Two curves or proper arcs are in minimal position when their number of transverse intersection points is the least possible among representatives of their isotopy classes.
Two transverse essential simple curves or proper arcs on a surface are in minimal position exactly when they form no bigon. For proper arcs, the proof uses the compactification of the universal cover to control their ends at punctures.

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  1. Essential proper arc on a punctured surface
  2. Mapping class group
  3. Topological surface
  4. Topology
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