Co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy

Abstract

Let $\Sigma$ be a compact orientable surface with nonempty boundary, let $\phi :\Sigma \to \Sigma$ be an orientation-preserving pseudo-Anosov homeomorphism, and let $M=\Sigma \times I/\stackrel{\phi }{\sim }$ be the mapping torus of $\Sigma$ over $\phi$. Let ${\mathcal{F}}^s$ denote the stable foliation of $\phi$ in $\Sigma$. Let $T_1,\dots ,T_k$ denote the boundary components of $M$. With respect to a canonical choice of meridian and longitude on each $T_i$, the degeneracy locus of the suspension flow of $\phi$ on $T_i$ can be identified with a pair of integers $(p_i;q_i)$ such that $p_i>0$ and $-\frac{1}{2}{p}_i<q_i⩽\frac{1}{2}{p}_i$. Let $c_i$ denote the number of components of $T_i\cap (\Sigma \times \{0\})$. Assume that ${\mathcal{F}}^s$ is co-orientable and $\phi$ reverses the co-orientation on ${\mathcal{F}}^s$. We show that the Dehn filling of $M$ along $\partial M$ with any multislope in $J_1\times \dots \times J_k$ admits a co-orientable taut foliation, where $J_i$ is one of the two open intervals in $\mathbb{R}\cup \{\infty \}\cong \mathbb{R}{P}^1$ between $\frac{p_i}{q_i+c_i},\frac{p_i}{q_i-c_i}$ which doesn't contain $\frac{p_i}{q_i}$. For some hyperbolic fibered knot manifolds, the slopes given above contain all slopes that yield non-L-space Dehn filllings. The examples include (1) the exterior of the $(-2,3,2q+1)$-pretzel knot in $S^3$ for each $q\in {\mathbb{Z}}_{⩾3}$ (see \hyperref[Kri]{[Kri]} for a previous proof), (2) the exteriors of many L-space knots in lens spaces.

23 pages, 10 figures. Comments are welcome; v2: accepted version. To appear in Advances in Mathematics