Exact Lagrangian fillings of twist-spun torus links

Abstract

We construct exact Lagrangian fillings of Legendrian torus links $\Lambda(k, n-k)$ that are fixed by a Legendrian loop that acts by $2\pi\ell/n$ rotation. Using these rotationally symmetric fillings, we produce fillings of the corresponding Legendrian twist-spun tori. Our construction is combinatorial in nature, relating symmetric weakly separated collections and plabic graphs to symmetric Legendrian weaves via the T-shift procedure of Casals, Le, Sherman-Bennett, and Weng. The main technical ingredient in this process is a necessary and sufficient condition for the existence of maximal weakly separated collections of $k$-element subsets of $\{1, \dots, n\}$ that are fixed by addition of $\ell$ modulo $n$.