  In this paper, we generalize a mathematical definition of Gopakumar-Vafa (GV)
invariants on Calabi-Yau 3-folds introduced by Maulik and the author, using an
analogue of BPS sheaves introduced by Davison-Meinhardt on the coarse moduli
spaces of one dimensional twisted semistable sheaves with arbitrary holomorphic
Euler characteristics. We show that our generalized GV invariants are
independent of twisted stability conditions, and conjecture that they are also
independent of holomorphic Euler characteristics, so that they define the same
GV invariants. As an application, we will show the flop transformation formula
of GV invariants.
