\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 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 interior arcs
%\draw (0,0) to (0,1);
\draw[in=0,out=45,looseness =0.9] (0,0) to (0,1.2);
\draw[in=180,out=135,looseness=0.9] (0,0) to (0,1.2);

%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);

%add labels
\node[left,xshift=1pt,scale=0.75] at (-0.3,1.1) {$2$};
\node [right,xshift=-1pt,scale=0.75] at (0.3,1.1) {$a$};
\node [right,xshift=-1pt,scale=0.75] at (-0.3,0.7) {$1$};
\draw[in=0,out=45,looseness =0.9] (0,0) to (0,1.2);
\draw[in=180,out=135,looseness=0.9] (0,0) to (0,1.2);

%draw crossing arc
\draw[orange, thick,out=0,in=0,looseness = 1.7,<-=0.5] (-0.4,1.3) to (-0.25,0.3);

\end{tikzpicture}

\end{document}