A Diagrammatic Approach for Determining the Braid Index of Alternating Links

TL;DR

This paper introduces a diagrammatic method to determine the braid index of alternating links, providing explicit formulas and applying them to classify braid indices of various classes including Montesinos links.

Contribution

It develops explicit formulas for the braid index of alternating links based on minimal projections, extending understanding to Montesinos, rational, and pretzel links.

Findings

Derived explicit formulas for braid index of many alternating links.

Determined braid index for all alternating Montesinos links.

Provided a new diagrammatic approach to link braid index calculation.

Abstract

It is well known that the braid index of a link equals the minimum number of Seifert circles among all link diagrams representing it. For a link with a reduced alternating diagram $D$, $s(D)$, the number of Seifert circles in $D$, equals the braid index $\textbf{b}(D)$ of $D$ if $D$ contains no {\em lone crossings} (a crossing in $D$ is called a {\em lone crossing} if it is the only crossing between two Seifert circles in $D$). If $D$ contains lone crossings, then $\textbf{b}(D)$ is strictly less than $s(D)$. However in general it is not known how $s(D)$ is related to $\textbf{b}(D)$. In this paper, we derive explicit formulas for many alternating links based on any minimum projections of these links. As an application of our results, we are able to determine the braid index for any alternating Montesinos link explicitly (which include all rational links and all alternating pretzel links).