The double-polarization observable $E$ was studied for the reaction $gamma pto pomega$ using the CEBAF Large Acceptance Spectrometer (CLAS) in Hall B at the Thomas Jefferson National Accelerator Facility and the longitudinally-polarized frozen-spin target (FROST). The observable was measured from the charged decay mode of the meson, $omegatopi^+pi^-pi^0$, using a circularly-polarized tagged-photon beam with energies ranging from the $omega$ threshold at 1.1 to 2.3 GeV. A partial-wave analysis within the Bonn-Gatchina framework found dominant contributions from the $3/2^+$ partial wave near threshold, which is identified with the sub-threshold $N(1720),3/2^+$ nucleon resonance. To describe the entire data set, which consisted of $omega$ differential cross sections and a large variety of polarization observables, further contributions from other nucleon resonances were found to be necessary. With respect to non-resonant mechanisms, $pi$ exchange in the $t$-channel was found to remain small across the analyzed energy range, while pomeron $t$-channel exchange gradually grew from the reaction threshold to dominate all other contributions above $W approx 2$ GeV.
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