Legendrian doubles, twist spuns, and clusters

TL;DR

The paper extends the cluster-geometry bridge between microlocal sheaves and exact Lagrangian fillings from Legendrian links to Legendrian surfaces in $(\mathbb{R}^5,\xi_{st})$ by studying Legendrian doubles $\Lambda(L_1,L_2)$ and twist-spuns $\Sigma_\varphi(\lambda)$. It shows that for doubles, exact fillability is obstructed when the induced toric charts from two fillings differ, and that certain doubles correspond to connected sums of standard and Clifford tori; for cubic planar doubles, mutation distance governs decomposability, with chromatic-polynomial invariants providing diagnostic power. For twist-spun Legendrians, the authors establish a cluster-ensemble structure via folding when the base moduli is globally foldable, deriving infinite families of exact fillings in favorable cases and obstructions in others through Grassmannian fixed points. The results connect Lagrangian surgery, Legendrian mutation, and folded cluster algebra mutations, offering conjectures on the count of fillings and highlighting the nuanced compatibility between folding and toric intersections. Overall, the work advances high-dimensional analogues of the Cassels–Gao–Casals–Zaslow framework, contributing tools to classify fillings and understand the role of cluster theory in higher-dimensional contact topology.

Abstract

Let $λ$ be a Legendrian link in standard contact $\mathbb{R}^3$, such that $L_1$, $L_2$ are two exact fillings of $λ$ and $\varphi$ is a Legendrian loop of $λ$. We study fillability and isotopy characterizations of Legendrian surfaces in standard contact $\mathbb{R}^5$ built from the above data by doubling or twist spinning; denoting them $Λ(L_1,L_2)$ or $Σ_\varphi(λ)$ respectively. In the case of doubles $Λ(L_1,L_2)$, if the sheaf moduli $\mathcal{M}_1(λ)$ admits a cluster structure, we introduce the notion of mutation distance and study its relationship with the isotopy class of the Legendrian surface. For twist spuns $Σ_\varphi(λ)$, when $\mathcal{M}_1(λ)$ admits a globally foldable cluster structure, we use the existence of a $\varphi$-symmetric filling of the Legendrian link to build a cluster structure on the sheaf moduli of the twist spun by folding. We then use that to motivate, and provide evidence for, conjectures on the number of embedded exact fillings of certain twist spuns. Further, we obstruct the exact fillability of certain twist spuns by analyzing fixed points of the cyclic shift action on Grassmanians.

Figures (15)

• Figure 1: Front projection of Legendrian given as the $(-1)$ closure of $\beta\Delta$.
• Figure 2: Legendrian double that does not decompose as a connect sum of standard and Clifford tori (left) and Legendrian double formed from fillings of $\lambda(2, 6)$ with a chromatic polynomial divisible by $q-2$ (right).
• Figure 3: Legendrian doubles formed from fillings of $\lambda(2, 10)$ with the same chromatic polynomial.
• Figure 4: A cubic planar Legendrian which is possibly not Legendrian isotopic to a double
• Figure 5: Singularities of front projections of Legendrian surfaces. Labels correspond to notation used by Arnold in his classification.

Theorems & Definitions (110)

• Theorem 1.1
• Lemma 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Example 1.8
• Remark 1.9
• Remark 1.10
• Theorem 1.11