We study contact structures compatible with genus one open book decompositions
with one boundary component. Any monodromy for such an open book can be
written as a product of Dehn twists around dual nonseparating curves in the
once-punctured torus. Given such a product, we supply an algorithm to determine
whether the corresponding contact structure is tight or overtwisted for all but a small
family of reducible monodromies. We rely on Ozsváth–Szabó Heegaard Floer
homology in our construction and, in particular, we completely identify the
–spaces
with genus one, one boundary component, pseudo-Anosov open book decompositions.
Lastly, we reveal a new infinite family of hyperbolic three-manifolds with no
co-orientable taut foliations, extending the family discovered by Roberts, Shareshian,
and Stein in [J. Amer. Math. Soc. 16 (2003) 639–679]
