Mirror Symmetry and smoothing Gorenstein toric affine 3-folds

We state two conjectures that together allow one to describe the set of smoothing components of a Gorenstein toric affine 3-fold in terms of a combinatorially defined and easily studied set of Laurent polynomials called 0-mutable polynomials. We explain the origin of the conjectures in mirror symmetry and present some of the evidence.Comment: 23 pages, 6 figures. The paper has been submitted to "Facets of Algebraic Geometry: A Volume in Honour of William Fulton's 80th Birthday