Journal of Symplectic Geometry

Volume 21 (2023)

Number 2

Simplified SFT moduli spaces for Legendrian links

Pages: 265 – 329



Russell Avdek (Department of mathematics, Uppsala University, Uppsala, Sweden; and Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, Orsay, France)


We study moduli spaces $\mathcal{M}$ of holomorphic maps $U$ to $\mathbb{R}^4$ with boundaries on the Lagrangian cylinder over a Legendrian link $\Lambda \subset (\mathbb{R}^3, \xi_{std})$. We allow our domains, $\dot{\Sigma}$ , to have non-trivial topology in which case $\mathcal{M}$ is the zero locus of an obstruction function $\mathcal{O}$, sending a moduli space of holomorphic maps in $\mathbb{C}$ to $H^1 (\dot{\Sigma})$. In general, $\mathcal{O}^{-1} (0)$ is not combinatorially computable. However after a Legendrian isotopy $\Lambda$ can be made left-right-simple, implying that any $U$

1) of index $1$ is a disk with one or two positive punctures for which $\pi_\mathbb{C} \circ U$ is an embedding.

2) of index $2$ is either a disk or an annulus with $\pi_\mathbb{C} \circ U$ simply covered and without interior critical points.

Therefore any SFT invariant of $\Lambda$ is combinatorially computable using only disks with $\leq 2$ positive punctures.

During the completion of this project the author was employed at Uppsala University and partly supported by the grant KAW 2016.0198 from the Knut and Alice Wallenberg Foundation.

Received 19 November 2021

Received revised 2 August 2022

Accepted 2 October 2022

Published 28 September 2023