  Let $\Lambda$ be a Legendrian link in the standard contact Euclidean space.
We are interested in studying holomorphic curves $U$ in $\mathbb{R} \times
\mathbb{R}^{3} \simeq \mathbb{R} \times \mathbb{R} \times \mathbb{C}$ with
boundary on $\mathbb{R} \times \Lambda$ whose domains are compact, connected
Riemann surfaces with boundary punctures removed $\Sigma'$, allowing for
positive genus or multiple boundary components. When $\chi(\Sigma') < 1$, the
moduli space can be seen as the zero locus of a function $\mathcal{O}$ whose
domain is a moduli space of holomorphic maps to $\mathbb{C}$ and whose target
is $H^{1}(\Sigma')$.
  In general, $\mathcal{O}^{-1}(0)$ is not combinatorially computable from the
Lagrangian projection, $\pi_{\mathbb{C}}$. However, after a Legendrian isotopy,
every such $\Lambda$ can be made to satisfy a condition we call
left-right-simple. With this condition in place, any holomorphic curve $U$ 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, the differential of any SFT
invariant defined by counting index $1$ curves on (possibly multiple parallel
copies of) $\Lambda$ is combinatorially computable using only the disks of
rational SFT.
