\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] (0.8,0.35) circle [radius=1.5pt] coordinate (A);
\draw[fill=black] (-0.7,0.25) circle [radius=1.5pt] coordinate (B);
\draw[fill=black] (0.7,1.75) circle [radius=1.5pt] coordinate (C);
\draw[fill=black] (-0.85,1.6) circle [radius=1.5pt] coordinate (D);

\draw[thick,out=150,in=-140] (A) to (C);
\draw[thick,out=50,in=150] (B) to (A);
\draw[thick,out=-20,in=50] (D) to (B);
\draw[thick,out=-20,in=-140] (D) to (C);

% draw vertices + arcs for each copy of the triangulation

%top copy
\draw[fill=black] (0.2,1.985) circle [radius=1pt];
\draw[fill=black] (-0.45,1.925 ) circle [radius=1pt];
\draw[out=-150,in=-70] (C) to (-0.45,1.925);

%left copy
\draw[fill=black] (-1.025,1.1) circle [radius=1pt];
\draw[fill=black] (-0.95,0.6) circle [radius=1pt];
\draw[out=-40,in=40] (D) to (-0.95,0.6);

%right copy
\draw[fill=black] (1,0.75) circle [radius=1pt];
\draw[fill=black] (0.98,1.3) circle [radius=1pt];
\draw[out=150,in=-160] (A) to (0.98,1.3);

%bottom copy
\draw[fill=black] (-0.2,0.02) circle [radius=1pt];
\draw[fill=black] (0.375,0.06) circle [radius=1pt];
\draw[out=40,in=120] (B) to (0.375,0.06);

%vertex labels
\node[right,scale=0.75,xshift=1pt] at (0.7,1.75) {$v_1$};
\node[above,scale=0.75] at (0.2,1.985) {$v_2$};
\node[above,scale=0.75] at (-0.45,1.925) {$v_3$};
\node[left,scale=0.75] at (-0.85,1.6) {$v_4$};
\node[left,scale=0.75] at (-1.025,1.1) {$v_5$};
\node[left,scale=0.75] at (-0.95,0.6) {$v_6$};
\node[left,scale=0.75,yshift=-2pt] at (-0.7,0.25) {$v_7$};
\node[below,scale=0.75] at (-0.2,0.02) {$v_8$};
\node[below,scale=0.75,xshift=1pt] at (0.375,0.06) {$v_9$};
\node[right,scale=0.75,xshift=1pt,yshift=-1pt] at (0.8,0.35) {$v_{10}$};
\node[right,scale=0.75] at (1,0.75) {$v_{11}$};
\node[right,scale=0.75] at (0.98,1.3) {$v_{12}$};

\end{tikzpicture}

\end{document}