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The LHCb data on $B^+\to J/\psi\phi K^+$ show four peaks and three dips inthe $J/\psi\phi$ invariant mass distribution, and the peaks are interpreted as$X(4140)$, $X(4274)$, $X(4500)$ and $X(4685)/X(4700)$ resonance contributions.Interestingly, all the peaks and dips are located at (or close to)$D^*_{s}\bar{D}^{(*)}_{s}$, $D_{s0}^*(2317)\bar{D}^{(*)}_{s}$,$D_{s1}(2536)\bar{D}^{(*)}_{s}$, and $\psi'\phi$ thresholds. These coincidencessuggest a close connection between the structures and the thresholds, whichhowever has not been seriously considered in previous theoretical studies onthe $X$ structures. In fact, if we utilize this connection and interpret the$X$ structures as common $s$-wave threshold cusps, we face a difficulty:$X(4274)$ and $X(4500)$ have spin-parity that conflict with the experimentallydetermined ones. In this work, we introduce double triangle mechanisms thatcause threshold cusps significantly sharper than the ordinary one-loop ones ofthe same spin-parity. We demonstrate that all the $X$ and dip structures arewell described by a combination of one-loop and double-triangle thresholdcusps, thereby proposing a novel interpretation of the $X$ and dip structures.

$X$ structures in $B^+\to J/\psi\phi K^+$ as one-loop and double-triangle threshold cusps