A rational homology solid torus is a Heegaard Floer homology solid torus if and only if the Dehn filling along its rational longitude contains a non-separating 2-sphere.
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Detecting Heegaard Floer homology solid tori
A rational homology solid torus is a Heegaard Floer homology solid torus if and only if the Dehn filling along its rational longitude contains a non-separating 2-sphere.