Skein traces from curve counting

Abstract

Given a 3-manifold $M$, and a branched cover arising from the projection of a Lagrangian 3-manifold $L$ in the cotangent bundle of $M$ to the zero-section, we define a map from the skein of $M$ to the skein of $L$, via the skein-valued counting of holomorphic curves. When $M$ and $L$ are products of surfaces and intervals, we show that wall crossings in the space of the branched covers obey a skein-valued lift of the Kontsevich-Soibelman wall-crossing formula. Holomorphic curves in cotangent bundles correspond to Morse flow graphs; in the case of branched double covers, this allows us to give an explicit formula for the the skein trace. After specializing to the case where $M$ is a surface times an interval, and additionally specializing the HOMFLYPT skein to the $\mathfrak{gl}(2)$ skein on $M$ and the $\mathfrak{gl}(1)$ skein on $L$, we recover an existing prescription of Neitzke and Yan.