A proper arc on a punctured surface approaches punctures at both ends. It is simple when its interior is embedded, and essential when it cannot be isotoped, relative to its ends, into a puncture.
Two transverse arcs form a bigon when subarcs between two consecutive intersection points together bound an embedded disc whose interior is disjoint from both arcs. Pushing one side across the disc removes the two corner intersections.
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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