The Braid Indices of Pretzel Links: A Comprehensive Study, Part II

TL;DR

This paper advances the understanding of the braid indices of pretzel links by providing formulas for Type 3 links, completing the classification for all pretzel link types, with most indices precisely determined and conjectures for the remaining cases.

Contribution

It extends previous work by deriving braid index formulas for Type 3 pretzel links, completing the classification for all pretzel link types.

Findings

Precise braid index formulas for Type 1 and Type 2 pretzel links.

Most Type 3 pretzel links have their braid indices determined exactly.

For remaining Type 3 links, braid indices are within two integers, with a conjecture on the exact value.

Abstract

This paper is the second part of our comprehensive study on the braid index problem of pretzel links. Our ultimate goal is to completely determine the braid indices of all pretzel links, alternating or non alternating. In our approach, we divide the pretzel links into three types as follows. Let $D$ be a standard diagram of an oriented pretzel link $\mathcal{L}$, $S(D)$ be the Seifert circle decomposition of $D$, and $C_1$, $C_2$ be the Seifert circles in $S(D)$ containing the top and bottom long strands of $D$ respectively, then $\mathcal{L}$ is classified as a Type 1 (Type 2) pretzel link if $C_1\not=C_2$ and $C_1$, $C_2$ have different (identical) orientations. In the case that $C_1=C_2$, then $\mathcal{L}$ is classified as a Type 3 pretzel link. In our previous paper, we succeeded in reaching our goal for all Type 1 and Type 2 pretzel links. That is, we successfully derived precise braid index formulas for all Type 1 and Type 2 pretzel links. In this paper, we present the results of our study on Type 3 pretzel links. In this case, we are very close to reaching our goal. More precisely, with the exception of a small percentage of Type 3 pretzel links, we are able to determine the precise braid indices for the majority of Type 3 pretzel links. Even for those exceptional ones, we are able to determine their braid indices within two consecutive integers. With some numerical evidence, we conjecture that in such a case, the braid index of the Type 3 pretzel link is given by the larger of the two consecutive integers given by our formulas.