Simplified SFT moduli spaces for Legendrian links

Abstract

We study moduli spaces $\mathcal{M}$ of holomorphic maps $U$ from Riemann surfaces 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, $\Sigma$, 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 $H^{1}(\Sigma)$. 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 $U$ of index $1$ is a disk with one or two positive punctures for which $\pi_{\mathbb{C}}\circ U$ is an embedding. Moreover, any $U$ 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.