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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
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2023
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| Accès en ligne: | https://arxiv.org/abs/2308.04654 |
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| _version_ | 1866915181826670592 |
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| author | Abramson, Heather Chesebro, Eric Cummins, Vivian Emlen, Cory Ke, Kenton Grady, Ryan |
| author_facet | Abramson, Heather Chesebro, Eric Cummins, Vivian Emlen, Cory Ke, Kenton Grady, Ryan |
| contents | This paper concerns the relationships between continued fractions and the geometry of the Stern-Brocot diagram. Each rational number can be expressed as a continued fraction $[a_0; a_1, \ldots, a_n]$ whose terms $a_i$ are integers and are positive if $i \geq 1$. Select an index $i \in \{ 1, \ldots, n \}$ and replace $a_i$ with an integer $m$ to obtain a continued fraction expansion for an extended rational $α_m \in \mathbb{Q} \cup \{ \infty \}$. This paper shows that the vertices of the Stern-Brocot diagram corresponding to the numbers $\{ α_m \}_{m \in \mathbb{Z}}$ lie on a pair of (extended) Euclidean lines across the diagram. The slopes of these two lines differ only by a sign change and they meet at the point $L=\left([a_0; a_1, \ldots, a_{i-1}], 0\right) \in \mathbb{R}^2$. Moreover, as $\lvert m \rvert \to \infty$, the associated vertices move down these lines and converge to $L$. This paper concludes with a discussion which interprets this result in the context of 2-bridge link complements and Thurston's work on hyperbolic Dehn surgery. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_04654 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Continued fractions and lines across the Stern--Brocot diagram Abramson, Heather Chesebro, Eric Cummins, Vivian Emlen, Cory Ke, Kenton Grady, Ryan Geometric Topology Combinatorics 57M50, 11B57 This paper concerns the relationships between continued fractions and the geometry of the Stern-Brocot diagram. Each rational number can be expressed as a continued fraction $[a_0; a_1, \ldots, a_n]$ whose terms $a_i$ are integers and are positive if $i \geq 1$. Select an index $i \in \{ 1, \ldots, n \}$ and replace $a_i$ with an integer $m$ to obtain a continued fraction expansion for an extended rational $α_m \in \mathbb{Q} \cup \{ \infty \}$. This paper shows that the vertices of the Stern-Brocot diagram corresponding to the numbers $\{ α_m \}_{m \in \mathbb{Z}}$ lie on a pair of (extended) Euclidean lines across the diagram. The slopes of these two lines differ only by a sign change and they meet at the point $L=\left([a_0; a_1, \ldots, a_{i-1}], 0\right) \in \mathbb{R}^2$. Moreover, as $\lvert m \rvert \to \infty$, the associated vertices move down these lines and converge to $L$. This paper concludes with a discussion which interprets this result in the context of 2-bridge link complements and Thurston's work on hyperbolic Dehn surgery. |
| title | Continued fractions and lines across the Stern--Brocot diagram |
| topic | Geometric Topology Combinatorics 57M50, 11B57 |
| url | https://arxiv.org/abs/2308.04654 |