Saturation of Newton polytopes of type A and D cluster variables

We study Newton polytopes for cluster variables in cluster algebras $\mathcal{A}(\mathrm{\Sigma })$ of types A and D. A famous property of cluster algebras is the Laurent phenomenon: each cluster variable can be written as a Laurent polynomial in the cluster variables of the initial seed $\mathrm{\Sigma }$. The cluster variable Newton polytopes are the Newton polytopes of these Laurent polynomials. We show that if $\mathrm{\Sigma }$ has principal coefficients or boundary frozen variables, then all cluster variable Newton polytopes are saturated. We also characterize when these Newton polytopes are \emph{empty}; that is, when they have no non-vertex lattice points.