The first full amplitude analysis of \(B^+\to J/\psi \phi K^+\) with \(J/\psi\to\mu^+\mu^-\), \(\phi\to K^+K^-\) decays is performed with a data sample of 3 fb\(^{-1}\) of \(pp\) collision data collected at \(\sqrt{s}=7\) and \(8\) TeV with the LHCb detector. The data cannot be described by a model that contains only excited kaon states decaying into \(\phi K^+\), and four \(J/\psi\phi\) structures are observed, each with significance over \(5\) standard deviations. The quantum numbers of these structures are determined with significance of at least \(4\) standard deviations. The lightest is best described as a \(D_s^{\pm}D_s^{*\mp}\) cusp, but a resonant interpretation is also possible with mass consistent with, but width much larger than, previous measurements of the claimed \(X(4140)\) state.