Skein traces from curve counting

I will describe a joint project with Tobias Ekholm, Pietro Longhi, and Vivek Shende constructing a map from the HOMFLYPT skein module of a 3-manifold M to that of its branched cover arising from the projection of a Lagrangian 3-manifold L in the cotangent bundle of M. The map is defined by counting holomorphic curves and is a vast generalization of the quantum UV-IR map of Neitzke and Yan, which is a close cousin of the quantum trace map of Bonahon and Wong. The existence of this map has some interesting consequences in the theory of skein-valued curve counts, and I will discuss some of them if time permits.