\begin{figure}[htbp]
\centering
\begin{tikzpicture}[line width=.65pt,
 every node/.style={font=\small},
 localvertex/.style={circle,fill=black,inner sep=1.35pt}]
  % Theta.
  \begin{scope}[xshift=-4.4cm]
    \coordinate (p) at (-.9,0); \coordinate (q) at (.9,0);
    \draw (p)--(q);
    \draw (p) .. controls (-.45,1) and (.45,1) .. (q);
    \draw (p) .. controls (-.45,-1) and (.45,-1) .. (q);
    \node[localvertex] at (p) {}; \node[localvertex] at (q) {};
    \node at (0,-1.25) {$\Theta$};
  \end{scope}
  % The two-sided prism: two bigons and two joining edges.
  \begin{scope}
    \coordinate (a) at (-.7,.6); \coordinate (b) at (-.7,-.6);
    \coordinate (c) at (.7,.6);  \coordinate (d) at (.7,-.6);
    \draw (a)--(c) (b)--(d);
    \draw (a) .. controls (-1.12,.25) and (-1.12,-.25) .. (b);
    \draw (a) .. controls (-.28,.25) and (-.28,-.25) .. (b);
    \draw (c) .. controls (.28,.25) and (.28,-.25) .. (d);
    \draw (c) .. controls (1.12,.25) and (1.12,-.25) .. (d);
    \foreach \p in {a,b,c,d} \node[localvertex] at (\p) {};
    \node at (0,-1.25) {$L_2$};
  \end{scope}
  % Triangular prism: nested triangles with corresponding corners joined.
  \begin{scope}[xshift=4.4cm]
    \coordinate (a) at (-1,-.7); \coordinate (b) at (1,-.7);
    \coordinate (c) at (0,1);
    \coordinate (d) at (-.38,-.35); \coordinate (e) at (.38,-.35);
    \coordinate (f) at (0,.32);
    \draw (a)--(b)--(c)--cycle;
    \draw (d)--(e)--(f)--cycle;
    \draw (a)--(d) (b)--(e) (c)--(f);
    \foreach \p in {a,b,c,d,e,f} \node[localvertex] at (\p) {};
    \node at (0,-1.25) {$L_3$};
  \end{scope}
  % Cube: nested squares with corresponding corners joined.
  \begin{scope}[xshift=-4.4cm,yshift=-3.25cm]
    \draw (-.85,-.85) rectangle (.85,.85);
    \draw (-.34,-.34) rectangle (.34,.34);
    \foreach \sx in {-1,1} \foreach \sy in {-1,1} {
      \draw (.85*\sx,.85*\sy)--(.34*\sx,.34*\sy);
      \node[localvertex] at (.85*\sx,.85*\sy) {};
      \node[localvertex] at (.34*\sx,.34*\sy) {};}
    \node at (0,-1.35) {$L_4$};
  \end{scope}
  % Double-bigon theta, with precisely the vertex labels used in the text.
  \begin{scope}[yshift=-3.25cm]
    \coordinate (p) at (-1.25,0); \coordinate (q) at (1.25,0);
    \coordinate (r) at (-.5,.6); \coordinate (u) at (.5,.6);
    \coordinate (s) at (-.5,-.6); \coordinate (t) at (.5,-.6);
    \draw (p)--(q) (p)--(r) (p)--(s) (q)--(t) (q)--(u);
    \draw (r) .. controls (-.22,.99) and (.22,.99) .. (u);
    \draw (r) .. controls (-.22,.21) and (.22,.21) .. (u);
    \draw (s) .. controls (-.22,-.99) and (.22,-.99) .. (t);
    \draw (s) .. controls (-.22,-.21) and (.22,-.21) .. (t);
    \foreach \v in {p,q,r,s,t,u} \node[localvertex] at (\v) {};
    \node[left,inner sep=2pt] at (p) {$p$};
    \node[right,inner sep=2pt] at (q) {$q$};
    \node[above left,inner sep=2pt] at (r) {$r$};
    \node[above right,inner sep=2pt] at (u) {$u$};
    \node[below left,inner sep=2pt] at (s) {$s$};
    \node[below right,inner sep=2pt] at (t) {$t$};
    \node at (0,-1.35) {$N$};
  \end{scope}
  % Planar handcuff, whose connecting edge is a bridge.
  \begin{scope}[xshift=4.4cm,yshift=-3.25cm]
    \coordinate (p) at (-.42,0); \coordinate (q) at (.42,0);
    \draw (p)--(q);
    \draw (p) .. controls (-1.7,1.2) and (-1.7,-1.2) .. (p);
    \draw (q) .. controls (1.7,1.2) and (1.7,-1.2) .. (q);
    \node[localvertex] at (p) {}; \node[localvertex] at (q) {};
    \node at (0,-1.35) {Handcuff};
  \end{scope}
\end{tikzpicture}
\caption{The local boundary webs. Here $L_n$ denotes the $n$-sided prism,
so $L_4$ is the cube; these labels are distinct from the octahedral
labels in Figure~\ref{fig:octahedron}. Collapsing the two bigons of $N$
gives $\Theta$. All pictures show planar embeddings. The Hopf link
used to define $\J$ is omitted.}
\label{fig:local-webs}
\end{figure}