Alternating links, rational balls, and cube tilings

When does the double cover of the three-sphere branched along an alternating link bound a rational homology ball? Heegaard Floer homology generates a necessary condition for it to bound: the link's chessboard lattice must be cubiquitous, implying that its normalized determinant is less than or equal to one. We conjecture that the converse holds and prove it when the normalized determinant equals one. The proof involves flows on planar graphs and the Haj\'os-Minkowski theorem that a lattice tiling of Euclidean space by cubes contains a pair of cubes which touch along an entire facet. We extend our main results to the study of ribbon cobordism and ribbon concordance.