We give new tightness criteria for positive surgeries along knots in the 3-sphere, generalising results of Lisca and Stipsicz, and Sahamie. The main tools will be Honda, Kazez and Matics, Ozsvath and Szabos Floer-theoretic contact invariants. We compute the Ozsvath and Szabos invariant of positive contact surgeries along Legendrian knots in the 3-sphere in terms of the classical invariants of the knot. We also combine a Legendrian cabling construction with contact surgeries to get results about rational contact surgeries.
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