The diffeotopy group of S¹ times S² via contact topology

As shown by H. Gluck in 1962, the diffeotopy group of S^1 \times S^2 is isomorphic to Z_2 + Z_2 + Z_2. Here an alternative proof of this result is given, relying on contact topology. We then discuss two applications to contact topology: (i) it is shown that the fundamental group of the space of contact structures on S^1 \times S^2, based at the standard tight contact structure, is isomorphic to the integers; (ii) inspired by previous work of M. Fraser, an example is given of an integer family of Legendrian knots in S^1 \times S^2 # S^1 \times S^2 (with its standard tight contact structure) that can be distinguished with the help of contact surgery, but not by the classical invariants (topological knot type, Thurston-Bennequin invariant, and rotation number).