Topology of toric gravitational instantons

For an asymptotically locally Euclidean (ALE) or asymptotically locally flat (ALF) gravitational instanton $(M,g)$ with toric symmetry, we express the signature of $(M,g)$ directly in terms of its rod structure. Applying Hitchin–Thorpe-type inequalities for Ricci-flat ALE/ALF manifolds, we formulate, as a step toward a classification of toric ALE/ALF instantons, necessary conditions that the rod structures of such spaces must satisfy. Finally, we apply these results to the study of rod structures with three turning points.