The distribution of braid indices of 2-bridge knots

TL;DR

The paper analyzes the distribution of braid indices for 2-bridge knots with fixed crossing number $c$, deriving asymptotic mean and variance, and obtaining exact counts $k_{c,b}$ for knots with crossing number $c$ and braid index $b$. Using even continued fraction representations and Schubert's classification, it provides recursive and closed formulas for $e(c,b)$, $e_p(c,b)$, and hence $k_{c,b}$, showing the mode lies at $b= frac{c}{3}+1$ (rounded). It further defines total braid index sums and derives recursive and closed forms for $E(\operatorname{Braid}_c)$ and $\operatorname{Var}(\operatorname{Braid}_c)$, proving that the mean grows linearly with $c$ and the variance grows as $\sim \tfrac{2}{27}c$, with precise constants and corrections. Overall, the work provides a detailed probabilistic understanding of braid indices in the 2-bridge knot family and establishes sharp asymptotics for their distribution.

Abstract

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 $b=\left\lceil \frac{c}{3}\right\rceil +1$.

Figures (2)

• Figure 1: Top. A 2-bridge knot represented by the tuple $(a_1,a_2,\dots,a_{2m})$. Bottom. The 2-bridge knot represented by the tuple $(a_1,a_2,\dots,a_{2m+1})$.
• Figure 2: The knot $8_{14}$ is the two bridge knot $K(2,-4,2,2)$.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Theorem 2.5
• Theorem 2.6