Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-05-22T10:09:12.634Z Has data issue: false hasContentIssue false
  • English
  • Français

Roll-spun knots

Published online by Cambridge University Press:  24 October 2008

Masakazu Teragaito
Masakazu Teragaito
Affiliation:
Department of Mathematics, Kobe University, Japan
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we will study 2-knots in the 4-sphere S4 which are obtained by roll-spinning 1-knots in the 3-sphere S3. The process of spinning was introduced by Artin[2] and generalized to twist-spinning by Fox [7] and Zeeman [20]. In [7] Fox introduced another variation of the spinning process, called roll-spinning. He only showed that the roll-spun figure-eight knot cannot be obtained by twist-spinning the figure-eight knot. Finally Litherland [14] described the general process of deform-spinning, and he gave a precise definition of roll-spinning. We remark that Fox's roll-spinning is not the same as Litherland's: Fox's roll-spun figure-eight knot is the symmetry-spun figure-eight knot in terms of [14].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 1 , January 1993 , pp. 91 - 96
Copyright
Copyright © Cambridge Philosophical Society 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Andrews, J. J. and Sumners, D. W.. On higher-dimensional fibered knots. Trans. Amer. Math. Soc. 153 (1971), 415–426.CrossRefGoogle Scholar
[2]Artin, E.. Zur Isotopie zwei dimensionaler Flächen im R4. Abh. Math. Sem. Univ. Hamburg 4 (1925), 47–72.CrossRefGoogle Scholar
[3]Burde, G. and Zieschang, H.. Knots. Studies in Math. no. 5 (de Gruyter, 1985).Google Scholar
[4]Cochran, T.. Ribbon knots in,S4. J. London Math. Soc. (2) 28 (1983), 563–576.CrossRefGoogle Scholar
[5]Crowell, R. H. and Fox, R. H.. Introduction to Knot Theory. Graduate Texts in Math. no. 57 (Springer-Verlag, 1977).CrossRefGoogle Scholar
[6]Feustel, C. D. and Whitten, W.. Groups and complements of knots. Canad. J. Math. 30 (1978), 1284–1295.CrossRefGoogle Scholar
[7]Fox, R. H.. Rolling. Bull. Amer. Math. Soc. 72 (1966), 162–164.CrossRefGoogle Scholar
[8]Gabai, D.. Foliations and the topology of 3-manifolds. III. J. Differential Geom. 26 (1987), 479–536.Google Scholar
[9]Heil, W.. Elementary surgery on Seifert fiber spaces. Yokohama Math. J. 22 (1974), 135–139.Google Scholar
[10]Hempel, J.. 3-Manifolds. Ann. of Math. Studies no. 86 (Princeton University Press, 1976).Google Scholar
[11]Hillman, J. A.. 2-Knots and their Groups. Austral. Math. Soc. Lecture Series no. 5 (Cambridge University Press, 1989).Google Scholar
[12]Jaco, W.. Lectures on Three-manifold Topology. Regional Conf. Ser. in Math. no. 43 (American Mathematical Society, 1980).CrossRefGoogle Scholar
[13]Kanenobu, T.. Untwisted deform-spun knots; Examples of symmetry-spun 2-knots. In Transformation Groups, Lecture Notes in Math. vol. 1375 (Springer-Verlag, 1990), pp. 145–167.CrossRefGoogle Scholar
[14]Litherland, R. A.. Deforming twist-spun knots. Trans. Amer. Math. Soc. 250 (1979), 311–331.CrossRefGoogle Scholar
[15]Rolfsen, D.. Knots and Links. Math. Lecture Series no. 7 (Publish or Perish, 1976).Google Scholar
[16]Scharlemann, M.. Sutured manifolds and generalized Thurston norms. J. Differential Geom. 29 (1989), 557–614.CrossRefGoogle Scholar
[17]Waldhausen, F.. On irreducible 3-manifolds which are sufficiently large. Ann. of Math. 87 (1968), 56–88.CrossRefGoogle Scholar
[18]Waldhausen, F.. Gruppen mit Zentrum und 3-dimensionaler Mannigfaltigkeiten. Topology 6 (1967), 505–517.CrossRefGoogle Scholar
[19]Yanagawa, T.. Knot-groups of higher dimensional ribbon knots. Math. Sem. Notes Kobe Univ. 8 (1980), 573–591.Google Scholar
[20]Zeeman, E. C.. Twisting spun knots. Trans. Amer. Math. Soc. 115 (1965), 471–495.CrossRefGoogle Scholar