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

Alternating links, rational balls, and cube tilings