\documentclass[tikz]{standalone}

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

\begin{document}

\begin{tikzpicture}

%%%draws bigon
%draw vertices and orbifold point
\draw[fill=black] (-3,1) circle [radius=2pt];
\draw[fill=black] (-3,2.5) circle [radius=2pt];
\draw[thick] (-3,2) node {$\mathbf{\times}$};

%draw exterior boundaries
\draw[out=30,in=-30] (-3,1) to (-3,2.5);
\draw[out=150,in=-150] (-3,1) to (-3,2.5);

%draws pending arc
\draw[in=0,out=45,looseness =0.9] (-3,1) to (-3,2.3);
\draw[in=180,out=135,looseness=0.9] (-3,1) to (-3,2.3);

%add labels
\node[left,xshift=3pt,scale=0.75] at (-3.4,2.1) {$\alpha$};
\node [right,xshift=-3pt,scale=0.75] at (-2.6,2.1) {$\beta$};
\node [above,xshift=-1pt,scale=0.75] at (-3.1,1.4) {$\rho$};

%draws crossing arc
\draw[orange,thick,->,out=-20,in=-160] (-3.35,1.8) to (-2.63,1.8);

%%% arrow between bigon and tiles
\draw[ultra thick,->] (-2.3,1.7) to (-1,1.7);

%%% left hexagonal tile

%diagonals for left hand hexagonal tile
\draw[gray,thick,dashed] (-0.7,1.7) to node[midway,below,scale=0.75,xshift=-2pt,yshift=2pt]{$\rho$} (1,1);
\draw[gray,thick,dashed] (0,2.4) to node[midway,below,scale=0.75,xshift=-2pt,yshift=2pt]{$\rho$} (1.7,1.7);

%left hand hexagonal tile
\draw[thick] (-0.7,1.7) to node[midway,above,xshift=-15pt, yshift=-1pt,scale=0.5]{$U_{\ell+1}\rho$} (1.7,1.7);
\draw[ultra thick,white] (0,2.4) to (1,1);
\draw[thick] (0,2.4) to node[midway,xshift=14pt,yshift=-7pt,scale=0.5]{$U_{\ell-1}\rho$} (1,1);
\draw[thick] (1,1) to node[midway,below,xshift=3pt,yshift=2pt,scale=0.75]{$\alpha$} (0,1);
\draw[thick] (0,1) to node[midway,below,,xshift=-1pt,scale=0.75]{$\beta$} (-0.7,1.7) to node[midway,above,xshift=-2pt,scale=0.5]{$U_\ell \rho$} (0,2.4) to node[midway,above,scale=0.75,yshift=-2pt]{$\beta$} (1,2.4) to node[midway,above,xshift=-1pt,yshift=2pt,scale=0.75]{$\alpha$}(1.7,1.7) to node[midway,below,scale=0.5,xshift=7pt]{$U_\ell \rho$} (1,1);

%%% right hexagonal tile

%diagonals for right hand hexagonal tile
\draw[gray,thick,dashed] (2.3,1.7) to node[midway,below,scale=0.75,xshift=-2pt,yshift=2pt]{$\rho$} (4,1);
\draw[gray,thick,dashed] (3,2.4) to node[midway,below,scale=0.75,xshift=-2pt,yshift=2pt]{$
\rho$} (4.7,1.7);

%right hand hexagonal tile
\draw[thick] (2.3,1.7) to node[midway,above,xshift=-15pt, yshift=-1pt,scale=0.5]{$U_{\ell-1}\rho$} (4.7,1.7);
\draw[ultra thick,white] (3,2.4) to (4,1);
\draw[thick] (3,2.4) to node[midway,xshift=14pt,yshift=-7pt,scale=0.5]{$U_{\ell+1}\rho$} (4,1);
\draw[thick] (4,1) to node[midway,below,xshift=3pt,yshift=2pt,scale=0.75]{$\beta$} (3,1);
\draw[thick] (3,1) to node[midway,below,,xshift=-1pt,scale=0.75]{$
\alpha$} (2.3,1.7) to node[midway,above,xshift=-2pt,scale=0.5]{$U_\ell \rho$} (3,2.4) to node[midway,above,scale=0.75,yshift=-1pt]{$
\alpha$} (4,2.4) to node[midway,above,xshift=-1pt,yshift=2pt,scale=0.75]{$
\beta$}(4.7,1.7) to node[midway,below,scale=0.5,xshift=7pt]{$U_\ell 
\rho$} (4,1);

\end{tikzpicture}

\end{document}