Tight contact structures and genus one fibered knots

Abstract

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}.