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In this paper we compute the bivariate generating function of$\gamma$-matchings over two backbones, filtered by the number of arcs and thetopological genus. $\gamma$-matchings over two backbones are chord-diagrams,obtained via concatenation and nesting of irreducible shapes of topologicalgenus $\le \gamma$. We show that the key information is contained in thepolynomials counting these shapes and provide recursions that allow to computethe latter. In particular we give a bijection between such irreducible shapesover one and two backbones. We present two applications of our results. Thefirst is concerned with RNA-RNA interaction structures, obtained from the$\gamma$-matchings via symbolic methods. We secondly show that, usinganalytic-combinatorial methods, the topological genus satisfies a central limittheorem.

On the genus filtration of diagrams over two backbones