Plummer Dark Matter Black Hole with Topological Defects: Shadow, Greybody Factors, Quasinormal Modes, and Thermodynamics

We construct a static, spherically symmetric black hole (BH) solution embedded in a cored Plummer dark matter (DM) halo and a Letelier cloud of strings (CoS). Starting from the Plummer-Schwarzschild metric of Senjaya et al.~\cite{Senjaya2026}, we incorporate the string-cloud tension parameter $\alpha$ into the lapse function, obtaining $A(r)=h_{\mathrm{P}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{r}}(r)-\alpha$. The resulting spacetime admits a single, non-degenerate event horizon (EH) for $\alpha <1$ and a naked singularity for $\alpha \ge 1$. We determine the photon sphere (PS) and BH shadow radii, compute the weak deflection angle via the Gauss-Bonnet theorem (GBT), and analyze the innermost stable circular orbit (ISCO). Scalar perturbations are studied through the effective potential, greybody factor (GF) bounds obtained from the Boonserm-Visser method, the Hawking emission spectrum, and quasinormal mode (QNM) frequencies computed with the WKB approximation. The thermodynamic analysis covers the Hawking temperature, Bekenstein-Hawking entropy, heat capacity, and Gibbs free energy; the heat capacity is found to be strictly negative for all parameter values, confirming the absence of any Davies-type phase transition. A consistent hierarchy emerges across all six analyses: the CoS tension $\alpha$ governs the leading-order modifications to every observable, while the Plummer halo density ${\rho }_0$ provides a subdominant, additive correction.