The Maximal Number of Transverse Self-Intersections of Geodesics ...

a
b
b
a
Now, we can represent curves as combinations of a’s and b’s by drawing them as directed
horizontal and vertical lines on this rectangle, as shown in the next two figures. Note that
A and B are lines of the same type, drawn in the opposite direction. Note that a k and b k
can be represented as in the diagrams with complete k − 1 horizontal or vertical segments
and two partial horizontal or vertical segments, respectively.
a
b
.
.
.
b
a
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