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

Abstract

Let $Σ$ be a compact orientable surface with nonempty boundary, let $\varphi: Σ\to Σ$ be an orientation-preserving pseudo-Anosov homeomorphism, and let $M = Σ\times I / \stackrel{\varphi}{\sim}$ be the mapping torus of $Σ$ over $\varphi$. Let $\mathcal{F}^{s}$ denote the stable foliation of $\varphi$ in $Σ$. 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 $T_i \cap (Σ\times \{0\})$. 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.

Figures (12)

• Figure 1: The dots are singularities of $\mathcal{F}^{s}$ or $\mathcal{F}^{u}$ on a boundary component of $\Sigma$. In $\operatorname{Int}(\Sigma)$, the solid lines are leaves of $\mathcal{F}^{s}$, and the dashed lines are leaves of $\mathcal{F}^{u}$.
• Figure 2: Local models of standard spines, where the shaded regions in (d) and (e) are contained in $\partial M$.
• Figure 3: Local models of branched surfaces, where the shaded regions in (d) and (e) are contained in $\partial M$.
• Figure 4: The fibered neighborhood $N(B)$.
• Figure 5: At a double intersection point of a union of paths positively transverse to $\mathcal{F}^{s}$, we smooth it with respect to the orientations on the paths. Then the paths are still positively transverse to $\mathcal{F}^{s}$.

Theorems & Definitions (53)

• Theorem 1.2: Roberts
• Theorem 1.3: Kalelkar-Roberts
• Theorem 1.4: Gabai
• Theorem 1.5
• Corollary 1.6
• Remark 1.7
• Remark 1.8
• Proposition 1.9
• Remark 1.10
• Corollary 1.11