The distribution of braid indices of 2-bridge knots
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914642610094080 |
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| author | Clark, Tobias Frank, Jeremy Lowrance, Adam M. |
| author_facet | Clark, Tobias Frank, Jeremy Lowrance, Adam M. |
| contents | In this article we study the braid indices of 2-bridge knots with a fixed crossing number $c$. We show that the average braid index of the set of $2$-bridge knots of crossing number $c$ is asymptotically linear, approaching $\frac{c}{3}+\frac{11}{9}$. Additionally, we show that the variance of the braid indices of the set of $2$-bridge knots of crossing number $c$ is also asymptotically linear, approaching $\frac{2c}{27} - \frac{10}{81}$. Finally, we find a formula for the number $k_{c,b}$ of $2$-bridge knots with crossing number $c$ and braid index $b$, and show that for any fixed $c$, the braid index where $k_{c,b}$ achieves its maximum is $b=\left\lceil \frac{c}{3}\right\rceil +1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_08388 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The distribution of braid indices of 2-bridge knots Clark, Tobias Frank, Jeremy Lowrance, Adam M. Geometric Topology 57K10 In this article we study the braid indices of 2-bridge knots with a fixed crossing number $c$. We show that the average braid index of the set of $2$-bridge knots of crossing number $c$ is asymptotically linear, approaching $\frac{c}{3}+\frac{11}{9}$. Additionally, we show that the variance of the braid indices of the set of $2$-bridge knots of crossing number $c$ is also asymptotically linear, approaching $\frac{2c}{27} - \frac{10}{81}$. Finally, we find a formula for the number $k_{c,b}$ of $2$-bridge knots with crossing number $c$ and braid index $b$, and show that for any fixed $c$, the braid index where $k_{c,b}$ achieves its maximum is $b=\left\lceil \frac{c}{3}\right\rceil +1$. |
| title | The distribution of braid indices of 2-bridge knots |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/2401.08388 |