We propose a novel topological vertex formalism for 5d \( \mathcal{N}=1 \) SU(N) gauge theory with a hypermultiplet in the symmetric tensor representation, whose Type IIB brane construction involves an NS5-brane attached to an O7+-plane. Inspired by the identification O7+ ∼ ℤ2 + 4D7, we introduce two new types of vertices: the ℤ2-vertex, which implements the ℤ2 orbifold action, and the FD-vertex, which encodes the monodromy cut induced by the O7+-plane. This formalism generalizes the framework presented in JHEP 04 (2025) 182 and establishes a systematic method for computing partition functions for 5-brane configurations that incorporate an O7+-plane. The resulting partition functions are expressed as sums over Young diagrams, providing a powerful computational tool for studying such gauge theories.
