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