Contact surgery numbers of Sigma(2,3,11) and L(4m+3,4)

Abstract

We classify all contact structures with contact surgery number one on the Brieskorn sphere Sigma(2,3,11) with both orientations. We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere. We further prove similar results for some lens spaces: We classify all contact structures with contact surgery number one on lens spaces of the form L(4m+3,4). Along the way, we present an algorithm and a formula for computing the Euler class of a contact structure from a general rational contact surgery description and classify which rational surgeries along Legendrian unknots are tight and which ones are overtwisted.