The first full amplitude analysis of B+→J/ψφK+ with J/ψ→μ+μ-, φ→K+K- decays is performed with a data sample of 3 fb-1 of pp collision data collected at 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 φK+, and four J/ψφ 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)+→φK+ transition.