Unity of Jones polynomials in the unit circle and the plane
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910053936660480 |
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| author | Jablonowski, Michal |
| author_facet | Jablonowski, Michal |
| contents | In this note, we study solutions of the equation $J_K(t)=1$ for the Jones polynomial of knots and links. For the family $K_n$ of double-twist knots, we show that every root of unity (except $-1$) satisfies $J_{K_n}(ζ)=1$ for some $n$. Consequently, the set of solutions to $J_{K_n}(t)=1$ arising from this family is dense in the unit circle. We further show that there exists a family of links for which the zeros of $J_L(t)-1$ are dense in the complex plane, adapting the density mechanism of Jin--Zhang--Dong--Tay for Jones polynomial zeros. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_14585 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unity of Jones polynomials in the unit circle and the plane Jablonowski, Michal Geometric Topology In this note, we study solutions of the equation $J_K(t)=1$ for the Jones polynomial of knots and links. For the family $K_n$ of double-twist knots, we show that every root of unity (except $-1$) satisfies $J_{K_n}(ζ)=1$ for some $n$. Consequently, the set of solutions to $J_{K_n}(t)=1$ arising from this family is dense in the unit circle. We further show that there exists a family of links for which the zeros of $J_L(t)-1$ are dense in the complex plane, adapting the density mechanism of Jin--Zhang--Dong--Tay for Jones polynomial zeros. |
| title | Unity of Jones polynomials in the unit circle and the plane |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2603.14585 |