  We introduce and study the Hermitian matrix model with potential
V(x)=x^2/2-stx/(1-tx), which enumerates the number of linear chord diagrams of
fixed genus with specified numbers of backbones generated by s and chords
generated by t. For the one-cut solution, the partition function, correlators
and free energies are convergent for small t and all s as a perturbation of the
Gaussian potential, which arises for st=0. This perturbation is computed using
the formalism of the topological recursion. The corresponding enumeration of
chord diagrams gives at once the number of RNA complexes of a given topology as
well as the number of cells in Riemann's moduli spaces for bordered surfaces.
The free energies are computed here in principle for all genera and explicitly
for genera less than four.
