Legendrian loops and cluster modular groups

Abstract

This work studies Legendrian loop actions on exact Lagrangian fillings of Legendrian links in $(\R^3, ξ_{\st})$. By identifying the induced action of Legendrian loops as generators of cluster modular groups, we establish the existence of faithful group actions on the exact Lagrangian fillings of several families of Legendrian positive braid closures, including all positive torus links. In addition, we leverage a Nielsen-Thurston-like classification of cluster automorphisms to provide new combinatorial and algebraic tools for proving that a Legendrian loop action has infinite order.

Figures (32)

• Figure 1: Front projections of the Legendrian isotopic links given as the rainbow closure (left) and $(-1)$-framed closure (right) of the positive braids $\beta$ and $\beta\Delta^2$. Here $\Delta$ denotes a half twist of the braid.
• Figure 2: Singularities of front projections of Legendrian surfaces. Labels correspond to notation used by Arnold in his classification.
• Figure 3: The weaving of singularities of fronts along the edges of the $N$-graph. Gluing these local models according to the $N$-graph $\Gamma$ yields the weave $\mathfrak{w}(\Gamma)$.
• Figure 4: Two local models of $\mathbb{L}$-compressing cycles in $H_1(\mathfrak{w})$.
• Figure 5: A pair of long I-cycles. The cycle on the left passes through an even number of hexavalent vertices, while the cycle on the right passes through an odd number.

Theorems & Definitions (79)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Definition 2.2
• Theorem 2.3: CasalsZaslow, Section 7.3