Liminal ${\rm SL}_2\mathbb{Z}_p$-representations and odd-th cyclic covers of genus one two-bridge knots
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917225412165632 |
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| author | Sakamoto, Honami Tange, Ryoto Ueki, Jun |
| author_facet | Sakamoto, Honami Tange, Ryoto Ueki, Jun |
| contents | Let $p$ be a prime number and let $K$ be a genus one two-bridge knot. In the spirit of arithmetic topology, we observe that if $p$ divides the size of the 1st homology group of some odd-th cyclic branched cover of the knot $K$, then its group $π_1(S^3-K)$ admits a liminal ${\rm SL}_2\mathbb{Z}_p$-character, where $\mathbb{Z}_p$ denotes the ring of $p$-adic integers. In addition, we discuss the existence of liminal ${\rm SL}_2\mathbb{Z}_p$-representations and give a remark on a general two-bridge knot. In the course of argument, we also point out a constraint for prime numbers dividing certain Lucas-type sequences by using the Legendre symbols. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00323 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Liminal ${\rm SL}_2\mathbb{Z}_p$-representations and odd-th cyclic covers of genus one two-bridge knots Sakamoto, Honami Tange, Ryoto Ueki, Jun Geometric Topology Number Theory Primary 20C12, 57M12, Secondary 57K10, 20E26 Let $p$ be a prime number and let $K$ be a genus one two-bridge knot. In the spirit of arithmetic topology, we observe that if $p$ divides the size of the 1st homology group of some odd-th cyclic branched cover of the knot $K$, then its group $π_1(S^3-K)$ admits a liminal ${\rm SL}_2\mathbb{Z}_p$-character, where $\mathbb{Z}_p$ denotes the ring of $p$-adic integers. In addition, we discuss the existence of liminal ${\rm SL}_2\mathbb{Z}_p$-representations and give a remark on a general two-bridge knot. In the course of argument, we also point out a constraint for prime numbers dividing certain Lucas-type sequences by using the Legendre symbols. |
| title | Liminal ${\rm SL}_2\mathbb{Z}_p$-representations and odd-th cyclic covers of genus one two-bridge knots |
| topic | Geometric Topology Number Theory Primary 20C12, 57M12, Secondary 57K10, 20E26 |
| url | https://arxiv.org/abs/2501.00323 |