Title: Persistently foliar knots

Abstract: A manifold with Heegard-Floer homology of minimal rank is called an L-space, since this is the case for lens spaces and other elliptic manifolds. A taut co-orientable foliation is associated with non-trivial elements of Heegard-Floer homology (by combined results of Eliashberg-Thurston, Ozsv´ath-Szab´o, Kazez-Roberts); hence, if a 3- manifold admits a taut, co-oriented foliation, it is not an L-space. Conjecturally (Ozsv´ath-Szab´o, Boyer-Gordon-Watson, Juhasz), the converse is also true for irreducible manifolds. Thus far, the evidence from Dehn surgery on knots is consistent with this conjecture. We posit the following 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 a general approach, applicable in a variety of settings, to constructing families of foliations realizing all boundary slopes. 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.