Converse fillability for twist-spun torus links

Establish that the Legendrian twist-spun torus Σ_{ψ^ℓ}((k, n−k)), formed as the mapping torus of the Kalman Legendrian loop ψ^ℓ acting on the Legendrian (k, n−k) torus link, is orientably exact Lagrangian fillable if and only if k is congruent to −1, 0, or 1 modulo d, where d = n / gcd(n, ℓ).

Background

The paper proves a sufficient condition (Theorem 1.1) for orientable exact Lagrangian fillings of twist-spun Legendrian torus links Σ_{ψ^ℓ}((k, n−k)) arising from rotationally symmetric Legendrian weave fillings constructed via T-shift from symmetric weakly separated collections. The combinatorial core (Theorem 1.2) provides necessary and sufficient conditions for the existence of maximal weakly separated collections invariant under the cyclic shift ρ^ℓ.

Motivated by cluster-theoretic obstructions and symmetry considerations, the authors formulate a converse conjecture asserting that the same congruence conditions on k modulo d are also necessary for orientable exact Lagrangian fillability of the twist-spun torus. This aims to match geometric fillability with the combinatorial symmetry criterion.