Arc index of pretzel knots of type $(-p,q,r)$

TL;DR

This paper calculates the arc index for certain pretzel knots of type $(-p,q,r)$, revealing specific relationships between arc index and minimal crossing number based on parameters.

Contribution

It provides explicit formulas for the arc index of pretzel knots $(-p,q,r)$ under various conditions, advancing understanding of their geometric properties.

Findings

Arc index equals minimal crossing number when $q=2$.

Arc index is one less than minimal crossing number when $p ext{ } ext{and} ext{ } q=3$.

Arc index is two less than minimal crossing number when $p ext{ } ext{and} ext{ } q=4$.

Abstract

We computed the arc index for some of the pretzel knots $K=P(-p,q,r)$ with $p,q,r\ge2$, $r\geq q$ and at most one of $p,q,r$ is even. If $q=2$, then the arc index $\alpha(K)$ equals the minimal crossing number $c(K)$. If $p\ge3$ and $q=3$, then $\alpha(K)=c(K)-1$. If $p\ge5$ and $q=4$, then $\alpha(K)=c(K)-2$.