  We propose a new way to compute the genus zero Gopakumar-Vafa invariants for
two families of non-toric non-compact Calabi-Yau threefolds that admit simple
flops: Reid's Pagodas, and Laufer's examples. We exploit the duality between
M-theory on these threefolds, and IIA string theory with D6-branes and
O6-planes. From this perspective, the GV invariants are detected as
five-dimensional open string zero modes. We propose a definition for genus zero
GV invariants for threefolds that do not admit small crepant resolutions. We
find that in most cases, non-geometric T-brane data is required in order to
fully specify the invariants.
