  Schreyer has proved that the graded Betti numbers of a canonical tetragonal
curve are determined by two integers $b_1$ and $b_2$, associated to the curve
through a certain geometric construction. In this article we prove that in the
case of a smooth projective tetragonal curve on a toric surface, these integers
have easy interpretations in terms of the Newton polygon of its defining
Laurent polynomial. We can use this to prove an intrinsicness result on Newton
polygons of small lattice width.
