2025/10/21 by Tobias Ekholm, Pietro Longhi, Ekholm, Tobias +5 · 1 citation
Mathematics · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.2510.19041
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 \mathfrakgl(2) skein on M and the \mathfrakgl(1) skein on L, we recover an existing prescription of Neitzke and Yan.