Co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy
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2023
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| _version_ | 1866915829137801216 |
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| author | Zhao, Bojun |
| author_facet | Zhao, Bojun |
| contents | Let $Σ$ be a compact orientable surface with nonempty boundary, let $φ: Σ\to Σ$ be an orientation-preserving pseudo-Anosov homeomorphism, and let $M = Σ\times I / \stackrelφ{\sim}$ be the mapping torus of $Σ$ over $φ$. Let $\mathcal{F}^{s}$ denote the stable foliation of $φ$ 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 $φ$ 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 $φ$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_01368 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy Zhao, Bojun Geometric Topology Let $Σ$ be a compact orientable surface with nonempty boundary, let $φ: Σ\to Σ$ be an orientation-preserving pseudo-Anosov homeomorphism, and let $M = Σ\times I / \stackrelφ{\sim}$ be the mapping torus of $Σ$ over $φ$. Let $\mathcal{F}^{s}$ denote the stable foliation of $φ$ 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 $φ$ 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 $φ$ 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. |
| title | Co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2310.01368 |