\documentclass[tikz]{standalone}

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

\standaloneconfig{margin=12pt}

\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 circular boundary of covering space
\draw[thick,in=0,out=0,looseness=1.75] (0,0) to (0,2);
\draw[thick,in=180,out=180,looseness=1.75] (0,0) to (0,2);

%draw vertices and arcs from the orbifold point / pending arc
\draw[fill=black] (0.5,0.1) circle [radius=1.5pt];
\draw[fill=black] (0.6,1.85) circle [radius=1.5pt];
\draw[fill=black] (-1.005,0.95) circle [radius=1.5pt];

\draw[thick,out=110,in=-130] (0.5,0.1) to (0.6,1.85);
\draw[thick,out=-130,in=0] (0.6,1.85) to (-1.005,0.95);
\draw[thick,out=0,in=110] (-1.005,0.95) to (0.5,0.1);

%draw vertices/arcs in each of the three triangulation copies

%right hand copy
\draw[fill=black] (0.95,0.6) circle [radius=1pt];
\draw[fill=black] (0.95,1.4) circle [radius=1pt];
\draw[out=100,in=-150] (0.5,0.1) to (0.95,1.4);

%upper left copy
\draw[fill=black] (-0.1,2) circle [radius=1pt];
\draw[fill=black] (-0.9,1.5) circle [radius=1pt];
\draw[out=-140,in=-10] (0.6,1.85) to (-0.9,1.5);

%lower left copy
\draw[fill=black] (-0.8,0.35) circle [radius=1pt];
\draw[fill=black] (-0.1,0) circle [radius=1pt];
\draw[out=-10,in=100] (-1.005,0.95) to (-0.1,0);

%add vertex labels
\node[above,scale=0.75] at (0.6,1.85) {$v_1$};
\node[above,scale=0.75] at (-0.1,2) {$v_2$};
\node[left,scale=0.75] at (-0.9,1.5) {$v_3$};
\node [left,scale=0.75] at (-1.005,0.95) {$v_4$};
\node [left,scale=0.75] at (-0.8,0.35) {$v_5$};
\node[below,scale=0.75] at (-0.1,0) {$v_6$};
\node[below,scale=0.75,yshift=-1pt] at (0.5,0.1) {$v_7$};
\node[right,scale=0.75] at (0.95,0.6) {$v_8$};
\node[right,scale=0.75] at (0.95,1.4) {$v_9$};


\end{tikzpicture}

\end{document}