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Given any smooth germ of a threefold flopping contraction, we first give acombinatorial characterisation of which Gopakumar-Vafa (GV) invariants arenon-zero, by prescribing multiplicities to the walls in the movable cone. Onthe Gromov-Witten (GW) side, this allows us to describe, and even draw, thecritical locus of the associated quantum potential. We prove that the criticallocus is the infinite hyperplane arrangement of Iyama and the second author,and moreover that the quantum potential can be reconstructed from a finitefundamental domain. We then iterate, obtaining a combinatorial description ofthe matrix which controls the transformation of the non-zero GV invariantsunder a flop. There are three main ingredients and applications: (1) aconstruction of flops from simultaneous resolution via cosets, which describeshow the dual graph changes, (2) a closed formula which describes the change indimension of the contraction algebra under flop, and (3) a direct and explicitisomorphism between quantum cohomologies of different crepant resolutions,giving a Coxeter-style, visual proof of the Crepant Transformation Conjecturefor isolated cDV singularities.

GV and GW invariants via the enhanced movable cone