The first full amplitude analysis of $B^+to J/psi phi K^+$ with $J/psitomu^+mu^-$, $phito K^+K^-$ 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 $J/psiphi$ 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 has mass consistent with, but width much larger than, previous measurements of the claimed $X(4140)$ state. The model includes significant contributions from a number of expected kaon excitations, including the first observation of the $K^{*}(1680)^+tophi K^+$ transition.
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