File:The full-twist braid is central.svg

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Original file ‎(SVG file, nominally 641 × 96 pixels, file size: 9 KB)

Captions

Captions

The full-twist braid commutes with any element of the braid group

Summary

[edit]
Description
English: The picture proves that the full-twist braid is a central element of the braid group. In other words, it commutes with any braid (shown as a rectangle). The right braid is obtained from the left one by moving the rectangle through the strands via twisting it twice.
Русский: Изображение доказывает, что полный оборот является центральным элементом группы кос. Иными словами, он коммутирует с произвольной косой (показана в виде прямоугольника). Правая коса получается из левой протаскиванием прямоугольника через нити путём проворачивания его дважды.
Date
Source Own work
Author MauveAccueil

LaTeX source

[edit]
\documentclass[tikz,border=10pt]{standalone}
\usetikzlibrary{braids}

\begin{document}
\begin{tikzpicture}[scale=0.4, every node/.style={scale=0.4}]
\pic[
  rotate=90,
  line width=1.5pt,
  braid/control factor=0,
  braid/nudge factor=0,
  braid/gap=0.11,
  braid/number of strands = 5,
  name prefix=braid,
] at (-0.25,0) {braid={
s_4^{-1}
s_3^{-1}
s_2^{-1}-s_4^{-1}
s_1^{-1}-s_3^{-1}
s_2^{-1}-s_4^{-1}
s_3^{-1}
s_4^{-1}
s_4^{-1}
s_3^{-1}
s_2^{-1}-s_4^{-1}
s_1^{-1}-s_3^{-1}
s_2^{-1}-s_4^{-1}
s_3^{-1}
s_4^{-1}
1
1111
}};
\draw[dashed,draw=black, very thick, fill=white] (15-0.3,0-0.25) rectangle ++(4,4.5);
\node at (20,2) {{\Huge $=$}};
\begin{scope}[shift={(21,0)}]
\pic[
  rotate=90,
  line width=1.5pt,
  braid/control factor=0,
  braid/nudge factor=0,
  braid/gap=0.11,
  braid/number of strands = 5,
  name prefix=braid,
] at (-0.25,0) {braid={
1111
1
s_4^{-1}
s_3^{-1}
s_2^{-1}-s_4^{-1}
s_1^{-1}-s_3^{-1}
s_2^{-1}-s_4^{-1}
s_3^{-1}
s_4^{-1}
s_4^{-1}
s_3^{-1}
s_2^{-1}-s_4^{-1}
s_1^{-1}-s_3^{-1}
s_2^{-1}-s_4^{-1}
s_3^{-1}
s_4^{-1}
}};
\draw[dashed,draw=black, very thick, fill=white] (0+0.3,0-0.25) rectangle ++(4,4.5);
\end{scope}
\end{tikzpicture}
\end{document}

Licensing

[edit]
I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

File history

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Date/TimeThumbnailDimensionsUserComment
current16:39, 6 February 2023Thumbnail for version as of 16:39, 6 February 2023641 × 96 (9 KB)MauveAccueil (talk | contribs)Uploaded own work with UploadWizard

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