The authors construct cluster structures on sheaf moduli of twist-spun Legendrian surfaces and use them to produce new exact Lagrangian fillings and obstructions in contact R^5.
Skein valued cluster transformation in enumerative geometry of Legendrian mutation
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abstract
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.
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math.SG 1years
2025 1verdicts
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Legendrian doubles, twist spuns, and clusters
The authors construct cluster structures on sheaf moduli of twist-spun Legendrian surfaces and use them to produce new exact Lagrangian fillings and obstructions in contact R^5.