\begin{tikzpicture}[baseline]
     \clip (-.9,-3) rectangle (1.2,3);
     \node at (0,1) (top) [ellipse,draw] {$ \hspace{.15 cm} S \hspace{.15cm} $};
     \node at (0,-1) (bottom) [ellipse,draw] {$ \hspace{.15 cm} S \hspace{.15cm} $};

     \draw (top.145)--++(0,1);
     \node at (-.1,1.5) {...};
     \node at (-.1,.5) {...};
     \draw (top.60)--++(0,1);
     \draw (top.35) arc (180:0:.2cm) node(anchor){} -- (bottom.-35 -| anchor) arc (0:-180:.2cm);
     
     \draw (top.-145)-- (bottom.145);
     \draw (top.-60)-- (bottom.60);
     \draw (top.-35) -- (bottom.35);
          
     \draw (bottom.-145)--++(0,-1);
     \node at (-.1,-1.5) {...};
     \node at (-.1,-.5) {...};
     \draw (bottom.-60)--++(0,-1);
     
     \node at (.1,0) [rectangle,draw,fill=white] {$\JW{4n-4-i}$};
     \node at (-.1,2) [rectangle,draw,fill=white] {$\JW{i}$};
      \node at (-.1,-2) [rectangle,draw,fill=white] {$\JW{i}$};
\end{tikzpicture}
=
\begin{tikzpicture}[baseline]
     \clip (-.8,-3) rectangle (.9,3);
     \node at (0,.7) (top) [ellipse,draw] {$ \hspace{.15 cm} S \hspace{.15cm} $};
     \node at (0,-.7) (bottom) [ellipse,draw] {$ \hspace{.15 cm} S \hspace{.15cm} $};

     \draw (top.145)--++(0,1);
     \node at (-.1,1.3) {...};
     \draw (top.60)--++(0,1);
     \draw (top.35) arc (180:0:.2cm) node(anchor){} -- (bottom.-35 -| anchor) arc (0:-180:.2cm);
     
     \draw (top.-145)-- (bottom.145);
     \node at (-.1,0) {...};
     \draw (top.-60)-- (bottom.60);
     \draw (top.-35) -- (bottom.35);
          
     \draw (bottom.-145)--++(0,-1);
     \node at (-.1,-1.3) {...};
     \draw (bottom.-60)--++(0,-1);
     
      \node at (-.1,2) [rectangle,draw,fill=white] {$\JW{i}$};
      \node at (-.1,-2) [rectangle,draw,fill=white] {$\JW{i}$};
\end{tikzpicture}
= \qi{2n-1}
\begin{tikzpicture}[baseline]
     \clip (-.8,-3) rectangle (2.3,3);
    \node at (.4,0) (mid) [rectangle,draw] {$ \hspace{.2 cm} \JW{4n-4} \hspace{.2cm} $};
    
     \draw (mid.157)--++(0,2);
     \node at (-.1,1) {...};
     \draw (mid.115)--++(0,2);
     \draw (mid.70) arc (180:0:.8cm) node(anchor){} -- (mid.-70 -| anchor) arc (0:-180:.8cm);
     \draw (mid.25) arc (180:0:.2cm) node(anchor){} -- (mid.-25 -| anchor) arc (0:-180:.2cm);       
     \draw (mid.0)++(.5,0) node {...};
          
     \draw (mid.-157)--++(0,-2);
     \node at (-.1,-1) {...};
     \draw (mid.-115)--++(0,-2);
     
      \node at (-.1,2) [rectangle,draw,fill=white] {$\JW{i}$};
      \node at (-.1,-2) [rectangle,draw,fill=white] {$\JW{i}$};
\end{tikzpicture}
= \frac{\qi{2n-1}}{\qi{i+1}}
\begin{tikzpicture}[baseline]
     \clip (-.7,-3) rectangle (.7,3);
     \node at (0,0) (mid) [rectangle,draw] {$\JW{i}$};
      \node at (0,2) (top) [rectangle,draw,fill=white] {$\JW{i}$};
      \node at (0,-2) (bottom) [rectangle,draw,fill=white] {$\JW{i}$};
      
      \draw (top.-50)--(mid.50);
      \node at (0,1) {...};
      \draw (top.-130)--(mid.130);
      
      \draw (mid.-50)--(bottom.50);
      \node at (0,-1) {...};
      \draw (mid.-130)--(bottom.130);
\end{tikzpicture}
= \frac{\qi{2n-1}}{\qi{i+1}}  \JW{i}