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

TL;DR

This paper investigates conditions under which Dehn fillings of certain pseudo-Anosov mapping tori admit co-orientable taut foliations, especially focusing on slopes that produce non-L-space manifolds, with applications to specific knot exteriors.

Contribution

It provides a new criterion for the existence of co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy, extending understanding of L-space fillings.

Findings

Dehn fillings along specific slopes admit co-orientable taut foliations.

Identifies slopes that yield non-L-space Dehn fillings for certain hyperbolic fibered knots.

Includes examples with pretzel knots and L-space knots in lens spaces.

Abstract

Let $\Sigma$ be a compact orientable surface with nonempty boundary, let $\varphi: \Sigma \to \Sigma$ be an orientation-preserving pseudo-Anosov homeomorphism, and let $M = \Sigma \times I / \stackrel{\varphi}{\sim}$ be the mapping torus of $\Sigma$ over $\varphi$. Let $\mathcal{F}^{s}$ denote the stable foliation of $\varphi$ in $\Sigma$. Let $T_1, \ldots, T_k$ 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 $\varphi$ on $T_i$ can be identified with a pair of integers $(p_i; q_i)$ such that $p_i > 0$ and $-\frac{1}{2}p_i < q_i \leqslant \frac{1}{2}p_i$. Let $c_i$ denote the number of components of $T_i \cap (\Sigma \times \{0\})$. Assume that $\mathcal{F}^{s}$ is co-orientable and $\varphi$ reverses the co-orientation on $\mathcal{F}^{s}$. We show that the Dehn filling of $M$ along $\partial M$ with any multislope in $J_1 \times \ldots \times J_k$ 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 $q \in \mathbb{Z}_{\geqslant 3}$ (see \hyperref[Kri]{[Kri]} for a previous proof), (2) the exteriors of many L-space knots in lens spaces.