  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 non-separating 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.
We rely on Ozsv{\'a}th-Szab{\'o} Heegaard Floer homology in our construction
and, in particular, we completely identify the $L$-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 in \cite{RSS}.
