  We study multisections of embedded surfaces in 4-manifolds admitting
effective torus actions. We show that a simply-connected 4-manifold admits a
genus one multisection if and only if it admits an effective torus action.
Orlik and Raymond showed that these 4-manifolds are precisely the connected
sums of copies of $\mathbb{CP}^2$, $\overline{\mathbb{CP}^2}$, and $S^2\times
S^2$. Therefore, embedded surfaces in these 4-manifolds can be encoded
diagrammatically on a genus one surface. Our main result is that every smooth,
complex curve in $\mathbb{CP}^1\times\mathbb{CP}^1$ can be put in efficient
bridge position with respect to a genus one 4-section. We also analyze the
algebraic topology of genus one multisections.

Geometric Topology