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We compute the local Gromov-Witten invariants of certain configurations ofrational curves in a Calabi-Yau threefold. These configurations are connectedsubcurves of the `minimal trivalent configuration', which is a particular treeof P^1's with specified formal neighborhood. We show that these localinvariants are equal to certain global or ordinary Gromov-Witten invariants ofa blowup of P^3 at points, and we compute these ordinary invariants using thegeometry of the Cremona transform. We also realize the configurations inquestion as formal toric schemes and compute their formal Gromov--Witteninvariants using the mathematical and physical theories of the topologicalvertex. In particular, we provide further evidence equating the vertexamplitudes derived from physical and mathematical theories of the topologicalvertex.Comment: This is the version published by Geometry & Topology on 7 March 200

The local Gromov–Witten invariants of configurations of rational curves