Skip to main content
SpringerLink
Log in
Book cover

Transformation Groups pp 145–167Cite as

Untwisted deform-spun knots: Examples of symmetry-spun 2-knots

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1375))

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J.J. Andrews and D.W. Sumners: On higher-dimensional fibered knots, Trans. Amer. Math. Soc. 153 (1971), 415–426.

    Article  MathSciNet  MATH  Google Scholar 

  2. E. Artin: Zur Isotopie zweidimensionaler Flächen im R4, Hamburg Abh. 4 (1925), 174–177.

    Article  MathSciNet  Google Scholar 

  3. F. Bonahon: Diffeotopies des espaces lenticulaires, Topology 22 (1983), 305–314.

    Article  MathSciNet  MATH  Google Scholar 

  4. J.H. Conway: An enumeration of knots and links, Computational problems in abstract algebra (J. Leech, ed.), Pergamon Press 1969, 329–358.

    Google Scholar 

  5. H.S.M. Coxeter and W.O.J. Moser: Generators and Relations for Discrete Groups, Springer-Verlag, Berlin-Heidelberg-New York, 1980.

    Book  MATH  Google Scholar 

  6. R.H. Crowell and R.H. Fox: Introduction to Knot Theory, Grad. Texts in Math., vol.57, Springer-Verlag, New York and Berlin, 1977.

    Book  MATH  Google Scholar 

  7. R.H. Fox: A quick trip through knot theory, Topology of 3-Manifolds and Related Topics (M.K. Fort, Jr. (ed.)), Prentice-Hall, Englewood Cliffs, 1962, pp.120–167.

    Google Scholar 

  8. R.H. Fox: Rolling, Bull. Amer. Math. Soc. 72 (1966), 162–164.

    Article  MathSciNet  MATH  Google Scholar 

  9. C.H. Giffen: The generalized Smith conjecture, Amer. J. Math. 88 (1966), 187–198.

    Article  MathSciNet  MATH  Google Scholar 

  10. D.L. Goldsmith: Symmetric fibered links, Knots, Groups and 3-Manifolds (L.P. Neuwirth, ed.), Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975, 3–23.

    Google Scholar 

  11. D.L. Goldsmith and L.H. Kauffman: Twist spinning revisited, Trans. Amer. Math. Soc. 239 (1978), 229–251.

    Article  MathSciNet  MATH  Google Scholar 

  12. C. McA. Gordon: On the higher-dimensional Smith conjecture, Proc. London Math. Soc. (3) 29 (1974), 98–110.

    Article  MathSciNet  MATH  Google Scholar 

  13. C.McA. Gordon: Knots in the 4-sphere, Comment. Math. Helv. 51 (1976), 585–596.

    Article  MathSciNet  MATH  Google Scholar 

  14. J.A. Hillman: 2-Knots and their groups, preprint.

    Google Scholar 

  15. L.R. Hitt: Examples of higher-dimensional slice knots which are not ribbon knots, Proc. Amer. Math. Soc. 77

    Google Scholar 

  16. C. Hodgson and J. H. Rubinstein: Involutions and isotopies of lens spaces, Knot Theory and Manifolds (D.Rolfsen, ed.), Lecture Notes in Mathematics No. 1144, Springer-Verlag, 1985, 60–96.

    Google Scholar 

  17. T. Kanenobu: 2-knot groups with elements of finite order, Math. Sem. Notes Kobe Univ. 8 (1980) 557–560.

    MathSciNet  MATH  Google Scholar 

  18. T. Kanenobu: Groups of higher dimensional satellite knots, J. Pure Appl. Algebra 28 (1983) 179–188.

    Article  MathSciNet  MATH  Google Scholar 

  19. T. Kanenobu: Higher dimensional cable knots and their finite cyclic covering spaces, Topology Appl. 19 (1985) 123–127.

    Article  MathSciNet  MATH  Google Scholar 

  20. T. Kanenobu: Deforming twist spun 2-bridge knots of genus one, Proc. Japan Acad. Ser. A 64 (1988) 98–101.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Kawauchi, T. Shibuya and S. Suzuki: Descriptions on surfaces in four-space I, Normal forms, Math. Sem. Notes Kobe Univ. 10 (1982), 75–125.

    MathSciNet  MATH  Google Scholar 

  22. R.A. Litherland: Deforming twist-spun knots, Trans. Amer. Math. Soc. 250 (1979), 311–331.

    Article  MathSciNet  MATH  Google Scholar 

  23. S.J. Lomonaco, Jr.: The homotopy groups of knots. I. How to compute the algebraic 2-type, Pacific J. Math. 95 (1981), 349–390.

    Article  MathSciNet  MATH  Google Scholar 

  24. W. Magnus, A. Karrass and D. Solitar: Combinatorial Group Theory, Pure and Applied Math., vol. 13, Interscience, 1966.

    Google Scholar 

  25. J. Milnor: On the 3-dimensional Brieskorn manifold M(p,q,r), Knots, Groups and 3-Manifolds (L.P. Neuwirth, ed.), Ann. of Math. Studies, No. 84, Princeton Univ. Press, Princeton, N.J., 1975, 175–225.

    Google Scholar 

  26. J.M. Montesinos: 4-Manifolds, 3-fold covering spaces and ribbons, Trans. Amer. Math. Soc., 245 (1978), 453–467.

    MathSciNet  MATH  Google Scholar 

  27. J.M. Montesinos: On twins in the four-sphere I, Quart. J. Math. Oxford (2), 4 (1983), 171–199; II, ibid., 35 (1984), 73–83.

    Article  MathSciNet  MATH  Google Scholar 

  28. L. Neuwirth: The algebraic determination of the genus of a knot, Amer. J. Math. 82 (1960), 791–798.

    Article  MathSciNet  MATH  Google Scholar 

  29. W.D. Newmann and F. Raymond: Seifert manifolds, plumbing, μ-invariant and orientation reversing maps, Algebraic and Geometric Topology (K.C.Millett, ed.), Lecture Notes in Mathematics No.664, Springer-Verlag, 1978, 163–196.

    Google Scholar 

  30. P.S. Pao: Nonlinear circle actions on the 4-sphere and twisting spun knots, Topology 17 (1978), 291–296.

    Article  MathSciNet  MATH  Google Scholar 

  31. S.P. Plotnick: Fibered knots in S4 — twisting, spinning, rolling, surgery, and branching, Contemporary Math. 35 (1984), 437–459.

    Article  MathSciNet  MATH  Google Scholar 

  32. D. Rolfsen: Knots and Links, Math. Lecture Series no.7, Publish or Perish Inc., Berkeley, 1976.

    MATH  Google Scholar 

  33. C.P. Rourke and B.J. Sanderson: Introduction to Piecewise-Linear Topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 69, Springer-Verlag, Berlin-Heidelberg-New York, 1972.

    Google Scholar 

  34. H. Seifert: Topologie dreidimensionaler gefaserter Räume, Acta Math. 60 (1933), 147–238.

    Article  MathSciNet  MATH  Google Scholar 

  35. D.W. Sumners: Smooth Zp-actions on spheres which leave knots pointwise fixed, Trans. Amer. Math. Soc. 205 (1975), 193–203.

    MathSciNet  MATH  Google Scholar 

  36. S. Suzuki: Knotting problems of 2-spheres in 4-sphere, Math. Sem. Notes Kobe Univ. 4 (1976), 241–375.

    MathSciNet  MATH  Google Scholar 

  37. M. Teragaito: Twisting and rolling, Proc. of a symp. held at the R.I.M.S., Kyoto Univ. No.636 (1988), 153–169 (in Japanese).

    Google Scholar 

  38. M. Teragaito: On fibered knots, Master Thesis, Kobe Univ. (1988) (in Japanese).

    Google Scholar 

  39. M. Teragaito: Fibered 2-knots and lens spaces, preprint.

    Google Scholar 

  40. A.M. Vinogradov and M.S. Kushel’man: The generalized Smith conjecture in four dimensions, Siberian Math. J. 13 (1972), 36–45 (Translated from Sibirskii Mathematicheskii Zhurnal 13 (1972), 52–62).

    Article  MATH  Google Scholar 

  41. K. Yoshikawa: On 2-knot groups with the finite commutator subgroup, Math. Sem. Notes, Kobe Univ. 8 (1980), 321–330.

    MathSciNet  MATH  Google Scholar 

  42. K. Yoshikawa: A letter to the author, 1981.

    Google Scholar 

  43. K. Yoshikawa: On a 2-knot group with nontrivial center, Bull. Australian Math. Soc. 25 (1982), 321–326.

    Article  MathSciNet  MATH  Google Scholar 

  44. E.C. Zeeman: Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471–495.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kyushu University, 812, Fukuoka

    Taizo Kanenobu

Authors
  1. Taizo Kanenobu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Katsuo Kawakubo

Additional information

Dedicated to Professor Junzo Tao on his 60th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag

About this paper

Cite this paper

Kanenobu, T. (1989). Untwisted deform-spun knots: Examples of symmetry-spun 2-knots. In: Kawakubo, K. (eds) Transformation Groups. Lecture Notes in Mathematics, vol 1375. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085606

Download citation

  • DOI: https://doi.org/10.1007/BFb0085606

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51218-9

  • Online ISBN: 978-3-540-46178-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation