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We consider the computation of the topological string partition function for5-brane web diagrams with an O7$^-$-plane. Since upon quantum resolution of theorientifold plane these diagrams become non-toric web diagrams without theorientifold we are able to apply the topological vertex to obtain the Nekrasovpartition function of the corresponding 5d theory. We apply this procedure tothe case of 5d $SU(N)$ theories with one hypermultiplet in the antisymmetricrepresentation and to the case of 5d pure $USp(2N)$ theories. For these caseswe discuss the dictionary between parameters and moduli of the 5d gauge theoryand lengths of 5-branes in the web diagram and moreover we perform comparisonof the results obtained via application of the topological vertex and the oneobtained via localisation techniques, finding in all instances we considerperfect agreement.

Partition functions of web diagrams with an O7$^-$-plane