Israel Journal of Mathematics

, Volume 51, Issue 1–2, pp 90–120 | Cite as

Intersections of curves on surfaces

  • Joel Hass
  • Peter Scott
Article

Abstract

The authors consider curves on surfaces which have more intersections than the least possible in their homotopy class.

Theorem 1.Let f be a general position arc or loop on an orientable surface F which is homotopic to an embedding but not embedded. Then there is an embedded 1-gon or 2-gon on F bounded by part of the image of f.

Theorem 2.Let f be a general position arc or loop on an orientable surface F which has excess self-intersection. Then there is a singular 1-gon or 2-gon on F bounded by part of the image of f.

Examples are given showing that analogous results for the case of two curves on a surface do not hold except in the well-known special case when each curve is simple.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. H. Freedman, J. Hass and P. Scott,Closed geodesics on surfaces, Bull. London Math. Soc.14 (1982), 385–391.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    J. Hass and J. H. Rubinstein,One-sided closed geodesic on surfaces, University of Melbourne, preprint.Google Scholar

Copyright information

© Hebrew University 1985

Authors and Affiliations

  • Joel Hass
    • 1
  • Peter Scott
    • 2
  1. 1.Mathematics DepartmentUniversity of MichiganUSA
  2. 2.Department of Pure MathematicsUniversity of LiverpoolLiverpoolEngland

Personalised recommendations