Intersections of curves on surfaces

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.