The braid indices of the reverse parallel links of alternating knots

Diao, Yuanan; Morton, Hugh

doi:10.2140/agt.2024.24.2957

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The braid indices of the reverse parallel links of alternating knots

Yuanan Diao and Hugh Morton

Algebraic & Geometric Topology 24 (2024) 2957–2970

DOI: 10.2140/agt.2024.24.2957

Abstract

The braid indices of most links remain unknown as there is no known universal method for determining the braid index of an arbitrary knot. This is also the case for alternating knots. We show that if $K$ is an alternating knot, then the braid index of any reverse parallel link of $K$ can be precisely determined. Specifically, if $D$ is a reduced diagram of $K$ , $v_{+} (D)$ (resp. $v_{-} (D)$ ) is the number of regions in the checkerboard shading of $D$ for which all crossings are positive (resp. negative) and $w (D)$ is the writhe of $D$ , then the braid index of a reverse parallel link of $K$ with framing $f$ , denoted by $𝕂_{f}$ , is given by the precise formula

b (𝕂_{f}) = {\begin{matrix} c (D) + 2 + a (D) - f & if f < a (D), \\ c (D) + 2 & if a (D) \leq f \leq b (D), \\ c (D) + 2 - b (D) + f & if f > b (D), \end{matrix}

where $a (D) = - v_{-} (D) + w (D)$ and $b (D) = v_{+} (D) + w (D)$ .

Keywords

knots, links, alternating knots and links, reverse parallels of alternating knots, braid index

Mathematical Subject Classification

Primary: 57K10, 57K31

References

Publication

Received: 4 April 2023

Revised: 8 August 2023

Accepted: 20 August 2023

Published: 19 August 2024

Authors

Yuanan Diao
Department of Mathematics University of North Carolina Charlotte Charlotte, NC United States

Hugh Morton
Department of Mathematical Sciences University of Liverpool Liverpool United Kingdom

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Volume 24, issue 5 (2024)