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Two transverse essential simple curves or proper arcs on a surface are in <minimal position of curves or arcs>[minimal position] exactly when they form no <bigon formed by two arcs>[bigon]. For proper arcs, the proof uses the compactification of the universal cover to control their ends at punctures.

Bigon criterion

The arcs $\alpha$ and $\beta$ bound a <bigon formed by two arcs>[bigon] if there are subarcs with common endpoints whose union is the boundary of an embedded disc and whose interiors are disjoint. They are in <minimal position of curves or arcs>[minimal position] if $|\alpha\cap\beta|$ is the least possible intersection count among proper arcs isotopic to $\alpha$ and $\beta$ relative to their ends.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 156 / 2 / a / Solution

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.


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 Incoming links: Bigon formed by two arcs