\documentclass[tikz]{standalone}

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

\begin{document}

\begin{tikzpicture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% No self-intersections %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

%draw exterior boundaries
\draw[out=30,in=-30] (-1,0) to (-1,1.5);
\draw[out=150,in=-150] (-1,0) to (-1,1.5);

%add pending arc
\draw[in=0,out=45,looseness =0.9] (-1,0) to (-1,1.3);
\draw[in=180,out=135,looseness=0.9] (-1,0) to (-1,1.3);

%draw generalized arc
\draw[orange, thick,out=-90,in=-90,looseness = 4.5,->=0.25] (-1.175,1.3) to (-0.825,1.3);
%\draw[orange, thick,out=150,in=275,looseness = 1] (0,0.8) to (-0.15,1.35);

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% 1 self-intersection   %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%draw exterior boundaries
\draw[out=30,in=-30] (0.5,0) to (0.5,1.5);
\draw[out=150,in=-150] (0.5,0) to (0.5,1.5);

% draw pending arc
\draw[in=0,out=45,looseness =0.9] (0.5,0) to (0.5,1.3);
\draw[in=180,out=135,looseness=0.9] (0.5,0) to (0.5,1.3);

%draw generalized arc
\draw[orange, thick,out=100,in=180,looseness = 1] (0.35,0.5) to (0.5,1.15);
\draw[orange, thick,out=0,in=-30,looseness = 2] (0.5,1.15) to (0.5,0.8);
\draw[orange, thick,in=150,out=275,looseness = 1,->] (0.35,1.35) to (0.5,0.8) ;

%draw vertices and orbifold point
\draw[fill=black] (0.5,0) circle [radius=2pt];
\draw[fill=black] (0.5,1.5) circle [radius=2pt];
\draw[thick] (0.5,1) node {$\mathbf{\times}$};

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% 2 self-intersection   %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% draw exterior boundaries
\draw[out=30,in=-30] (2,0) to (2,1.5);
\draw[out=150,in=-150] (2,0) to (2,1.5);

% draw pending arc
\draw[in=0,out=45,looseness =0.9] (2,0) to (2,1.3);
\draw[in=180,out=135,looseness=0.9] (2,0) to (2,1.3);

% draw generalized arc
\draw[orange, thick,out=100,in=180,looseness = 1] (2,0) to (2,1.15);
\draw[orange, thick,out=0,in=-30,looseness = 2] (2,1.15) to (2,0.8);
\draw[orange,thick,out=180,in=150,->=0.5] (2,1.2) to (2,0.8);
\draw[orange,thick,out=0,in=10,looseness=1.2] (2,1.2) to (2.1,0.6);
\draw[orange,thick,out=190,in=-90] (2.1,0.6) to (1.85,1.35);

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% no self-intersections (CW)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%draw exterior boundaries
\draw[out=30,in=-30] (3.5,0) to (3.5,1.5);
\draw[out=150,in=-150] (3.5,0) to (3.5,1.5);

% draw pending arc
\draw[in=0,out=45,looseness =0.9] (3.5,0) to (3.5,1.3);
\draw[in=180,out=135,looseness=0.9] (3.5,0) to (3.5,1.3);

% draw generalized arc
\draw[orange, thick,out=-90,in=-90,looseness = 4.5,->] (3.675,1.3) to (3.325,1.3);
%\draw[orange, thick,out=150,in=275,looseness = 1] (0,0.8) to (-0.15,1.35);

%draw vertices and orbifold point
\draw[fill=black] (3.5,0) circle [radius=2pt];
\draw[fill=black] (3.5,1.5) circle [radius=2pt];
\draw[thick] (3.5,1) node {$\mathbf{\times}$};

%%% place congruence symbol
\node[scale=1.5] at (2.75,0.75) {$\approx$};

\end{tikzpicture}

\end{document}