Question: Is there a classification of almost contact or contact structures on $S^3$? What is it and references?
The motivation of this question is as follows:
(1) There is one paper showing that a supersymmetric gauge theory can be defined on a 3-manifold $M$ for any almost contact structure that $M$ admits.
(2) On String 2013 conference, there is a talk on supersymmetric partition functions in three dimension, which claims that they are invariants of almost contact structures.
(3) Many papers have computed partition functions on $S^3$ with different metrics (with $U(1) \times U(1)$, $SU(2)\times U(1)$ and $SU(2) \times SU(2)$ isometries, with various auxiliary fields), and the results are all similar, with only one parameter call $b$.
So it seems there is only 1-parameter family of almost contact structures on $S^3$, and I would like to know if that is the true story.
Thank you!
