Lattices, graphs, and Conway mutation

The d-invariant of an integral, positive definite lattice Λ records the minimal norm of a characteristic covector in each equivalence class $({\textup{mod} }2\varLambda)$ . We prove that the 2-isomorphism type of a connected graph is determined by the d-invariant of its lattice of integral flows (or cuts). As an application, we prove that a reduced, alternating link diagram is determined up to mutation by the Heegaard Floer homology of the link’s branched double-cover. Thus, alternating links with homeomorphic branched double-covers are mutants.