The distribution of genera of 2-bridge knots

Moshe Cohen; Abigail DiNardo; Adam M. Lowrance; Steven Raanes; Izabella M. Rivera; Andrew J. Steindl; Ella S. Wanebo

doi:10.4064/fm230719-2-10

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arXiv · 2307.09399

The distribution of genera of 2-bridge knots

Moshe Cohen·Abigail DiNardo·Adam M. Lowrance·Steven Raanes·Izabella M. Rivera·Andrew J. Steindl·Ella S. Wanebo

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Abstract

The average genus of a 2-bridge knot with crossing number $c$ approaches $\frac{c}{4} + \frac{1}{12}$ as $c$ approaches infinity, as proven by Suzuki and Tran and independently Cohen and Lowrance. In this paper, for the genera of $2$-bridge knots of a fixed crossing number $c$, we show that the median and mode are both $\lfloor \frac{c+2}{4} \rfloor$ and that the variance approaches $\frac{c}{16}-\frac{17}{144}$ as $c$ approaches infinity. We prove that the distribution of genera of 2-bridge knots is asymptotically normal.

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Moshe Cohen, Abigail DiNardo, Adam M. Lowrance, Steven Raanes, Izabella M. Rivera, Andrew J. Steindl, Ella S. Wanebo. 2023-07-17. The distribution of genera of 2-bridge knots. https://doi.org/10.4064/fm230719-2-10

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