Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-28T15:38:19.433Z Has data issue: false hasContentIssue false
  • English
  • Français

Ribbon knot families via stallings' twists

Part of: Low-dimensional topology

Published online by Cambridge University Press:  09 April 2009

L. Richard Hitt  and
Daniel S. Silver
L. Richard Hitt
Affiliation:
University of South AlabamaMobile, Alabama 36688, U.S.A.
Daniel S. Silver
Affiliation:
University of South AlabamaMobile, Alabama 36688, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We investigate the effect on the Jones polynomial of a ribbon knot when two of its bands are twisted together. We use our results to prove that each of the three S-equivalence classes of genus 2 fibered doubly slice knots in S3 can be represented by infinitely many distinct prime fibered doubly slice ribbon knots.

Keywords

knotJones polynomialfibered knotdoubly slice knot

MSC classification

Secondary: 57M25: Knots and links in $S^3$
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 3 , June 1991 , pp. 356 - 372
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Aitchison, I. R., ‘Isotoping and twisting knots and ribbons’, (Ph.D. Dissertation, Univ. of California, Berkeley, 1984).Google Scholar
[2]Aitchison, I. R. and Rubinstein, J. H., ‘Fibered knots and involutions on homotopy spheres’, Four-Manifold Theory, pp. 1–74, (Amer. Math. Soc., Providence, R.I., 1985).Google Scholar
[3]Aitchison, I.R. and Silver, D.S., ‘On fibering certain ribbon disk pairs’, Trans. Amer. Math. Soc. 306 (1988), 529–551.CrossRefGoogle Scholar
[4]Akbulut, S. and Kirby, R., ‘An involution of S4’, Topology 18 (1970), 75–81.CrossRefGoogle Scholar
[5]Bonahon, F., ‘Ribbon fibred knots, cobordism of surface diffeomorphisms, and pseudo-Anosov diffeomorphisms,’ Math. Proc. Cambridge Phil. Soc. 94 (1983), 235–251.CrossRefGoogle Scholar
[6]Casson, A. J. and Gordon, C. M., ‘A loop theorem for duality spaces and fibered ribbon knots,’ Invent. Math. 74 (1983), 119–137.CrossRefGoogle Scholar
[7]Freyd, P., Yetter, D., Hoste, J., Lickorish, W. B. R., Millett, K., and Ocneanu, A., ‘A new polynomial invariant of knots and links’, Bull. Amer. Math. Soc. 12 (1985), 239–246.CrossRefGoogle Scholar
[8]Gordon, C. M., ‘Some aspects of classical knot theory,’ Knot Theory, (Lecture Notes in Math. 685, Springer-Verlag, 1978).Google Scholar
[9]Jones, V. F. R., ‘A new knot polynomial and von Neumann algebras,’, Notices Amer. Math. Soc. 33 (1986), 219–225.Google Scholar
[10]Kanenobu, T., ‘Module d'Alexander des noeuds fibrés et polynôme de Hosokawa des lacements fibrés,’ Math. Sem. Notes Kobe Univ. 9 (1981), 75–84.Google Scholar
[11]Kanenobu, T., ‘Infinitely many knots with the same polynomial invariant,’ Proc. Amer. Math. Soc. 97 (1986), 158–162.CrossRefGoogle Scholar
[12]Kanenobu, T., ‘Examples on polynomial invariants of knots and links,’ Math. Ann. 275 (1986), 555–572.CrossRefGoogle Scholar
[13]Kauffman, L. H., ‘State models for knot polynomials,’ preprint.Google Scholar
[14]Levine, J., ‘Doubly slice knots and doubled disk knots,’ Michigan Math. J. 30 (1983), 249–256.CrossRefGoogle Scholar
[15]Long, D. D. and Morton, H. R., ‘Hyperbolic 3-manifolds and surface automorphisms,’ Topology 25 (1986), 575–583.CrossRefGoogle Scholar
[16]Morton, H. R., ‘Infinitely many fibered knots having the same Alexander polynomial,’ Topology 17 (1978), 101–104.CrossRefGoogle Scholar
[17]Morton, H. R., ‘Fibered knots with a given Alexander polynomial,’ L'Enseignment Math. 31 (1983), 205–222.Google Scholar
[18]Neuwirth, L. P., ‘The status of some problems related to knot groups,’ Topology Conference VPISU, (Lecture Notes Math 375, pp. 209–230, Springer-Verlag, 1974).CrossRefGoogle Scholar
[19]Rolfsen, D., Knots and Links, (Math. Lecture Ser. 7, Publish or Perish, Berkeley, 1976).Google Scholar
[20]Stallings, J., ‘Constructions of fibered knots and links,’ Proc. Sympos. Pure Math., vol. 32, pp. 55–60, (Amer. Math. Soc., Providence, R.I., 1978).Google Scholar
[21]Sumners, D. W., ‘Invertible knot cobordisms,’ Comment. Math. Helv. 46 (1971), 240–256.CrossRefGoogle Scholar