Taut foliations from double-diamond replacements

Abstract

A 3-manifold is foliar if it supports a codimension-one co-oriented taut foliation. Suppose M is an oriented 3-manifold with connected boundary a torus, and suppose M contains a properly embedded, compact, oriented, surface R with a single boundary component that is Thurston norm minimizing in H 2 (M, ∂M ). We define a readily recognizable type of sutured manifold decomposition, which for notational reasons we call double-diamond taut, and show that if R admits a double-diamond taut sutured manifold decomposition, then every boundary slope except one is strongly realized by a co-oriented taut foliation; that is, the foliation intersects ∂M transversely in a foliation by curves of that slope. In the case that M is the complement of a knot κ in S 3 , the exceptional filling is the meridional one, and hence κ is persistently foliar, by which we mean that every non-trivial slope is strongly realized; hence, restricting attention to rational slopes, every manifold obtained by non-trivial Dehn surgery along κ is foliar. In particular, if R is a Murasugi sum of surfaces R 1 and R 2 , where R 2 is an unknotted band with an even number 2m ≥ 4 of half-twists, then κ = ∂R is persistently foliar. Definition 1.2. A group G is left-orderable if its elements can be given a strict total ordering < which is left invariant, meaning that g < h implies f g < f h for all f, g, h ∈ G.

Euler class of taut foliations and Dehn filling

“…Since b 1 (X) > 0, essential surfaces contained in X often serve as a convenient starting point for sutured manifold decomposition [Gab83,Gab87] and for building branched surfaces. As a result, taut foliations on manifolds obtained by Dehn filling are usually constructed by first building one on X and then extending it to the entire closed manifold using either meridional disks or stacking chairs (see [Rob95,Rob01a,Rob01b,LR14,Kri18,DR19a,DR19b] for instance). These foliations are transverse to the core of N .…”