  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.
