{\red\bf A faster program,} in which the Feynman diagrams are ``pre-computed'' (see theta.nb at \web{ap}):

\noindent\nbpdfInput{Snips/theta/1.pdf}

\noindent\nbpdfInput{Snips/theta/2.pdf}

\noindent\nbpdfInput{Snips/theta/3.pdf}

We call the invariant computed $\theta$:

\noindent\nbpdfInput{Snips/theta/4.pdf}

\needspace{15mm}
{\bf\red Some Knots.}

\noindent\nbpdfInput{Snips/theta/5.pdf}

\noindent\nbpdfMessage{Snips/theta/6.pdf}

\noindent\nbpdfOutput{Snips/theta/7.pdf}

\noindent\nbpdfInput{Snips/theta/8.pdf}

\needspace{25mm}
\parpic[r]{$
  {\includegraphics[height=0.5in]{../Beijing-2407/K11n34.png}
    \atop\text{\tiny K11n34}}
  {\includegraphics[height=0.5in]{../Beijing-2407/K11n42.png}
    \atop\text{\tiny K11n42}}
$}

\noindent\nbpdfInput{Snips/theta/9.pdf}

\noindent\nbpdfOutput{Snips/theta/10.pdf}

\parpic[r]{$
  {\includegraphics[height=0.6in]{../../Projects/Gallery/Conway.png}
    \atop\text{\scriptsize Conway}}
  {\includegraphics[height=0.6in]{../../Projects/Gallery/PhotoNotAvailable.png}
    \atop\text{\scriptsize Kinoshita}}
  {\includegraphics[height=0.6in]{../../Projects/Gallery/Terasaka.jpg}
    \atop\text{\scriptsize Terasaka}}
$}
So $\theta$ detects knot mutation and separates the Conway knot K11n34 from the Kinoshita-Terasaka knot K11n42!

\noindent\nbpdfInput{Snips/theta/11.pdf}

\noindent\nbpdfOutput{Snips/theta/12.pdf}

The torus knot $T_{22/7}$:
\newline\includegraphics[width=\linewidth]{T227.pdf}

\needspace{5cm}
Next, a random 300 crossing knot from~\cite{DHOEBL:Random} (more\newline
at \web{DK}):
\[ \includegraphics[width=0.8\linewidth]{../UBC-241004/DK300.pdf} \]

\newcolumn
{\red\bf Unproven Fact.} For any knot $K$, twice its genus $g(K)$ bounds the $T_1$ degree of $\theta$: $\deg_{T_1}\theta(K)\leq 2g(K)$.
\parpic[r]{$
  {\includegraphics[height=0.6in]{../../Projects/Gallery/Gompf.jpg}
    \atop\text{\scriptsize Gompf}}
  {\includegraphics[height=0.6in]{../../Projects/Gallery/Scharlemann.jpg}
    \atop\text{\scriptsize Scharlemann}}
  {\includegraphics[height=0.6in]{../../Projects/Gallery/Thompson.jpg}
    \atop\text{\scriptsize Thompson}}
$}
The 48-crossing Gompf-Scharlemann-Thompson $\mathit{GST}_{48}$ knot \cite{GompfScharlemannThompson:Counterexample} is significant because it may be a counterexample to the slice-ribbon conjecture:

\[ \resizebox{\linewidth}{!}{\import{../Waco-2203/}{GST48-Marked.pdf_t}} \]

\noindent\nbpdfInput{Snips/theta/13.pdf}

\noindent\nbpdfOutput{Snips/theta/14.pdf}

\noindent\nbpdfInput{Snips/theta/15.pdf}

\noindent\nbpdfOutput{Snips/theta/16.pdf}

\vfill\hfill So $\Theta$ knows things about $\mathit{GST}_{48}$ that $\Delta$ doesn't!

