Block–Göttsche invariants from wall-crossing

We show how some of the refined tropical counts of Block and Göttsche emerge from the wall-crossing formalism. This leads naturally to a definition of a class of putativeq-deformed Gromov–Witten invariants. We prove that this coincides with another naturalq-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined.