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The string corrections of tree-level open-string amplitudes can be describedby Selberg integrals satisfying a Knizhnik-Zamolodchikov (KZ) equation. Thisallows for a recursion of the $\alpha'$-expansion of tree-level stringcorrections in the number of external states using the Drinfeld associator.While the feasibility of this recursion is well-known, we provide amathematical description in terms of twisted de Rham theory and intersectionnumbers of twisted forms. In particular, this leads to purely combinatorialexpressions for the matrix representation of the Lie algebra generatorsappearing in the KZ equation in terms of directed graphs. This, in turn, admitsefficient algorithms for symbolic and numerical computations using adjacencymatrices of directed graphs and is a crucial step towards analogous recursionsand algorithms at higher genera.

A note on the Drinfeld associator for genus-zero superstring amplitudes in twisted de Rham theory