Legendrian doubles, twist spuns, and clusters

TL;DR

This paper explores the construction and classification of Legendrian surfaces in contact 5-space using doubling and twist spinning techniques, linking cluster structures in sheaf moduli to fillability and isotopy properties.

Contribution

It introduces new concepts like mutation distance and studies cluster structures in sheaf moduli to understand Legendrian surface isotopy and fillability.

Findings

Cluster structures relate to Legendrian isotopy classes.

Mutation distance measures differences between doubles.

Obstructions to fillability are identified via Grassmannian actions.

Abstract

Let $\lambda$ be a Legendrian link in standard contact $\mathbb{R}^3$, such that $L_1$, $L_2$ are two exact fillings of $\lambda$ and $\varphi$ is a Legendrian loop of $\lambda$. 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 $\Lambda(L_1,L_2)$ or $\Sigma_\varphi(\lambda)$ respectively. In the case of doubles $\Lambda(L_1,L_2)$, if the sheaf moduli $\mathcal{M}_1(\lambda)$ 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 $\Sigma_\varphi(\lambda)$, when $\mathcal{M}_1(\lambda)$ 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.