We prove that every closed, connected contact 3-manifold can be obtained from S 3 with its standard contact structure by contact (±1)-surgery along a Legendrian link. As a corollary, we derive a result of Etnyre and Honda about symplectic cobordisms (in slightly stronger form). Mathematics Subject Classification (2000). Primary 53D35; Secondary 57R65, 57R90.

We study weak versus strong symplectic fillability of some tight contact structures on torus bundles over the circle. In particular, we prove that almost all of these tight contact structures are weakly, but not strongly symplectically fillable. For the 3-torus this theorem was established by Eliashberg.

It is shown that Legendrian (resp. transverse) cable links in S 3 with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston-Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space J 1 (S 1 ) with its standard tight contact structure.

Abstract. According to Giroux, contact manifolds can be described as open books whose pages are Stein manifolds. For 5-dimensional contact manifolds the pages are Stein surfaces, which permit a description via Kirby diagrams. We introduce handle moves on such diagrams that do not change the corresponding contact manifold. As an application, we derive classification results for subcritically Stein fillable contact 5-manifolds and characterise the standard contact structure on the 5-sphere in terms of such fillings. This characterisation is discussed in the context of the Andrews-Curtis conjecture concerning presentations of the trivial group. We further illustrate the use of such diagrams by a covering theorem for simply connected spin 5-manifolds and a new existence proof for contact structures on simply connected 5-manifolds.

Abstract. We describe various handle moves in contact surgery diagrams, notably contact analogues of the Kirby moves. As an application of these handle moves, we discuss the respective classifications of long and loose Legendrian knots.