Tunnel number one knots satisfy the Berge Conjecture
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2017
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| _version_ | 1866909816909201408 |
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| author | Li, Tao Moriah, Yoav Pinsky, Tali |
| author_facet | Li, Tao Moriah, Yoav Pinsky, Tali |
| contents | Let $K$ be a tunnel number one knot in $M$ with irreducible knot exterior, where $M$ is either $S^3$, or a connected sum of $S^2\times S^1$ with any lens space. (In particular, this includes $M = S^2\times S^1$.) We prove that if a non-trivial Dehn surgery on $K$ yields a lens space, then $K$ is a doubly primitive knot in $M$. For $M = S^3$ this resolves the tunnel number one Berge Conjecture. For $M = S^2\times S^1$ this resolves a conjecture of Greene and Baker-Buck-Lecuona for tunnel number one knots. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1701_01421 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Tunnel number one knots satisfy the Berge Conjecture Li, Tao Moriah, Yoav Pinsky, Tali Geometric Topology 57M99, 57K10, 57K30 Let $K$ be a tunnel number one knot in $M$ with irreducible knot exterior, where $M$ is either $S^3$, or a connected sum of $S^2\times S^1$ with any lens space. (In particular, this includes $M = S^2\times S^1$.) We prove that if a non-trivial Dehn surgery on $K$ yields a lens space, then $K$ is a doubly primitive knot in $M$. For $M = S^3$ this resolves the tunnel number one Berge Conjecture. For $M = S^2\times S^1$ this resolves a conjecture of Greene and Baker-Buck-Lecuona for tunnel number one knots. |
| title | Tunnel number one knots satisfy the Berge Conjecture |
| topic | Geometric Topology 57M99, 57K10, 57K30 |
| url | https://arxiv.org/abs/1701.01421 |