2023/12/17 by Matthias Scharitzer, Vivek Shende, Scharitzer, Matthias +1 · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2312.10625
openalex publication_date 2023/12/17 · openalex created_date 2023/12/20 · openalex updated_date 2026/07/28
Under certain hypotheses, we show that Legendrian surfaces related by disk surgery will have q-deformed augmentation spaces that are related by q-deformed cluster transformation. The proof is geometric, via considerations of moduli of holomorphic curves. In fact, our results naturally give a more general HOMFLYPT "skein-valued cluster transformation", of which the q-cluster transformation is the U(1) specialization. We apply our methods to the Legendrians associated to cubic planar graphs, where mutation of graphs lifts to Legendrian disk surgery. We show that their skein-valued mirrors transform by skein-valued cluster transformation, and give a formula for the skein-valued curve counts on their fillings.