Curve complex ($\mathcal{C}(S)$)

The vertices of the curve complex are isotopy classes of essential simple closed curves. Distinct vertices span a simplex when they have pairwise disjoint representatives.

For every connected orientable surface of complexity above one, the one-skeleton of the curve complex is connected. Surgery replaces one curve by an essential curve disjoint from it while reducing intersection with a fixed target; induction on geometric intersection number gives a path.

A collection of curves fills a surface when every essential simple closed curve intersects at least one member. For curves in minimal position on a closed surface, this is equivalent to every complementary component being a disc.

Split a closed genus-two surface along a separating curve $\alpha$ into two one-holed tori. In each torus choose two disjoint proper arcs that cut it into a disc, and match their four endpoints across $\alpha$ so that the four arcs join into one simple closed curve $\beta$. Then $i(\alpha ,\beta )=4$ and the complement of $\alpha \cup \beta$ is two discs, so the pair fills.