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