Contact homology of left-handed Dehn-Seidel twists and connected sums

We show that on any closed contact manifold of dimension >1 a contact structure with vanishing contact homology can be constructed. The basic idea for the construction comes from Giroux. We use a special open book decomposition for spheres. The page is the cotangent bundle of a sphere and the monodromy is given by a left-handed Dehn twist. In the resulting contact manifold we exhibit a closed Reeb orbit that bounds a single finite energy plane. As a result, the unit element of the contact homology algebra is exact and so the contact homology vanishes. We extend this result to other contact manifolds by using connected sums. Under suitable conditions we can show that finite energy planes cannot pass the middle of the ``connecting tube'' of the connected sum. This is then used to show that the unit element of contact homology is also exact in these situations. An additional application of the connecting tube argument is a long exact sequence for cylindrical contact homology of connected sums.