Weave-realizability of all cluster seeds beyond finite and affine Dynkin types

Determine whether Legendrian weaves realize all possible cluster seeds for cluster algebras beyond finite and affine Dynkin types; specifically, ascertain whether every cluster seed (e.g., for Grassmannian cluster algebras C[ẐGr(k, n)]) can be obtained from a Legendrian weave construction such as T-shift, not only in finite or affine Dynkin settings.

Background

The paper uses Legendrian weaves produced via the T-shift procedure to construct exact Lagrangian fillings corresponding to cluster seeds of Grassmannians and their symmetric loci. While plabic graphs do not realize all seeds, Legendrian weaves are known to realize strictly more, and complete realizability is established in finite and affine Dynkin types in prior work.

The authors note that, outside of these types, it remains unresolved whether all cluster seeds can be realized by Legendrian weaves, and they explicitly do not address this question in the current work. Clarifying this would significantly strengthen the bridge between contact-geometric constructions and cluster algebra combinatorics.

References

However, the question of whether Legendrian weaves realize all possible cluster seeds is still unknown outside of finite and affine Dynkin types [Hughes2021, ABL22]. We do not attempt to answer it in this setting.

— Exact Lagrangian fillings of twist-spun torus links  (2509.19095 - Chen et al., 23 Sep 2025) in Section 1.2 (Main results), final paragraphs of the introduction