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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.

The geometric intersection number $i(\alpha ,\beta )$ is the minimum number of transverse intersection points among representatives of the isotopy classes of $\alpha$ and $\beta$. Representatives in minimal position realize it.

A sufficient condition is that both collections are Alexander systems: within each collection the essential simple closed curves are pairwise nonisotopic, are in pairwise minimal position of curves or arcs, have no triple intersection points, and no three curves intersect pairwise. If ${\alpha }_i$ is isotopic to ${\beta }_i$ for every $i$, the simultaneous-isotopy lemma for Alexander systems gives an ambient isotopy $H_t$ with $H_1({\alpha }_i)={\beta }_i$ for all $i$. Successive applications of the bigon criterion prove the lemma while preserving the curves already matched.

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.