Exact Lagrangian fillability of 3-braid closures

TL;DR

This work determines exact Lagrangian fillability for closures of quasipositive $3$-braids in the standard contact $\mathbb{R}^3$. By leveraging Orevkov's quasipositivity criterion and constructing max-$tb$ Legendrian representatives $\Lambda_k(\beta)$ for $m\le -2$, the authors establish that orientable fillability holds precisely when $\hat{\beta}$ is quasipositive and the Garside exponent $m$ satisfies $m\ge -2$, while non-orientable fillability occurs for $m\le -2$ with obstructions at $m=-1$. They also derive an explicit formula for the maximum Thurston–Bennequin number in terms of the signed word length $|\beta|$ and $m$, and provide concrete rulings to realize these bounds, including decomposable non-orientable fillings. Furthermore, the paper connects these constructions to Legendrian weaves and augmentation varieties, contributing evidence in support of the Hayden–Sabloff orientable fillability conjecture for braid index $3$ and completing the orientable-fillability classification in this family. Overall, the results advance understanding of how positivity, TB bounds, and ruling invariants govern the existence of exact Lagrangian fillings for simple braid closures.

Abstract

We determine when a Legendrian quasipositive 3-braid closure in standard contact $\mathbb{R}^3$ admits an orientable or non-orientable exact Lagrangian filling. Our main result provides evidence for the orientable fillability conjecture of Hayden and Sabloff, showing that a 3-braid closure is orientably exact Lagrangian fillable if and only if it is quasipositive and the HOMFLY bound on its maximum Thurston-Bennequin number is sharp. Of possible independent interest, we construct explicit Legendrian representatives of quasipositive 3-braid closures with maximum Thurston-Bennequin number.

Figures (21)

• Figure 1: Max-tb Legendrian representative of the closure of $\sigma_1^3\sigma_2^2\sigma_1^2\sigma_2^2\sigma_1^3\sigma_2^3\sigma_1^2\Delta^{-7}$. This knot has no orientable exact Lagrangian fillings, but is non-orientably fillable.
• Figure 2: Rainbow closure of a positive braid $\beta$ (left) and $(-1)$-closure of $\beta_+\Delta$ (right).
• Figure 3: Local picture of stabilization.
• Figure 4: Initial fronts $\Lambda_0(\beta)$ for $m$ even (left) and odd (right).
• Figure 5: All possible configurations of ruling disks in a neighborhood of switches (top), departures (middle), and returns (bottom).

Theorems & Definitions (30)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Corollary 1.4
• Remark 1.5
• Corollary 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 2.1: Orevkov04, Proposition 3.1
• Theorem 2.2: Rutherford06