Refined topological vertex and instanton counting

It has been proposed recently that topological A-model string amplitudes for toric Calabi-Yau 3-folds in non self-dual graviphoton background can be calculated by a diagrammatic method that is called the refined topological vertex. We compute the extended A-model amplitudes for SU(N)-geometries using the proposed vertex. If the refined topological vertex is valid, these computations should give rise to the Nekrasov's partition functions of N = 2 SU(N) gauge theories via the geometric engineering. In this article, we verify the proposal by confirming the equivalence between the refined A-model amplitude and the K-theoretic version of the Nekrasov's partition function by explicit computation.