Simplified SFT moduli spaces for Legendrian links

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.