Work

Lagrangian fillings for Legendrian links of finite or affine Dynkin type

Year: 2025

Type: article

Abstract: We prove that there are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of finite type \mathsf{ADE} or affine type \widetilde{\mathsf{D}}\widetilde{\mathsf{E}} . We also provide as many Lagrangian fillings with rotational symmetry as seeds of type \mathsf{B} , \mathsf{G}_{2} , \widetilde{\mathsf{G}}_{2} , \widetilde{\mathsf{B}} , or \widetilde{\mathsf{C}}_{2} , and with conjugation symmetry as seeds of type \mathsf{F}_{4} , \mathsf{C} , \mathsf{E}_{6}^{(2)} , \widetilde{\mathsf{F}}_{4} , or \mathsf{A}_{5}^{(2)} . These families are the first known Legendrian links with (infinitely many) exact Lagrangian fillings (with symmetry) that exhaust all seeds in the corresponding cluster structures beyond type \mathsf{A}\mathsf{D} . Furthermore, we show that the N -graph realization of (twice of) Coxeter mutation of type \widetilde{\mathsf{D}}\widetilde{\mathsf{E}} corresponds to a Legendrian loop of the corresponding Legendrian links. Especially, the loop of type \widetilde{\mathsf{D}} coincides with the one considered by Casals and Ng.

Source: Quantum Topology

DOI: 10.4171/qt/233

Authors: Byung Hee An, Youngjin Bae, Eunjeong Lee

Institutions: Kyungpook National University, Incheon National University, Chungbuk National University, Institute for Basic Science

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond