If we have α and β be simple closed curves $\alpha$ and $\beta$ on a surface Σg. The$\Sigma_g$, the intersection number i(α,β)$i(\alpha ,\beta)$ is defined to be the minimal cardinality of α1 ∩ β1$\alpha_1\cap\beta_1$ as α1$\alpha_1$ and β1 ranges$\beta_1$ range over all simple closed curves isotopic to α$\alpha$ and β$\beta$, respectively. We say α$\alpha$ and β$\beta$ intersect minimally if i(α, β) = |α ∩ β| $i(\alpha ,\beta) = |\alpha\cap\beta|\,$.
How to see thatαthat $\alpha$ and β$\beta$ intersect minimally if there are no pairs of p, q ∈ α ∩ β$p,q\in\alpha\cap\beta$ such that the arc joining p$p$ to q$q$ along α$\alpha$ followed by the arc from q$q$ back to p$p$ along β$\beta$ bounds a disk in Σg? (maybe$\Sigma_g$?
Maybe a sketch of the idea of proof idea?)
I think the converse is also true : "thatα"that $\alpha$ and β$\beta$ intersect minimally only if there are no pairs of p, q ∈ α ∩ β$p,q\in\alpha\cap\beta$ such that the arc joining p$p$ to q$q$ along α$\alpha$ followed by the arc from q$q$ back to p$p$ along β$\beta$ bounds a disk in Σg$\Sigma_g$."
