Journal of Symplectic Geometry

Volume 21 (2023)

Number 2

Simplified SFT moduli spaces for Legendrian links

Pages: 265 – 329

DOI: https://dx.doi.org/10.4310/JSG.2023.v21.n2.a2

Author

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

Abstract

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.

Received 19 November 2021

Received revised 2 August 2022

Accepted 2 October 2022

Published 28 September 2023