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{á}th-Szab{ó} 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}.

30 pages, 10 figures. Added figures, extended result to all monodromies, and added sections 5 and 6

References in corpus (12)

• Holomorphic disks and genus bounds
• Laminations and groups of homeomorphisms of the circle
• Heegaard Floer homologies and contact structures
• Overtwisted open books from sobering arcs
• Holomorphic triangles and invariants for smooth four-manifolds
• Planar open book decompositions and contact structures
• Monopoles and lens space surgeries
• Tight contact structures on Seifert manifolds over T^2 with one singular fibre
• Planar open books and Floer homology
• Tight contact structures with no symplectic fillings
• The topology and geometry of contact structures in dimension three
• A note on genus one fibered knots in lens spaces

Cited by in corpus (5)

• Heegaard Floer homology and genus one, one boundary component open books
• On L-spaces and non left-orderable 3-manifold groups
• Dehn surgery, rational open books and knot Floer homology
• Khovanov homology and tight contact structures
• A note on genus one fibered knots in lens spaces