Trace Relations via Binor Identity

Snippets Page

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);
    }
$$