  Understanding the nature of the hidden charm pentaquark(like) signals in the
LHCb data for $\Lambda_b^0\to J/\psi p K^-$ is a central problem of hadron
spectroscopy. We propose a scenario completely different from previous ones
such as hadron molecules and compact pentaquarks. We identify relevant double
triangle mechanisms with leading or lower-order singularities. The associated
anomalous threshold cusps at the $\Sigma_c^{(*)}\bar{D}^{(*)}$ thresholds are
significantly more singular than the ordinary ones. Then we demonstrate that
the double triangle amplitudes reproduce the peak structures of $P_c(4312)^+$,
$P_c(4380)^+$, and $P_c(4457)^+$ in the LHCb data, through an interference with
other common mechanisms. Only the $P_c(4440)^+$ peak is due to a resonance with
width and strength significantly smaller than previously estimated. $P_c^+$
signals are expected in other processes and the proposed model (partly)
explains the current data such as: the GlueX $J/\psi$ photoproduction data with
no $P_c^+$ signals; the LHCb $\Lambda_b^0\to J/\psi p \pi^-$ data with a
possible signal only from $P_c(4440)^+$. The double triangle singularity is now
a possible option to interpret resonancelike structures near thresholds in
general.
