Laurent polynomials and deformations of non-isolated Gorenstein toric singularities

We establish a correspondence between one-parameter deformations of an affine Gorenstein toric pair $(X_P, partial X_P)$, defined by a polytope $P$, and mutations of a Laurent polynomial $f$ with Newton polytope $Delta(f) = P$. For a Laurent polynomial $f$ in two variables, we construct a formal deformation of the three-dimensional Gorenstein toric pair $(X_{Delta(f)}, partial X_{Delta(f)})$ over ${mathbb C}[[{mathbf T}_f]]$, where ${mathbf T}_f$ is the set of deformation parameters arising from mutations. The general fibre of this deformation is smooth if and only if $f$ is $0$-mutable. The Kodaira-Spencer map of the constructed deformation is injective, and if $f$ is maximally mutable, then the deformation cannot be nontrivially extended to a larger smooth base space.