Gravitational Wave Signatures from Periodic Orbits around a non-commutative inspired black hole surrounded by quintessence

We study gravitational wave emission from periodic orbits of a test particle around a noncommutative-inspired black hole surrounded by quintessence. Using the zoom-whirl taxonomy, which is characterized by three topological numbers $(z,w,v)$, we classify these orbits and calculate several representative gravitational waveforms for certain periodic orbits. We find that the noncommutative parameter $\mathrm{\Theta }$ and the quintessence field significantly modify both the orbital structure and the emitted waveforms. In particular, increasing $\mathrm{\Theta }$ leads to a phase shift and a change in amplitude in the waveform, while higher zoom numbers produce more complicated substructures. The characteristic strain spectra peak in the millihertz range, lying within the sensitivity band of the LISA detector. Moreover, the presence of the quintessence field introduces significant modifications to these waveforms, imprinting measurable deviations that could be tested or constrained by future space-based gravitational wave detectors. These results suggest that future space-based gravitational wave missions could probe or constrain noncommutative effects in strong gravitational fields.