Appears in collection : Foliations and Diffeomorphism Groups / Feuilletages et Groupes de Difféomorphisme

Taut co-orientable foliations are associated with non-trivial elements of Heegard-Floer homology, hence, if a 3-manifold admits a taut, co-oriented foliation, it is not an L-space (Kronheimer-Mrowka-Ozsváth-Szabó). Conjecturally (Boyer-Gordon-Watson, Juhász), the converse is also true for irreducible manifolds. Thus far, the evidence from Dehn surgery on knots in S3 is consistent with this conjecture. We consider the L-space Knot Conjecture: if a knot has no reducible or L-space surgeries, then it is persistently foliar, meaning that for each boundary slope there is a taut, co-oriented foliation meeting the boundary of the knot complement in curves of that slope. For rational slopes, these foliations may be capped off by disks to obtain a taut, co-oriented foliation in every manifold obtained by Dehn surgery on that knot. I will describe an approach, applicable in a variety of settings, to constructing families of foliations realizing all boundary slopes. Recalling the work of Ghiggini, Hedden, Ni, Ozsváth-Szabó (and more recently, Juhász and Baldwin-Sivek) revealed that Dehn surgery on a knot in S3 can yield an L-space only if the knot is fibered and strongly quasipositive, we note that this approach seems to apply more easily when the knot is far from being fibered. As applications of this approach, we find that among the alternating and Montesinos knots, all those without reducible or L-space surgeries are persistently foliar. In addition, we find that any connected sum of alternating knots, Montesinos knots, or fibered knots is persistently foliar. Furthermore, any composite knot with a persistently foliar summand is easily shown to be persistently foliar. This work is joint with Charles Delman.