Filling pair on a closed genus-two surface

A concrete filling set of curves consisting of two simple closed curves can be built from four unit squares. Label the squares $1,2,3,4$. Glue the right side of square $i$ to the left side of square $h(i)$ and its top side to the bottom side of square $v(i)$ by translations, whereThis gives a connected oriented square-tiled surface. The horizontal midlines form one simple closed curve $\alpha$, because $h$ has one cycle of a permutation of length four; the vertical midlines form one simple closed curve $\beta$, because $v$ has one cycle of a permutation of length four. They meet once at each square center.

The corner identifications have two classes, corresponding to the cycles of a permutation ofwith $1$ fixed. Thus the square CW complex has $V=2$, $E=8$, $F=4$, and Euler characteristic $2-8+4=-2$. The classification of compact surfaces gives genus two. The complement of $\alpha \cup \beta$ consists of the two open discs around these vertex classes, so the pair fills.

Orient both curves in their positive coordinate directions. All four crossings have the same local sign, giving algebraic intersection number of curves on an oriented surface $\alpha \cdot \beta =4$. Therefore $i(\alpha ,\beta )\ge 4$, and the displayed representatives attain equality. In particular both curves are nonseparating; a separating curve could not have this nonzero signed pairing.