Arithmetic Kei Theory
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913466584924160 |
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| author | Davis, Ariel Schlank, Tomer M |
| author_facet | Davis, Ariel Schlank, Tomer M |
| contents | A kei, or 2-quandle, is an algebraic structure one can use to produce a numerical invariant of links, known as coloring invariants. Motivated by Mazur's analogy between prime numbers and knots, we define for every finite kei $\mathcal{K}$ an analogous coloring invariant $\textrm{col}_{\mathcal K}(n)$ of square-free integers. This is achieved by defining a fundamental kei for every such $n$. We conjecture that the asymptotic average order of $\textrm{col}_{\mathcal K}$ can be predicted to some extent by the colorings of random braid closures. This conjecture is fleshed out in general, building on previous work, and then proven for several cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05489 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arithmetic Kei Theory Davis, Ariel Schlank, Tomer M Number Theory Geometric Topology A kei, or 2-quandle, is an algebraic structure one can use to produce a numerical invariant of links, known as coloring invariants. Motivated by Mazur's analogy between prime numbers and knots, we define for every finite kei $\mathcal{K}$ an analogous coloring invariant $\textrm{col}_{\mathcal K}(n)$ of square-free integers. This is achieved by defining a fundamental kei for every such $n$. We conjecture that the asymptotic average order of $\textrm{col}_{\mathcal K}$ can be predicted to some extent by the colorings of random braid closures. This conjecture is fleshed out in general, building on previous work, and then proven for several cases. |
| title | Arithmetic Kei Theory |
| topic | Number Theory Geometric Topology |
| url | https://arxiv.org/abs/2408.05489 |