Lagrangian fillings for Legendrian links of affine type

Abstract

We prove that there are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of affine type $\tilde{\mathsf{D}}\tilde{\mathsf{E}}$. We also provide as many Lagrangian fillings with certain symmetries as seeds of type ${\tilde{\mathsf{B}}}_n$, ${\tilde{\mathsf{F}}}_4$, ${\tilde{\mathsf{G}}}_2$, and ${\mathsf{E}}_6^{(2)}$. These families are the first known Legendrian links with infinitely many fillings that exhaust all seeds in the corresponding cluster structures. Furthermore, we show that Legendrian realization of Coxeter mutation of type $\tilde{\mathsf{D}}$ corresponds to the Legendrian loop considered by Casals and Ng.

arXiv comment46pages

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Cited by in corpus (4)

• Weave Realizability for D-type
• A note on infinite number of exact Lagrangian fillings for spherical spuns
• On folded cluster patterns of affine type
• Non-fillable augmentations of twist knots