Simplified SFT moduli spaces for Legendrian links

We study moduli spaces M of holomorphic maps U from Riemann surfaces to R⁴ with boundaries on the Lagrangian cylinder over a Legendrian link Λ ⊂ (R³, ξstd). We allow our domains, Σ, to have non-trivial topology in which case M is the zero locus of an obstruction function O, sending a moduli space of holomorphic maps in C to H¹(Σ). In general, O⁻¹(0) is not combinatorially computable. However after a Legendrian isotopy, Λ can be made left-right-simple, implying that any U of index $1$ is a disk with one or two positive punctures for which πC∘ U is an embedding. Moreover, any U of index $2$ is either a disk or an annulus with πC ∘ U simply covered and without interior critical points. Therefore any SFT invariant of Λ is combinatorially computable using only disks with ≤ 2 positive punctures.