Control of transcription presides over a vast array of biological processes. This control typically manifests through a web of regulatory circuits with different genes interacting under a range of feedback architectures, often exhibiting multistability as a result. Our work uses a geometric approach grounded in bifurcation theory to study the stability profile of a mutually repressing three-gene network across different regions of an unconstrained parameter space. The symmetric network exhibits a distinct dynamic topology as the relative repressive strengths among the genes change, with greater complexity as the genes become more similar in their regulatory activity. We also observe transitions across topologies in the bifurcation plane, and the parameter thresholds under which they occur. These boundaries broaden in parameter space as coupling sensitivity rises via the Hill coefficient and as a higher growth rate implies more available energy. We then show that the model extends to higher-order dynamic networks in which one or more genes from the core three-gene motif drive each downstream gene. The work highlights dynamic relationships between a system's parametrization and the resulting gene expression phenotypes (i.e., fixed point profiles) that may improve understanding of the mechanisms through which complex networks evolve in nature.