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We study cosmetic contact surgeries along transverse knots in the standardcontact 3-sphere, i.e. contact surgeries that yield again the standard contact3-sphere. The main result is that we can exclude non-trivial cosmetic contactsurgeries along all transverse knots not isotopic to the transverse unknot withself-linking number -1. As a corollary it follows that every transverse knot inthe standard contact 3-sphere is determined by the contactomorphism type of itsexteriors. Moreover, we give counterexamples to this for transverse links inthe standard contact 3-sphere.

Cosmetic contact surgeries along transverse knots and the knot complement problem