Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 156 / 4 / c / Solution

Write the genus-two surface aswhere $\alpha$ is separating and each $T_i$ is a one-holed torus. In each $T_i$, choose two disjoint essential proper arcs $p_i,q_i$ from the boundary to itself that form a cut system, so $T_i\setminus (p_i\cup q_i)$ is a disc. Arrange their four endpoints on each copy of $\alpha$ and glue the boundaries so thatjoin cyclically into one simple closed curve $\beta$. After smoothing at the four gluing points, $i(\alpha ,\beta )=4$. Cutting along $\alpha$ and these four arcs leaves one disc from each $T_i$, so $S\setminus (\alpha \cup \beta )$ consists of two discs. Hence $(\alpha ,\beta )$ is the filling pair on a closed genus-two surface.

If their curve-complex distance were at most two, there would be an essential curve $\gamma$ disjoint from both: it would be the intermediate vertex of a length-two path. This contradicts filling. Therefore