Every lens space that supports a non-free toric contact action has exactly one tight and two overtwisted contact structures up to contactomorphism, and all are concave boundaries of linear plumbings.
Classification of overtwisted contact structures on 3-manifolds
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Contact 3-manifolds that admit a non-free toric action
Every lens space that supports a non-free toric contact action has exactly one tight and two overtwisted contact structures up to contactomorphism, and all are concave boundaries of linear plumbings.