The Braid Indices of the Reverse Parallel Links of Alternating Knots

TL;DR

This paper provides an exact formula for the braid index of reverse parallel links of alternating knots, based on properties of their diagrams, filling a gap in understanding the braid indices of such links.

Contribution

It introduces a precise method to determine the braid index of reverse parallel links of alternating knots, which was previously unknown.

Findings

Derived a formula for braid index based on diagram properties

Established the relationship between braid index and framing for alternating knots

Provided a computational approach for specific classes of links

Abstract

The braid indices of most links remain unknown as there is no known universal method that can be used to determine the braid index of an arbitrary knot. This is also the case for alternating knots. In this paper, we show that if $K$ is an alternating knot, then the braid index of any reverse parallel link of $K$ can be precisely determined. More precisely, if $D$ is a reduced diagram of $K$, $v_+(D)$ ($v_-(D)$) is the number of regions in the checkerboard shading of $D$ for which all crossings are positive (negative), $w(D)$ is the writhe of $D$, then the braid index of a reverse parallel link of $K$ with framing $f$, denoted by $\mathbb{K}_f$, is given by the following precise formula $$\textbf{b}(\mathbb{K}_f)=\left\{ \begin{array}{ll} c(D)+2+a(D)-f, &\ {\rm if}\ f < a(D),\\ c(D)+2, &\ {\rm if}\ a(D)\le f \le b(D),\\ c(D)+2-b(D)+f, &\ {\rm if}\ f > b(D),\\ \end{array} \right. $$ where $a(D)=-v_-(D)+w(D)$ and $b(D)=v_+(D)+w(D)$.