Illustrating permutations as braided group - TiKZ-PGF

Question

I would like to illustrate the permutation group using the braid package in TikZ-PGF, such as this image (from Wikipedia):

In fact I would like to illustrate the permutation group for N = 3, which should have 6 total configurations. Having followed the example provided, I so far have:

\documentclass[tikz]{standalone}
\usepackage{pgfplots} 
\usepackage{braids}

\pgfplotsset{compat=newest}

\begin{document}

\begin{tikzpicture}
\braid[style strands={1}{red},style
strands={2}{blue},style strands={3}{green}]  (TEST2) at (5,0) s_1
s_3^{-1} s_1 s_2^{−1} s_1 s_2^{−1}; 
\end{tikzpicture}
\end{document}

However, I am finding it difficult to understand how to change the ordering. Any help would be appreciated.

Answer 1

My apologies for being a bit late to the party, but here's the permutations of 3 objects represented as braids using the braids package.

\documentclass[tikz,border=1cm]{standalone}
%\url{https://tex.stackexchange.com/q/455488/86}
\usepackage{tikz}
\usepackage{braids}

\begin{document}

\begin{tikzpicture}[
  /pgf/braid/.cd,
  style strands={1}{red},
  style strands={2}{blue},
  style strands={3}{green},
  number of strands=3
  ]
\braid (identity) at (0,0) 1 1 1;
\node at ([yshift=-1cm]identity-rev-2-e) {Identity};

\braid (123) at ([xshift=2cm]identity-3-s) 1 s_2^{-1} s_1;
\node at ([yshift=-1cm]123-rev-2-e) {\((123)\)};

\braid (321) at ([xshift=2cm]123-3-s) 1 s_1^{-1} s_2;
\node at ([yshift=-1cm]321-rev-2-e) {\((321)\)};

\braid (12) at ([yshift=-2cm]identity-rev-1-e) 1 s_1^{-1} 1;
\node at ([yshift=-1cm]12-rev-2-e) {\((12)\)};

\braid (23) at ([yshift=-2cm]123-rev-1-e) 1 s_2 1;
\node at ([yshift=-1cm]23-rev-2-e) {\((23)\)};

\braid (31) at ([yshift=-2cm]321-rev-1-e) s_1^{-1} s_2 s_1^{-1};
\node at ([yshift=-1cm]31-rev-2-e) {\((31)\)};


\end{tikzpicture}
\end{document}

Obviously, I had free choice on the over-under crossings.

Result: