Chebyshev polynomials and Gram determinants from the Möbius band
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866910917199921152 |
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| author | Christiana, Anthony Ibarra, Dionne Montoya-Vega, Gabriel |
| author_facet | Christiana, Anthony Ibarra, Dionne Montoya-Vega, Gabriel |
| contents | This article explores the connection between Chebyshev polynomials and knot theory, specifically in relation to Gram determinants. We reveal intriguing formulae involving the Chebyshev polynomial of the first and second kind. In particular we show that for Mersenne numbers, $M_k=2^k-1$ where $k\geq 2$, the $M_k$-th Chebyshev polynomial of the second kind is the product of Chebyshev polynomials of the first kind. We then discuss the Gram determinant of type $(Mb)_1$, restate the conjecture of its closed formula in terms of mostly products of Chebyshev polynomials of the second kind, and prove a factor of the determinant that supports the conjecture. We also showcase an algorithm for calculating the Gram determinant's corresponding matrix. Furthermore, we restate Qi Chen's conjectured closed formula for the Gram determinant of type Mb and discuss future directions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16439 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chebyshev polynomials and Gram determinants from the Möbius band Christiana, Anthony Ibarra, Dionne Montoya-Vega, Gabriel Geometric Topology Primary: 57K10. Secondary: 57K31 This article explores the connection between Chebyshev polynomials and knot theory, specifically in relation to Gram determinants. We reveal intriguing formulae involving the Chebyshev polynomial of the first and second kind. In particular we show that for Mersenne numbers, $M_k=2^k-1$ where $k\geq 2$, the $M_k$-th Chebyshev polynomial of the second kind is the product of Chebyshev polynomials of the first kind. We then discuss the Gram determinant of type $(Mb)_1$, restate the conjecture of its closed formula in terms of mostly products of Chebyshev polynomials of the second kind, and prove a factor of the determinant that supports the conjecture. We also showcase an algorithm for calculating the Gram determinant's corresponding matrix. Furthermore, we restate Qi Chen's conjectured closed formula for the Gram determinant of type Mb and discuss future directions. |
| title | Chebyshev polynomials and Gram determinants from the Möbius band |
| topic | Geometric Topology Primary: 57K10. Secondary: 57K31 |
| url | https://arxiv.org/abs/2504.16439 |