Molecular emission was imaged with ALMA from numerous components near and within bright H2-emitting knots and absorbing dust globules in the Crab Nebula. These observations provide a critical test of how energetic photons and particles produced in a young supernova remnant interact with gas, cleanly differentiating between competing models. The four fields targeted show contrasting properties but within them, seventeen distinct molecular clouds are identified with CO emission; a few also show

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... ission from HCO+, SiO, and/or SO. These observations are compared with Cloudy models of these knots. It has been suggested that the Crab filaments present an exotic environment in which H2 emission comes from a mostly neutral zone probably heated by cosmic rays produced in the supernova surrounding a cool core of molecular gas. Our model is consistent with the observed CO J = 3 − 2 line strength. These molecular line emitting knots in the Crab Nebula present a novel phase of the ISM representative of many important astrophysical environments.