On detection probabilities of link invariants
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912737312899072 |
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| author | Kelomäki, Tuomas Lacabanne, Abel Tubbenhauer, Daniel Vaz, Pedro Zhang, Victor L. |
| author_facet | Kelomäki, Tuomas Lacabanne, Abel Tubbenhauer, Daniel Vaz, Pedro Zhang, Victor L. |
| contents | We prove that the detection rate of n-crossing alternating links by many standard link invariants decays exponentially in n, implying that they detect alternating links with probability zero. This phenomenon applies broadly, in particular to the Jones and HOMFLYPT polynomials and integral Khovanov homology. We also use a big-data approach to analyze knots and provide evidence that, for knots as well, these invariants exhibit the same asymptotic failure of detection. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05574 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On detection probabilities of link invariants Kelomäki, Tuomas Lacabanne, Abel Tubbenhauer, Daniel Vaz, Pedro Zhang, Victor L. Geometric Topology Machine Learning Quantum Algebra Primary: 57K16, 62R07, secondary: 57K18, 68P05 We prove that the detection rate of n-crossing alternating links by many standard link invariants decays exponentially in n, implying that they detect alternating links with probability zero. This phenomenon applies broadly, in particular to the Jones and HOMFLYPT polynomials and integral Khovanov homology. We also use a big-data approach to analyze knots and provide evidence that, for knots as well, these invariants exhibit the same asymptotic failure of detection. |
| title | On detection probabilities of link invariants |
| topic | Geometric Topology Machine Learning Quantum Algebra Primary: 57K16, 62R07, secondary: 57K18, 68P05 |
| url | https://arxiv.org/abs/2509.05574 |