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

TL;DR

This paper classifies contact structures with contact surgery number one on certain Brieskorn spheres and lens spaces, revealing infinitely many non-isotopic structures not obtainable from the standard tight contact 3-sphere, and provides algorithms for computing Euler classes.

Contribution

It introduces a classification of contact structures with surgery number one on specific manifolds and develops algorithms for computing Euler classes from rational contact surgeries.

Findings

Classified all contact structures with contact surgery number one on Sigma(2,3,11) and L(4m+3,4).

Proved existence of infinitely many non-isotopic contact structures not derived from the standard tight sphere.

Presented an algorithm and formula for Euler class computation from rational contact surgeries.

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.