\documentclass[tikz]{standalone}

\usetikzlibrary{decorations.pathreplacing,decorations.markings}
\usetikzlibrary{patterns}


\begin{document}

\begin{tikzpicture}

% sets up command for arrows in the middle of arcs
\tikzset{->-/.style={decoration={
  markings,
  mark=at position #1 with {\arrow{>}}},postaction={decorate}}}
  
 %%% draw version where the pending arc ends at the orbifold point

%draw vertices and orbifold point
\draw[fill=black] (0,0) circle [radius=2pt];
\draw[fill=black] (0,1.5) circle [radius=2pt];
\draw[thick] (0,1) node {$\mathbf{\times}$};

%draw pending arc
\draw (0,0) to (0,1);

%draw exterior boundaries
\draw[out=30,in=-30] (0,0) to (0,1.5);
\draw[out=150,in=-150] (0,0) to (0,1.5);


%%% draw version where the pending arc cuts out a monogon around the orbifold point

%draw vertices and orbifold point
\draw[fill=black] (1.5,0) circle [radius=2pt];
\draw[fill=black] (1.5,1.5) circle [radius=2pt];
\draw[thick] (1.5,1) node {$\mathbf{\times}$};

%draw pending arc
\draw[out=45,in=0] (1.5,0) to (1.5,1.2);
\draw[out=180,in=135] (1.5,1.2) to (1.5,0);

%draw exterior boundaries
\draw[out=30,in=-30] (1.5,0) to (1.5,1.5);
\draw[out=150,in=-150] (1.5,0) to (1.5,1.5);



\end{tikzpicture}

\end{document}