\def\FigScale{0.67}
\begin{tikzpicture}[scale=\FigScale]
%\node (a) at (0,0) [fill,circle,inner sep=1pt] {};
\draw (0,0) circle (1.25cm);
\foreach \x in {-144,-72,0,72,144} {
   \begin{scope}[rotate=\x]
    \node (\x) at (0,-1.25) [fill,circle,inner sep=1pt] {};
   \end{scope}
}
\filldraw[fill=gray!20](0,0) circle (.25cm);
\draw (144) node [right] {$j$};
%\draw (72) node [right] {$3$};
\draw (0) node [above] {$i$};
%\draw (-72) node [left] {$1$};
%\draw (-144) node [above left] {$5$};

\draw [ thick] (-.5,.0) to[out=90,in=125] (.4,.4) to[out=-180+135,in=30] (0);
\draw [ thick, ->-=.8] (-.5,.0) to[out=270,in=220] (.4,-.4) to[out=30,in=-55] (144);

\draw[thin, red] (.62,-.2) circle (.2cm);

\draw (2,0) node[] {$=$}; 
\end{tikzpicture}
\begin{tikzpicture}[scale=\FigScale]
%\node (a) at (0,0) [fill,circle,inner sep=1pt] {};
\draw (0,0) circle (1.25cm);
\foreach \x in {-144,-72,0,72,144} {
   \begin{scope}[rotate=\x]
    \node (\x) at (0,-1.25) [fill,circle,inner sep=1pt] {};
   \end{scope}
}

\filldraw[fill=gray!20](0,0) circle (.25cm);
\draw (144) node [ right] {$j$};
%\draw (72) node [right] {$3$};
\draw (0) node [above] {$i$};
%\draw (-72) node [left] {$1$};
%\draw (-144) node [above left] {$5$};

\draw[ thick,blue] (0,0) circle (.5cm);
\draw[ thick, ->-=.8] (0) .. controls (.8,-.6) and (.8,-.2) .. (144);

\draw (2,0) node[] {$+$}; 
\end{tikzpicture}
\begin{tikzpicture}[scale=\FigScale]
%\node (a) at (0,0) [fill,circle,inner sep=1pt] {};
\draw (0,0) circle (1.25cm);
\foreach \x in {-144,-72,0,72,144} {
   \begin{scope}[rotate=\x]
    \node (\x) at (0,-1.25) [fill,circle,inner sep=1pt] {};
   \end{scope}
}

\filldraw[fill=gray!20](0,0) circle (.25cm);
\draw (144) node [ right] {$j$};
%\draw (72) node [right] {$3$};
\draw (0) node [above] {$i$};
%\draw (-72) node [left] {$1$};
%\draw (-144) node [above left] {$5$};

\draw[ thick, ->-=.8] (144) .. controls (-.8,.6) and (-.8,-.4) .. (0);
\end{tikzpicture}