Saturation of Newton polytopes of type A and D cluster variables

Cluster algebras are rings with distinguished generators called cluster variables, grouped into clusters. Each cluster variable can be written as a Laurent polynomial in any cluster. In this paper, we study the Newton polytopes of these Laurent polynomials. We focus on cluster algebras of types $A$ and $D$, with frozen variables corresponding to the boundary segments of a polygon and punctured polygon, respectively. For these cluster algebras, we show the cluster variable Newton polytopes are saturated. For type $A$, we additionally show that the cluster variable Newton polytopes have no non-vertex lattice points. Our main tool is the snake graph expansion formula of Musiker-Schiffler-Williams for cluster algebras from surfaces.