Tobias Clark | Searcharxiv

arXiv2024

The Enumeration of Alternating Pretzel Links

In this paper, we tabulate the set of alternating pretzel links. Specifically, for any given crossing number $c$, we derive a closed formula that would allow us to compute $\mathcal{P}(c)$, the total number of alternating pretzel links with crossing number $c$. Numerical computation suggests that $\mathcal{P}(c)\approx 0.155e^{0.588c}$. That is, the number of alternating pretzel links with a given crossing number $c$ grows exponentially in terms of $c$.

math.GT↗

arXiv2024

The distribution of braid indices of 2-bridge knots

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

math.GT↗