Augmentations, Fillings, and Clusters for 2-Bridge Links

TL;DR

This paper connects non-orientable Lagrangian fillings of Legendrian 2-bridge links with cluster theory, revealing isomorphisms between augmentation varieties and cluster varieties, and relating stratifications to knot invariants.

Contribution

It establishes the first link between Lagrangian fillings of Legendrian links and cluster varieties, providing new insights into their geometric and algebraic structures.

Findings

Augmentation variety is isomorphic to a product of A_n cluster varieties.

Surjective map from fillings to cluster seeds produces Catalan number counts.

Ruling stratification matches Lam and Speyer's anticlique stratification.

Abstract

We produce the first examples relating non-orientable exact Lagrangian fillings of Legendrian links to cluster theory, showing that the ungraded augmentation variety of certain max-tb representatives of Legendrian $2$-bridge links is isomorphic to a product of $A_n$-type cluster varieties. As part of this construction, we describe a surjective map from the set of (possibly non-orientable) exact Lagrangian fillings to cluster seeds, producing a product of Catalan numbers of distinct fillings. We also relate the ruling stratification of the ungraded augmentation variety to Lam and Speyer's anticlique stratification of acyclic cluster varieties, showing that the two coincide in this context. As a corollary, we apply a result of Rutherford to show that the cluster-theoretic stratification encodes the information of the lowest $a$-degree term of the Kauffman polynomial of the smooth isotopy class of the $2$-bridge links we study.