Stable collective dynamics are often observed in complex biochemical networks, such as in emergent oscillations. How these robust dynamics arise remains unclear, given the large reaction space and stochasticity demonstrated by underlying components. We propose a topological model that demonstrates emergent oscillations at the network boundary, effectively reducing the system dynamics to a lower-dimensional space. Inspired by topological band theory, we introduce a predictor of oscillation coherence from the analysis of spectral gaps. Using this to model KaiC, which regulates the circadian rhythm in cyanobacteria, we compare the coherence of oscillations produced to that in other KaiC models. We find that localization of currents on the system edge in the topological model supports a regime with simultaneously decreased cost and increased precision. This model also saturates a global thermodynamic bound. We conclude with a discussion of the model’s relevance to experimental data and propose testable predictions. Our work highlights a new paradigm for robust dynamics in complex biological networks and the robustness of biological function despite pervasive stochasticity.