Augmentations, Fillings, and Clusters for 2-Bridge Links

TL;DR

The paper establishes a precise bridge between non-orientable exact Lagrangian fillings of Legendrian 2-bridge links and cluster algebras by proving the ungraded augmentation variety Aug_u(Λ[n_1,...,n_k]) decomposes as a product of $A$-type cluster varieties and by constructing cluster seeds from admissible pinching sequences. It shows that the coordinate ring of Aug_u is isomorphic to $A_{n_1-2}\times A_{n_2-3}\times \dots \times A_{n_{(k-1)}-3}\times A_{n_k-2}$, with at least $C_{n_1-1}C_{n_2-2}\dots C_{n_k-1}$ distinct fillings arising from combinatorial pinching data, and that the ruling stratification coincides with the Lam–Speyer anticlique stratification. A corollary via Rutherford connects the cluster-theoretic stratification to the lowest $a$-degree term of the Kauffman polynomial for the associated smooth link type, tying cluster geometry to classical knot invariants. The work further develops a robust framework for using non-orientable cobordisms to generate and distinguish fillings, and it raises open questions about intrinsically contact-geometric realizations of cluster structures and broader applicability to other Legendrian links.

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.

Figures (18)

• Figure 1: Front projections of $2$-bridge links $\Lambda[n_1, \ldots, n_{2l}]$ and $\Lambda[n_1, \ldots, n_{2l+1}]$, where each box labelled $n_i$ contains $n_i$ crossings.
• Figure 2: A pinch move at a Reeb chord $a$, where the crossing $a$ in the Lagrangian projection is replaced with its $0$ resolution and two co-oriented base points $s$.
• Figure 3: Topological interpretation of co-oriented base points as relative cycles in a saddle cobordism.
• Figure 4: A pinching sequence: pinch $a_2$ and then $a_3.$
• Figure 5: Lagrangian projections of $2$-bridge knots $\Lambda[n_1, \ldots, n_{2k}]$ and $\Lambda[n_1, \ldots, n_{2k+1}]$, where each box labeled $n_i$ contains $n_i$ crossings where the strand with more negative slope is the overstrand. Both co-oriented base points $t_1$ and $t_2$ are oriented counterclockwise.

Theorems & Definitions (147)

• Theorem 1.1: Theorems \ref{['thm: cluster structure on augmentation variety']} and \ref{['thm: cluster torus chart from pinching sequence']}
• Theorem 1.2: Propositions \ref{['prop:admissiblefilling']} and \ref{['prop: every cluster chart is achieved']}
• Theorem 1.3: Theorem \ref{['thm: stratification coincide']}
• Remark 2.1
• Remark 2.2
• Theorem 2.3: Chekanov
• Theorem 2.4: ng_2003
• Definition 2.5
• Corollary 2.6: ng_2003
• Definition 2.7