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 skein trace. After specializing to the case where M is a surface times an interval, and additionally specializing the HOMFLYPT skein to the gl(2) skein on M and the gl(1) skein on L, we recover an existing prescription of Neitzke and Yan. This talk presents joint work with Tobias Ekholm, Pietro Longhi, and Sunghyuk Park.
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