Knots in circle bundles are determined by their complements
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913187071262720 |
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| author | Cremaschi, Tommaso Yarmola, Andrew |
| author_facet | Cremaschi, Tommaso Yarmola, Andrew |
| contents | We resolve a case of the oriented knot complement conjecture by showing that knots in an orientable circle bundle $N$ over a genus $g \geq 2$ surface $S$ are determined by their complements. We apply this to the setting of canonical knots in the unit and projective tangent bundles, which are knots that are the set of tangents to a closed curve on $S$. We show that canonical knots have homeomorphic complements if and only if their shadows differ by Reidemeister moves, (de)stabilizations, loops/cusps added by transvections, and mapping classes of $S$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02895 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Knots in circle bundles are determined by their complements Cremaschi, Tommaso Yarmola, Andrew Geometric Topology 57K10, 57K30 We resolve a case of the oriented knot complement conjecture by showing that knots in an orientable circle bundle $N$ over a genus $g \geq 2$ surface $S$ are determined by their complements. We apply this to the setting of canonical knots in the unit and projective tangent bundles, which are knots that are the set of tangents to a closed curve on $S$. We show that canonical knots have homeomorphic complements if and only if their shadows differ by Reidemeister moves, (de)stabilizations, loops/cusps added by transvections, and mapping classes of $S$. |
| title | Knots in circle bundles are determined by their complements |
| topic | Geometric Topology 57K10, 57K30 |
| url | https://arxiv.org/abs/2401.02895 |