Lagrangian fillings for Legendrian links of finite or affine Dynkin type
Byung Hee An
Kyungpook National University, Daegu, South KoreaYoungjin Bae
Incheon National University, Incheon, South KoreaEunjeong Lee
Institute for Basic Science, Pohang, South Korea; Chungbuk National University, Cheongju, South Korea

Abstract
We prove that there are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of finite type or affine type . We also provide as many Lagrangian fillings with rotational symmetry as seeds of type , , , , or , and with conjugation symmetry as seeds of type , , , , or . 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 . Furthermore, we show that the -graph realization of (twice of) Coxeter mutation of type corresponds to a Legendrian loop of the corresponding Legendrian links. Especially, the loop of type coincides with the one considered by Casals and Ng.
Cite this article
Byung Hee An, Youngjin Bae, Eunjeong Lee, Lagrangian fillings for Legendrian links of finite or affine Dynkin type. Quantum Topol. (2025), published online first
DOI 10.4171/QT/233