNet  Article  MATH  Google Scholar 

  16. [N1]A. Nabutovsky,Einstein structures: existence versus uniqueness, Geom. Funct. Analysis,5 (1) (1995) 76–91.

    Article  MathSciNet  MATH  Google Scholar 

  17. [N2]A. Nabutovsky,Non-recursive functions, knots “with thick ropes” and self-clenching “thick” hypersurfaces, Comm. on Pure and Appl. Math.48 (1995), 381–428.

    MathSciNet  MATH  Article  Google Scholar 

  18. [N3]A. Nabutovsky,Geometry of the space of triangulations of a compact manifolds, to appear in Comm. Math. Phys.

  19. [N4]A. Nabutovsky,Disconnectedness of sublevel sets of some Riemannian functionals, to appear in Geom. Funct. Analysis.

  20. [N5]A. Nabutovsky,Funndamental group and contractible closed geodesics, to appear in Comm. on Pure and Appl. Math.

  21. [R]J. J. Rotman,An Introduction to the Theory of Groups, Allyn and Bacon, Boston, 1984.

    MATH  Google Scholar 

  22. [Ros]J. Rosenberg,Algebraic K-theory and its Applications, Springer, 1994.

  23. [S] J. L. Shaneson, Wall's surgery obstruction groups forG × ℤ, Ann. Math.90 (1969), 296–334.

    Article  MathSciNet  Google Scholar 

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Both authors were partially supported by NSF grants.

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Nabutovsky, A., Weinberger, S. Algorithmic unsolvability of the triviality problem for multidimensional knots. Commentarii Mathematici Helvetici 71, 426–434 (1996).

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  • Fundamental Group
  • Word Problem
  • Turing Machine
  • Normal Closure
  • Homology Sphere