Boundary slopes (nearly) bound exceptional slopes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866915355550547968 |
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| author | Ichihara, Kazuhiro Mattman, Thomas W. |
| author_facet | Ichihara, Kazuhiro Mattman, Thomas W. |
| contents | For a hyperbolic knot in $S^3$, Dehn surgery along slope $r \in \Q \cup \{\frac10\}$ is {\em exceptional} if it results in a non-hyperbolic manifold. We say meridional surgery, $r = \frac10$, is {\em trivial} as it recovers the manifold $S^3$. We provide evidence in support of two conjectures. The first (inspired by a question of Professor Motegi) states that there are boundary slopes $b_1 < b_2$ such that all non-trivial exceptional surgeries occur, as rational numbers, in the interval $[b_1,b_2]$. We say a boundary slope is {\em NIT} if it is non-integral or toroidal. Second, when there are non-trivial exceptional surgeries, we conjecture there are NIT boundary slopes $b_1 \leq b_2$ so that the exceptional surgeries lie in $[\floor{b_1},\ceil{b_2}]$. Moreover, if $\ceil{b_1} \leq \floor{b_2}$, the integers in the interval $[ \ceil{b_1}, \floor{b_2} ]$ are all exceptional surgeries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_09918 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Boundary slopes (nearly) bound exceptional slopes Ichihara, Kazuhiro Mattman, Thomas W. Geometric Topology Primary 57K32, Secondary K7K35, 57K10 For a hyperbolic knot in $S^3$, Dehn surgery along slope $r \in \Q \cup \{\frac10\}$ is {\em exceptional} if it results in a non-hyperbolic manifold. We say meridional surgery, $r = \frac10$, is {\em trivial} as it recovers the manifold $S^3$. We provide evidence in support of two conjectures. The first (inspired by a question of Professor Motegi) states that there are boundary slopes $b_1 < b_2$ such that all non-trivial exceptional surgeries occur, as rational numbers, in the interval $[b_1,b_2]$. We say a boundary slope is {\em NIT} if it is non-integral or toroidal. Second, when there are non-trivial exceptional surgeries, we conjecture there are NIT boundary slopes $b_1 \leq b_2$ so that the exceptional surgeries lie in $[\floor{b_1},\ceil{b_2}]$. Moreover, if $\ceil{b_1} \leq \floor{b_2}$, the integers in the interval $[ \ceil{b_1}, \floor{b_2} ]$ are all exceptional surgeries. |
| title | Boundary slopes (nearly) bound exceptional slopes |
| topic | Geometric Topology Primary 57K32, Secondary K7K35, 57K10 |
| url | https://arxiv.org/abs/2309.09918 |