Pretty Simple Loop Recurrence

Snippets Page

This diagram shows the strategy for obtaining a simple loop recurrence. The diagram is very symmetric, making it visually appealing.

Source Code

\tikz[shift={(0,.3)},rotate=22.5]{
    \draw[blue,trivalent](0,0)circle(.8)node[shift={(0:.35)}]{$1$};
    \draw[trivalent](0,0)circle(1)
    \foreach\xa/\xb in {67.5/n,22.5/1,-22.5/2,-67.5/3,-112.5/4,-157.5/5}{
        node[basiclabel,shift={([rotate=22.5]\xa:1.1)}]{$b_\xb$}
    };
    \draw[trivalent]
    \foreach\xa/\xb in {45/1,0/2,-45/3,-90/4,-135/5,-180/6}{
        (\xa:1)--(\xa:1.5)node[basiclabel,shift={(\xa:.25)}]{$a_\xb$}
    };
    \draw[draw=none,rotate=15](135:1.3)to[bend left](90:1.3)node[pos=0]{.}node[pos=.5]{.}node[pos=1]{.};
}
\longrightarrow
\tikz[shift={(0,.3)},rotate=22.5]{
    \draw[trivalent]
    \foreach\xa/\xb in {45/1,0/2,-45/3,-90/4,-135/5,-180/6}{
        (\xa:1)--(\xa:1.5)node[basiclabel,shift={(\xa:.25)}]{$a_\xb$}
    };
    \foreach\xa in {45,0,-45,-90,-135,-180}{
        \draw[blue,trivalent,rotate=\xa](10:1)to[bend left=45](35:1);
        \draw[trivalent,rotate=\xa](10:1)to[bend right=45](35:1);
        \draw[purple,trivalent,rotate=\xa](-10:1)to[bend right](10:1);
    }
    \draw[purple,trivalent,rotate=90](-10:1)..controls+(80:.3)and+(100:-.3)..(10:1);
    \draw(0,0)
    \foreach\xa/\xb in {67.5/n,22.5/1,-22.5/2,-67.5/3,-112.5/4,-157.5/5}{
        node[basiclabel,shift={([rotate=22.5]\xa:1.15)}]{$b_\xb$}
    };
    \draw[draw=none,rotate=15](135:1.3)to[bend left](90:1.3)node[pos=0]{.}node[pos=.5]{.}node[pos=1]{.};
}
\longrightarrow
\tikz[shift={(0,.3)},rotate=22.5]{
    \draw[trivalent]
    \foreach\xa/\xb in {45/1,0/2,-45/3,-90/4,-135/5,-180/6}{
        (\xa:1)--(\xa:1.5)node[basiclabel,shift={(\xa:.25)}]{$a_\xb$}
    };
    \draw[purple,trivalent](0,0)circle(1)
    \foreach\xa/\xb in {67.5/n,22.5/1,-22.5/2,-67.5/3,-112.5/4,-157.5/5}{
        node[basiclabel,shift={([rotate=22.5]\xa:1.1)}]{$b'_\xb$}
    };
    \draw[draw=none,rotate=15](135:1.3)to[bend left](90:1.3)node[pos=0]{.}node[pos=.5]{.}node[pos=1]{.};
}

This looks good, but the math requires you to be very cautious about the orientation of the vertices.