JSJ decompositions of knot exteriors, Dehn surgery and the $L$-space conjecture
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917780630011904 |
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| author | Boyer, Steven Gordon, Cameron McA. Hu, Ying |
| author_facet | Boyer, Steven Gordon, Cameron McA. Hu, Ying |
| contents | In this article, we apply slope detection techniques to study properties of toroidal $3$-manifolds obtained by performing Dehn surgeries on satellite knots in the context of the $L$-space conjecture. We show that if $K$ is an $L$-space knot or admits an irreducible rational surgery with non-left-orderable fundamental group, then the JSJ graph of its exterior is a rooted interval. Consequently, any rational surgery on a composite knot has a left-orderable fundamental group. This is the left-orderable counterpart of Krcatovich's result on the primeness of $L$-space knots, which we reprove using our methods. Analogous results on the existence of co-orientable taut foliations are proved when the knot has a fibred companion. Our results suggest a new approach to establishing the counterpart of Krcatovich's result for surgeries with co-orientable taut foliations, on which partial results have been achieved by Delman and Roberts. Finally, we prove results on left-orderable $p/q$-surgeries on knots with $p$ small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06815 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | JSJ decompositions of knot exteriors, Dehn surgery and the $L$-space conjecture Boyer, Steven Gordon, Cameron McA. Hu, Ying Geometric Topology 57M05, 57M50 (Primary) 57M99 (Secondary) In this article, we apply slope detection techniques to study properties of toroidal $3$-manifolds obtained by performing Dehn surgeries on satellite knots in the context of the $L$-space conjecture. We show that if $K$ is an $L$-space knot or admits an irreducible rational surgery with non-left-orderable fundamental group, then the JSJ graph of its exterior is a rooted interval. Consequently, any rational surgery on a composite knot has a left-orderable fundamental group. This is the left-orderable counterpart of Krcatovich's result on the primeness of $L$-space knots, which we reprove using our methods. Analogous results on the existence of co-orientable taut foliations are proved when the knot has a fibred companion. Our results suggest a new approach to establishing the counterpart of Krcatovich's result for surgeries with co-orientable taut foliations, on which partial results have been achieved by Delman and Roberts. Finally, we prove results on left-orderable $p/q$-surgeries on knots with $p$ small. |
| title | JSJ decompositions of knot exteriors, Dehn surgery and the $L$-space conjecture |
| topic | Geometric Topology 57M05, 57M50 (Primary) 57M99 (Secondary) |
| url | https://arxiv.org/abs/2307.06815 |