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Lattices, graphs, and Conway mutation

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Abstract

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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Notes

  1. As remarked on [20, p. 3], the results of that book are stated for simple graphs, i.e. those without parallel edges, but most results (including this one) apply, mutatis mutandis, to graphs with parallel edges.

  2. The conventions in [5] differ somewhat from those appearing here and in the other references.

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Acknowledgements

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). https://doi.org/10.1007/s00222-012-0421-4

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