$
\qi{4n-3}
\begin{tikzpicture}[baseline]
    \clip (-1,-1.3) rectangle (1,1.3);
    \node at (0,0) (box) [rectangle,draw] {$\hspace{.1cm} \JW{4n -3} \hspace{.1cm} $};

    \draw (box.-50)-- ++(-90:1cm);
    \draw (box.-90) -- ++(-90:1cm);
    \draw (box.-150)--++(-90:1cm);
    \node at (-.25,-.6) {...};
    \draw (box.-30) -- ++(-90:1cm);

    \draw (box.50)-- ++(90:1cm);
    \draw (box.90) -- ++(90:1cm);
    \draw (box.150)--++(90:1cm);
    \node at (-.25,.6) {...};
    \draw (box.30) -- ++(90:1cm);
\end{tikzpicture}
=
\qi{4n-3}
\begin{tikzpicture}[baseline]
    \clip (-1,-1.3) rectangle (1,1.3);
    \node at (0,0) (box) [rectangle,draw] {$\JW{4n-4}$};

    \draw (box.-145)--++(-90:1cm);
    \node at (-.1,-.6) {...};
    \draw (box.-35) -- ++(-90:1cm);
    \draw (box.-60) -- ++(-90:1cm);

    \draw (box.35) -- ++(90:1cm);
    \draw (box.60) -- ++(90:1cm);
    \node at (-.1,.6) {...};
    \draw (box.145) -- ++(90:1cm);

    \draw (.9,1.5)--(.9,-1.5);
\end{tikzpicture}
+
\qi{4n-4}
\begin{tikzpicture}[baseline]
    \clip (-1,-1.3) rectangle (1,1.3);
    \node at (0,.7) (top) [rectangle,draw] {$\JW{4n-4}$};
    \node at (0,-.7) (bottom) [rectangle,draw]{$\JW{4n-4}$};

    \draw (top.-60)--(bottom.60);
    \draw (top.-145)--(bottom.145);
    \node at (-.1,0) {...};
    \draw (top.-35) arc (-180:0:.2cm) -- ++(90:1cm);
    \draw (bottom.35) arc (180:0:.2cm) -- ++(-90:1cm);

    \draw (top.35) -- ++(90:1cm);
    \draw (top.60) -- ++(90:1cm);
    \node at (-.1,1.2) {...};
    \draw (top.145) -- ++(90:1cm);

    \draw (bottom.-35) -- ++(-90:1cm);
    \draw (bottom.-60) -- ++(-90:1cm);
    \node at (-.1,-1.2) {...};
    \draw (bottom.-145) -- ++(-90:1cm);
\end{tikzpicture}
$

$
=
\frac{1}{\qi{2n-1}}
\begin{tikzpicture}[baseline]
    \clip (-1,-1.3) rectangle (1,1.3);

    \node at (0,.6) (top) [ellipse,draw] {$ \hspace{.15 cm} S \hspace{.15cm} $};
    \node at (0,-.6) (bottom) [ellipse,draw]{$ \hspace{.15 cm} S \hspace{.15cm} $};

	\node at (top.0) (invisible) [right] {};
	\draw (invisible.90)--++(90:2cm);
	\draw (invisible.90) -- ++ (-90:2cm);

    \draw (top.35) -- ++(90:1cm);
    \draw (top.60) -- ++(90:1cm);
    \node at (-.1,1.2) {...};
    \draw (top.145) -- ++(90:1cm);

    \draw (bottom.-35) -- ++(-90:1cm);
    \draw (bottom.-60) -- ++(-90:1cm);
    \node at (-.1,-1.2) {...};
    \draw (bottom.-145) -- ++(-90:1cm);
\end{tikzpicture}
+
\frac{\qi{2}}{\qi{2n-2}^2}
\begin{tikzpicture}[baseline]
    \clip (-1,-2.2) rectangle (1,2.2);

    \node at (0,.6) (top) [ellipse,draw] {$ \hspace{.15 cm} S \hspace{.15cm} $};
    \node at (0,-.6) (bottom) [ellipse,draw]{$ \hspace{.15 cm} S \hspace{.15cm} $};
    \node at (0,-1.6) (bottom2) [ellipse,draw]{$ \hspace{.15 cm} S \hspace{.15cm} $};
    \node at (0,1.6) (top2) [ellipse,draw]{$ \hspace{.15 cm} S \hspace{.15cm} $};
    

    \draw (top.-60)--(bottom.60);
    \draw (top.-145)--(bottom.145);
    \node at (-.1,0) {...};
    \draw (top.-35) arc (-180:0:.2cm) -- ++(90:2cm);
    \draw (bottom.35) arc (180:0:.2cm) -- ++(-90:2cm);

    \draw (top2.35) -- ++(90:1cm);
    \draw (top2.60) -- ++(90:1cm);
    \node at (-.1,2.1) {...};
    \draw (top2.145) -- ++(90:1cm);

    \draw (bottom2.-35) -- ++(-90:1cm);
    \draw (bottom2.-60) -- ++(-90:1cm);
    \node at (-.1,-2.1) {...};
    \draw (bottom2.-145) -- ++(-90:1cm);
\end{tikzpicture}
$

$=
\frac{1}{\qi{2n-1}}
\begin{tikzpicture}[baseline]
    \clip (-1,-1.3) rectangle (1,1.3);

    \node at (0,.6) (top) [ellipse,draw] {$ \hspace{.15 cm} S \hspace{.15cm} $};
    \node at (0,-.6) (bottom) [ellipse,draw]{$ \hspace{.15 cm} S \hspace{.15cm} $};

	\node at (top.0) (invisible) [right] {};
	\draw (invisible.90)--++(90:2cm);
	\draw (invisible.90) -- ++ (-90:2cm);

    \draw (top.35) -- ++(90:1cm);
    \draw (top.60) -- ++(90:1cm);
    \node at (-.1,1.2) {...};
    \draw (top.145) -- ++(90:1cm);

    \draw (bottom.-35) -- ++(-90:1cm);
    \draw (bottom.-60) -- ++(-90:1cm);
    \node at (-.1,-1.2) {...};
    \draw (bottom.-145) -- ++(-90:1cm);
\end{tikzpicture}
+
\frac{\qi{2}}{\qi{2n-2}}
\begin{tikzpicture}[baseline]
    \clip (-1,-2.2) rectangle (3,2.2);

    \node at (1.25,0) (JW) [rectangle, draw] {$ \JW{4n-4}$};
    \node at (0,-1.6) (bottom2) [ellipse,draw]{$ \hspace{.15 cm} S \hspace{.15cm} $};
    \node at (0,1.6) (top2) [ellipse,draw]{$ \hspace{.15 cm} S \hspace{.15cm} $};
    	\node at (top2.0) (invisible) [right] {};
	\node at (bottom2.0) (invisible2) [right] {};

    
    \draw (JW.35) arc (180:0:.2cm) -- ++(-90:.7cm) arc (0:-180:.2cm) --(JW.-35);
    \node at (JW.75) [above] {...};
        \node at (JW.-75) [below] {...};
    \draw (JW.135) arc (180:0:1cm) -- ++(-90:.7cm) arc (0:-180:1cm) --(JW.-135);
	\draw (invisible.90)--++(90:1cm);
	\draw (invisible.90) -- (JW.145);
	\draw (invisible2.90)--++(-90:1cm);
	\draw (invisible2.90) -- (JW.-145);

    \draw (top2.35) -- ++(90:1cm);
    \draw (top2.60) -- ++(90:1cm);
    \node at (-.1,2.1) {...};
    \draw (top2.145) -- ++(90:1cm);

    \draw (bottom2.-35) -- ++(-90:1cm);
    \draw (bottom2.-60) -- ++(-90:1cm);
    \node at (-.1,-2.1) {...};
    \draw (bottom2.-145) -- ++(-90:1cm);
\end{tikzpicture}
$

$=\frac{1}{\qi{2n-1}} + \frac{\qi{2} (-1)^{4n-5} }{\qi{2n-1} \qi{2}}
\begin{tikzpicture}[baseline]
    \clip (-1,-1.3) rectangle (1,1.3);

    \node at (0,.6) (top) [ellipse,draw] {$ \hspace{.15 cm} S \hspace{.15cm} $};
    \node at (0,-.6) (bottom) [ellipse,draw]{$ \hspace{.15 cm} S \hspace{.15cm} $};

	\node at (top.0) (invisible) [right] {};
	\draw (invisible.90)--++(90:2cm);
	\draw (invisible.90) -- ++ (-90:2cm);

    \draw (top.35) -- ++(90:1cm);
    \draw (top.60) -- ++(90:1cm);
    \node at (-.1,1.2) {...};
    \draw (top.145) -- ++(90:1cm);

    \draw (bottom.-35) -- ++(-90:1cm);
    \draw (bottom.-60) -- ++(-90:1cm);
    \node at (-.1,-1.2) {...};
    \draw (bottom.-145) -- ++(-90:1cm);
\end{tikzpicture}
=0$.
