Torus knots, the A-polynomial, and SL(2,C)
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908830351228928 |
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| author | Baldwin, John A. Sivek, Steven |
| author_facet | Baldwin, John A. Sivek, Steven |
| contents | The A-polynomial of a knot is defined in terms of SL(2,C) representations of the knot group, and encodes information about essential surfaces in the knot complement. In 2005, Dunfield-Garoufalidis and Boyer-Zhang proved that it detects the unknot using Kronheimer-Mrowka's work on the Property P conjecture. Here we use more recent results from instanton Floer homology to prove that a version of the A-polynomial detects whether a knot is a torus knot. We moreover completely determine which individual torus knots are detected by this A-polynomial. These results enable progress towards a folklore conjecture about boundary slopes of non-torus knots. Finally, we use similar ideas to prove that a knot in the 3-sphere admits infinitely many SL(2,C)-abelian Dehn surgeries if and only if it is a torus knot, affirming a variant of a conjecture due to Sivek-Zentner. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19197 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torus knots, the A-polynomial, and SL(2,C) Baldwin, John A. Sivek, Steven Geometric Topology The A-polynomial of a knot is defined in terms of SL(2,C) representations of the knot group, and encodes information about essential surfaces in the knot complement. In 2005, Dunfield-Garoufalidis and Boyer-Zhang proved that it detects the unknot using Kronheimer-Mrowka's work on the Property P conjecture. Here we use more recent results from instanton Floer homology to prove that a version of the A-polynomial detects whether a knot is a torus knot. We moreover completely determine which individual torus knots are detected by this A-polynomial. These results enable progress towards a folklore conjecture about boundary slopes of non-torus knots. Finally, we use similar ideas to prove that a knot in the 3-sphere admits infinitely many SL(2,C)-abelian Dehn surgeries if and only if it is a torus knot, affirming a variant of a conjecture due to Sivek-Zentner. |
| title | Torus knots, the A-polynomial, and SL(2,C) |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2405.19197 |