Trace Relations via Binor Identity

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Diagram code for sum relations of three matrices in \$SL(2,\mathbb{C})\$.

# Source Code

``````\$\$
\tikz[heighttwo,xscale=.5]{
\coordinate(top)at(0,2.5){};\coordinate(topr)at(1,2.5){};\coordinate(bot)at(0,-1){};\coordinate(botr)at(1,-1){};
\coordinate(aa)at(0,2){};\coordinate(aar)at(1,2){};\coordinate(bb)at(0,1){};\coordinate(bbr)at(1,1){};
\coordinate(cc)at(0,0){};\coordinate(ccr)at(1,0){};
\draw(bot)to[wavyup](ccr)(botr)to[wavyup](cc);
\draw(cc)node[small matrix]{\$C\$}to[wavyup](bbr)(ccr)to[wavyup](bb);
\draw(bb)node[small matrix]{\$B\$}to[wavyup](aar)(bbr)to[wavyup](aa);
\draw(aa)node[small matrix]{\$A\$}to[wavyup](top)(aar)to[wavyup](topr);
}
=\tikz[heighttwo,xscale=.5]{
\draw(bot)to[wavyup](cc)(botr)to[wavyup](ccr);
\draw(cc)node[small matrix]{\$C\$}to[wavyup](bb)(ccr)to[wavyup](bbr);
\draw(bb)node[small matrix]{\$B\$}to[wavyup](aa)(bbr)to[wavyup](aar);
\draw(aa)node[small matrix]{\$A\$}to[wavyup](top)(aar)to[wavyup](topr);
}
-\tikz[heighttwo,xscale=.5]{
\draw(bot)to[wavyup](cc)(botr)to[wavyup](ccr);
\draw(cc)node[small matrix]{\$C\$}to[wavyup](bb)(ccr)to[wavyup](bbr);
\draw(bb)node[small matrix]{\$B\$}to[bend left=90](bbr)(aa)to[bend right=90](aar);
\draw(aa)node[small matrix]{\$A\$}to[wavyup](top)(aar)to[wavyup](topr);
}
-\tikz[heighttwo,xscale=.5]{
\draw(bot)to[wavyup](cc)(botr)to[wavyup](ccr);
\draw(cc)node[small matrix]{\$C\$}to[bend left=90](ccr)(bb)to[bend right=90](bbr);
\draw(bb)node[small matrix]{\$B\$}to[wavyup](aa)(bbr)to[wavyup](aar);
\draw(aa)node[small matrix]{\$A\$}to[wavyup](top)(aar)to[wavyup](topr);
}
-\tikz[heighttwo,xscale=.5]{
\draw(bot)to[bend left=90](botr)(cc)to[bend right=90](ccr);
\draw(cc)node[small matrix]{\$C\$}to[wavyup](bb)(ccr)to[wavyup](bbr);
\draw(bb)node[small matrix]{\$B\$}to[wavyup](aa)(bbr)to[wavyup](aar);
\draw(aa)node[small matrix]{\$A\$}to[wavyup](top)(aar)to[wavyup](topr);
}
+\tikz[heighttwo,xscale=.5]{
\draw(bot)to[bend left=90](botr)(cc)to[bend right=90](ccr);
\draw(cc)node[small matrix]{\$C\$}to[bend left=90](ccr)(bb)to[bend right=90](bbr);
\draw(bb)node[small matrix]{\$B\$}to[wavyup](aa)(bbr)to[wavyup](aar);
\draw(aa)node[small matrix]{\$A\$}to[wavyup](top)(aar)to[wavyup](topr);
}
+\tikz[heighttwo,xscale=.5]{
\draw(bot)to[bend left=90](botr)(cc)to[bend right=90](ccr);
\draw(cc)node[small matrix]{\$C\$}to[wavyup](bb)(ccr)to[wavyup](bbr);
\draw(bb)node[small matrix]{\$B\$}to[bend left=90](bbr)(aa)to[bend right=90](aar);
\draw(aa)node[small matrix]{\$A\$}to[wavyup](top)(aar)to[wavyup](topr);
}
+\tikz[heighttwo,xscale=.5]{
\draw(bot)to[wavyup](cc)(botr)to[wavyup](ccr);
\draw(cc)node[small matrix]{\$C\$}to[bend left=90](ccr)(bb)to[bend right=90](bbr);
\draw(bb)node[small matrix]{\$B\$}to[bend left=90](bbr)(aa)to[bend right=90](aar);
\draw(aa)node[small matrix]{\$A\$}to[wavyup](top)(aar)to[wavyup](topr);
}
-\tikz[heighttwo,xscale=.5]{
\draw(bot)to[bend left=90](botr)(cc)to[bend right=90](ccr);
\draw(cc)node[small matrix]{\$C\$}to[bend left=90](ccr)(bb)to[bend right=90](bbr);
\draw(bb)node[small matrix]{\$B\$}to[bend left=90](bbr)(aa)to[bend right=90](aar);
\draw(aa)node[small matrix]{\$A\$}to[wavyup](top)(aar)to[wavyup](topr);
}
\$\$```
```
page revision: 0, last edited: 28 Oct 2008 11:53