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.
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References
M. H. Freedman, J. Hass and P. Scott,Closed geodesics on surfaces, Bull. London Math. Soc.14 (1982), 385–391.
J. Hass and J. H. Rubinstein,One-sided closed geodesic on surfaces, University of Melbourne, preprint.
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Hass, J., Scott, P. Intersections of curves on surfaces. Israel J. Math. 51, 90–120 (1985). https://doi.org/10.1007/BF02772960
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DOI: https://doi.org/10.1007/BF02772960