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Lattices, graphs, and Conway mutation

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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.

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Thanks to Lucia Caporaso, Nathan Dunfield, Farbod Shokrieh, Oleg Viro, and Filippo Viviani for helpful correspondence; to the referees for helpful remarks; to Liam Watson for an inspiring conversation; and to the Venice Beach Institute for its hospitality. The author carried out the present work while supported by an NSF Postdoctoral Fellowship.

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Correspondence to Joshua Evan Greene.

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Greene, J.E. Lattices, graphs, and Conway mutation. Invent. math. 192, 717–750 (2013).

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