  We define symplectic fractional twists, which generalize Dehn twists, and use
these in open books to investigate contact structures. The resulting contact
structures are invariant under a circle action, and share several similarities
with the invariant contact structures that were studied by Lutz and Giroux. We
show that left-handed fractional twists often give rise to non-fillable contact
manifolds. These manifolds are in fact "algebraically overtwisted", yet they do
not seem to contain bLobs, nor are they directly related to negative
stabilizations. We also show that the Weinstein conjecture holds for the
non-fillable contact manifolds we construct, and we investigate the symplectic
isotopy problem for fractional twists.
