Skip to main content
SpringerLink
Log in

Legendrian Knots in Overtwisted Contact Structures on S3

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We study the problem of classifying Legendrian knots in overtwisted contact structures on S 3. The question is whether topologically isotopic Legendrian knots have to be Legendrian isotopic if they have equal values of the well-known invariants rot and tb. We give positive answer in the case that there is an overtwisted disc intersecting none of the knots and we construct an example of a knot intersecting each overtwisted disc (this provides a counterexample to the conjecture of Eliashberg). Our proof needs some results on the structure of the group of contactomorphisms of S 3. We divide the subgroup Cont+(S 3, ξ) of coorientation-preserving contactomorphisms for an overtwisted contact distribution ξ into two classes.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bennequin, D.: Entrelacements et équations de Pfaff, Asterisque 107-108 (1983), 87-161.

    Google Scholar 

  2. Eliashberg, Y: Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989), 623-637.

    Google Scholar 

  3. Eliashberg, Y. and Fraser, M.: Classification of topologically trivial Legendrian knots, in: F. Lalonde (ed.), Geometry, Topology and Dynamics, CRM Proc. Lecture Notes, Amer.Math. Soc., Providence, 1998, pp. 17-51.

    Google Scholar 

  4. Fuchs, D. and Tabachnikov, S.: Invariants of Legendrian and transverse knots in the standard contact space, Topology 36 (1997), 1025-1053.

    Google Scholar 

  5. Gray, J. W.: Some global properties of contact structures, Ann. Math. 69 (1959), 421-450.

    Google Scholar 

  6. Hatcher, A.: A proof of Smale conjecture, Diff(S 3) ? O(4), Ann. Math. 117 (1983), 553-607.

    Google Scholar 

  7. Lutz, R.: Structures de contact sur les fibrés principaux en cercles de dimension 3. Ann. Inst. Fourier 3 (1977), 1-15.

    Google Scholar 

  8. ?wiatkowski, J.: On the isotopy of Legendrian knots. Ann. Global Anal. Geom. 10 (1992), 195-207.

    Google Scholar 

  9. Tchernov, V.: Finite order invariants of Legendrian, transverse, and framed knots in contact 3-manifolds, http://xxx.lanl.gov/abs/math.SG/9907118 (1999).

Download references

Author information

Authors and Affiliations

  1. Instytut Matematyczny, Uniwersytet Wrocławski, Plac Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Katarzyna Dymara

Authors
  1. Katarzyna Dymara
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dymara, K. Legendrian Knots in Overtwisted Contact Structures on S3. Annals of Global Analysis and Geometry 19, 293–305 (2001). https://doi.org/10.1023/A:1010706529508

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1010706529508

Search

Navigation