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

The arcs $\alpha$ and $\beta$ bound a 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 if $|\alpha \cap \beta |$ is the least possible intersection count among proper arcs isotopic to $\alpha$ and $\beta$ relative to their ends.