We propose a novel scenario for the peak structures, usually interpreted as hidden charm pentaquark ($P_c$) contributions, in the LHCb's $\Lambda_b^0\to J/\psi p K^-$ data. The key idea is to utilize leading or lower-order singularities from double triangle mechanisms. The singularities cause anomalous threshold cusps, which are significantly more singular than the ordinary ones, at the $\Sigma_c^{(*)}\bar{D}^{(*)}$ threshold. We demonstrate that the double triangle amplitudes interfere with other common mechanisms to create peak structure that fit well the $P_c(4312)^+$, $P_c(4380)^+$, and $P_c(4457)^+$ peaks in the data. This picture is completely different from commonly used ones such as hadron molecules and compact pentaquarks. Meanwhile, $P_c(4440)^+$ is included in the proposed model as a resonance with width and strength significantly smaller than previously estimated. The proposed model can (partly) explain the current data for other processes where $P_c^+$ signals are expected such as: no $P_c^+$ signals in the GlueX $J/\psi$ photoproduction data; a possible signal only from $P_c(4440)^+$ in the LHCb's $\Lambda_b^0\to J/\psi p \pi^-$ data.