Intrinsicness of the Newton polygon for smooth curves on $\mathbb{P}^1\times \mathbb{P}^1$

Abstract: 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.