  Let $C$ be a smooth projective curve in $\mathbb{P}^1\times \mathbb{P}^1$ of
genus $g\neq 4$, and assume that it is birationally equivalent to a curve
defined by a Laurent polynomial that is non-degenerate with respect to its
Newton polygon $\Delta$. Then we show that the convex hull $\Delta^{(1)}$ of
the interior lattice points of $\Delta$ is a standard rectangle, up to a
unimodular transformation. Our main auxiliary result, which we believe to be
interesting in its own right, is that the first scrollar Betti numbers of
$\Delta$-non-degenerate curves are encoded in the combinatorics of
$\Delta^{(1)}$, if $\Delta$ satisfies some mild conditions.
