The distribution of genera of 2-bridge knots

TL;DR

This paper investigates the distribution of genera in 2-bridge knots, showing that as crossing number increases, the distribution becomes normally distributed with predictable median, mode, and variance.

Contribution

It provides the first detailed analysis of the genus distribution for 2-bridge knots, including median, mode, variance, and asymptotic normality results.

Findings

Median and mode of genus are both approximately (c+2)/4.

Variance of genus approaches c/16 - 17/144.

Distribution of genera is asymptotically normal.

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.