Lattice 3-polytopes of lattice width 2 and corresponding toric hypersurfaces

The Kodaira dimension of a nondegenerate toric hypersurface can be computed from the dimension of the Fine interior of its Newton polytope according to recent work of Victor Batyrev, where the Fine interior of the Newton polytope is the subpolytope consisting of all points which have an integral distance of at least $1$ to all integral supporting hyperplanes. In particular, if we have a Fine interior of codimension $1$, then the hypersurface is of general type and the Newton polytope has lattice width $2$. In this article we study this situation for lattice $3$-polytopes and the corresponding surfaces of general type. In particular, we classify all $2$-dimensional Fine interiors of those lattice $3$-polytopes which have at most $40$ interior lattice points, thus obtaining many examples of surfaces of general type and genus at most $40$.