If we have α and β be simple closed curves on a surface Σg. The intersection number i(α,β) is defined to be the minimal cardinality of α1 ∩ β1 as α1 and β1 ranges over all simple closed curves isotopic to α and β, respectively. We say α and β intersect minimally if i(α, β) = |α ∩ β|.

How to see thatα and β intersect minimally if there are no pairs of p, q ∈ α ∩ β such that the arc joining p to q along α followed by the arc from q back to p along β bounds a disk in Σg? (maybe a sketch of the idea of proof?)

I think the converse is also true : “thatα and β intersect minimally only if there are no pairs of p, q ∈ α ∩ β such that the arc joining p to q along α followed by the arc from q back to p along β bounds a disk in Σg.”