Obstructions to free periodicity and symmetric L-space knots
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866915767492018176 |
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| author | Boyle, Keegan Rouse, Nicholas Williams, Ben |
| author_facet | Boyle, Keegan Rouse, Nicholas Williams, Ben |
| contents | We investigate a polynomial factorization problem that naturally arises from Hartley's factorization condition on the Alexander polynomial of freely periodic knots. We give a number-theoretic interpretation of this factorization condition, which allows for efficient computation. As an application, we prove that any polynomial which is not a product of cyclotomic polynomials can be the Alexander polynomial of a freely p-periodic knot for only finitely many p. As a demonstration of the computational efficiency of these methods, we also show that the Alexander polynomial of any freely-periodic L-space knot with genus at most 16 must be a product of cyclotomic polynomials. We conjecture that any periodic or freely periodic L-space knot must be an iterated torus knot. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01705 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Obstructions to free periodicity and symmetric L-space knots Boyle, Keegan Rouse, Nicholas Williams, Ben Geometric Topology 57K10, 57M60, 11R09 We investigate a polynomial factorization problem that naturally arises from Hartley's factorization condition on the Alexander polynomial of freely periodic knots. We give a number-theoretic interpretation of this factorization condition, which allows for efficient computation. As an application, we prove that any polynomial which is not a product of cyclotomic polynomials can be the Alexander polynomial of a freely p-periodic knot for only finitely many p. As a demonstration of the computational efficiency of these methods, we also show that the Alexander polynomial of any freely-periodic L-space knot with genus at most 16 must be a product of cyclotomic polynomials. We conjecture that any periodic or freely periodic L-space knot must be an iterated torus knot. |
| title | Obstructions to free periodicity and symmetric L-space knots |
| topic | Geometric Topology 57K10, 57M60, 11R09 |
| url | https://arxiv.org/abs/2310.01705 |