The Enumeration of Alternating Pretzel Links

TL;DR

This paper provides a closed-form formula for counting alternating pretzel links with a given crossing number, revealing exponential growth in their enumeration as crossing number increases.

Contribution

It introduces a closed formula for enumerating alternating pretzel links based on crossing number, advancing understanding of their combinatorial complexity.

Findings

Number of alternating pretzel links grows exponentially with crossing number.

Derived a closed formula for counting these links.

Numerical estimates suggest exponential growth rate.

Abstract

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 $c$.