\section{Fun} The word ``fun'' rarely appears in the title of a math
paper, so let us start with a brief justification.

$\Theta$ is a pair of polynomials. The first, $\Delta$, is old news, the
Alexander polynomial~\cite{Alexander:TopologicalInvariants}. It is
a one-variable Laurent polynomial in a variable $T$. For example,
$\Delta(\lefttrefoil)=T^{-1}-1+T$.  We turn such a polynomial to
a list of coefficients (for $\lefttrefoil$, it is $(1\ -1\ 1)$),
and then to a chain of bars of varying colours: white for the zero
coefficients, and red and blue for the positive and negative coefficients
(with intensity proportional to the magnitude of the coefficients).
The result is a ``bar code'', and for the trefoil $\lefttrefoil$ is it
{\red\rule{2mm}{3mm}}{\blue\rule{2mm}{3mm}}{\red\rule{2mm}{3mm}}.

\parpic[r]{\includegraphics[width=1.2in]{PPDemo.pdf}}
Similiarly, $\theta$ is a 2-variable Laurent polynomial, in variables
$T_1$ and $T_2$. We can turn such a polynomial into a 2D array
of coefficients and then using the same rules, into a 2D array of
colours, namely into a picture! To highlight a certain conjectured
hexagonal symmetry of the resulting pictures, we apply a certain shear
transformation to the plane before printing. So the colour of a monomial
$cT_1^{n_1}T_2^{n_2}$ gets drawn at position $\begin{pmatrix} 1 & -1/2
\\ 0 & \sqrt{3}/2 \end{pmatrix}\begin{pmatrix} n_1 \\ n_2 \end{pmatrix}$
instead of the more traditional $\begin{pmatrix} n_1 \\ n_2 \end{pmatrix}$. On the right is the 2D picture
corresponding to the polynomial $2+T_1-T_1T_2+T_2-T_1^{-1}+T_1^{-1}T_2^{-1}-T_2^{-1}$.

Thus $\Theta$ becomes a pair of pictures: a bar code, and a 2D
picture that we call a ``hexagonal QR code''. For the knots in the
Rolfsen table (with the unknot prepended at the start), they are in
Figure~\ref{fig:Rolfsen}. In addition, the hexagonal QR codes of some
15 knots with $\geq 300$ crossings are in Figure~\ref{fig:300}.

Clearly there are patterns in Figures~\ref{fig:Rolfsen}
and~\ref{fig:300}. There is a hexaginal symmetry and the QR codes are
hexagons (these are independent properties). Much more can be seen
in Figure~\ref{fig:Rolfsen}. In Figure~\ref{fig:300} there seem to be
large-scale ``sand table patterns'' or ``diffraction patterns''. We can't
prove any of these things, and the last one, we can't even formulate
properly. Yet they are clearly there, too clear to be the result of
chance alone.

We plan to have fun over the next few years observing and proving these patterns. We hope
that others will join us too.

\begin{figure}
\adjustbox{valign=m}{\resizebox*{!}{8.7in}{
\def\k#1#2{{\parbox{0.5in}{\centering
  \vskip 2pt
  \includegraphics[width=\linewidth,height=\linewidth]{KnotFigs/#1_#2.pdf}
  \newline#1\_#2
  \vskip 2pt
}}}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
  \hline \k{0}{1} & \k{3}{1} & \k{4}{1} & \k{5}{1} & \k{5}{2} & \k{6}{1} & \k{6}{2} & \k{6}{3} & \k{7}{1} & \k{7}{2} \\
  \hline \k{7}{3} & \k{7}{4} & \k{7}{5} & \k{7}{6} & \k{7}{7} & \k{8}{1} & \k{8}{2} & \k{8}{3} & \k{8}{4} & \k{8}{5} \\
  \hline \k{8}{6} & \k{8}{7} & \k{8}{8} & \k{8}{9} & \k{8}{10} & \k{8}{11} & \k{8}{12} & \k{8}{13} & \k{8}{14} & \k{8}{15} \\
  \hline \k{8}{16} & \k{8}{17} & \k{8}{18} & \k{8}{19} & \k{8}{20} & \k{8}{21} & \k{9}{1} & \k{9}{2} & \k{9}{3} & \k{9}{4} \\
  \hline \k{9}{5} & \k{9}{6} & \k{9}{7} & \k{9}{8} & \k{9}{9} & \k{9}{10} & \k{9}{11} & \k{9}{12} & \k{9}{13} & \k{9}{14} \\
  \hline \k{9}{15} & \k{9}{16} & \k{9}{17} & \k{9}{18} & \k{9}{19} & \k{9}{20} & \k{9}{21} & \k{9}{22} & \k{9}{23} & \k{9}{24} \\
  \hline \k{9}{25} & \k{9}{26} & \k{9}{27} & \k{9}{28} & \k{9}{29} & \k{9}{30} & \k{9}{31} & \k{9}{32} & \k{9}{33} & \k{9}{34} \\
  \hline \k{9}{35} & \k{9}{36} & \k{9}{37} & \k{9}{38} & \k{9}{39} & \k{9}{40} & \k{9}{41} & \k{9}{42} & \k{9}{43} & \k{9}{44} \\
  \hline \k{9}{45} & \k{9}{46} & \k{9}{47} & \k{9}{48} & \k{9}{49} & \k{10}{1} & \k{10}{2} & \k{10}{3} & \k{10}{4} & \k{10}{5} \\
  \hline \k{10}{6} & \k{10}{7} & \k{10}{8} & \k{10}{9} & \k{10}{10} & \k{10}{11} & \k{10}{12} & \k{10}{13} & \k{10}{14} & \k{10}{15} \\
  \hline \k{10}{16} & \k{10}{17} & \k{10}{18} & \k{10}{19} & \k{10}{20} & \k{10}{21} & \k{10}{22} & \k{10}{23} & \k{10}{24} & \k{10}{25} \\
  \hline \k{10}{26} & \k{10}{27} & \k{10}{28} & \k{10}{29} & \k{10}{30} & \k{10}{31} & \k{10}{32} & \k{10}{33} & \k{10}{34} & \k{10}{35} \\
  \hline \k{10}{36} & \k{10}{37} & \k{10}{38} & \k{10}{39} & \k{10}{40} & \k{10}{41} & \k{10}{42} & \k{10}{43} & \k{10}{44} & \k{10}{45} \\
  \hline \k{10}{46} & \k{10}{47} & \k{10}{48} & \k{10}{49} & \k{10}{50} & \k{10}{51} & \k{10}{52} & \k{10}{53} & \k{10}{54} & \k{10}{55} \\
  \hline \k{10}{56} & \k{10}{57} & \k{10}{58} & \k{10}{59} & \k{10}{60} & \k{10}{61} & \k{10}{62} & \k{10}{63} & \k{10}{64} & \k{10}{65} \\
  \hline \k{10}{66} & \k{10}{67} & \k{10}{68} & \k{10}{69} & \k{10}{70} & \k{10}{71} & \k{10}{72} & \k{10}{73} & \k{10}{74} & \k{10}{75} \\
  \hline \k{10}{76} & \k{10}{77} & \k{10}{78} & \k{10}{79} & \k{10}{80} & \k{10}{81} & \k{10}{82} & \k{10}{83} & \k{10}{84} & \k{10}{85} \\
  \hline \k{10}{86} & \k{10}{87} & \k{10}{88} & \k{10}{89} & \k{10}{90} & \k{10}{91} & \k{10}{92} & \k{10}{93} & \k{10}{94} & \k{10}{95} \\
  \hline \k{10}{96} & \k{10}{97} & \k{10}{98} & \k{10}{99} & \k{10}{100} & \k{10}{101} & \k{10}{102} & \k{10}{103} & \k{10}{104} & \k{10}{105} \\
  \hline \k{10}{106} & \k{10}{107} & \k{10}{108} & \k{10}{109} & \k{10}{110} & \k{10}{111} & \k{10}{112} & \k{10}{113} & \k{10}{114} & \k{10}{115} \\
  \hline \k{10}{116} & \k{10}{117} & \k{10}{118} & \k{10}{119} & \k{10}{120} & \k{10}{121} & \k{10}{122} & \k{10}{123} & \k{10}{124} & \k{10}{125} \\
  \hline \k{10}{126} & \k{10}{127} & \k{10}{128} & \k{10}{129} & \k{10}{130} & \k{10}{131} & \k{10}{132} & \k{10}{133} & \k{10}{134} & \k{10}{135} \\
  \hline \k{10}{136} & \k{10}{137} & \k{10}{138} & \k{10}{139} & \k{10}{140} & \k{10}{141} & \k{10}{142} & \k{10}{143} & \k{10}{144} & \k{10}{145} \\
  \hline \k{10}{146} & \k{10}{147} & \k{10}{148} & \k{10}{149} & \k{10}{150} & \k{10}{151} & \k{10}{152} & \k{10}{153} & \k{10}{154} & \k{10}{155} \\
  \hline \k{10}{156} & \k{10}{157} & \k{10}{158} & \k{10}{159} & \k{10}{160} & \k{10}{161} & \k{10}{162} & \k{10}{163} & \k{10}{164} & \k{10}{165} \\
  \hline
\end{tabular}
}}
$\underset{\Theta}{\longrightarrow}$
\includegraphics[height=8.7in,valign=m]{Theta4Rolfsen.pdf}
\caption{$\Theta$ as a bar code and a hexagonal QR code, for all the knots in the Rolfsen table.}
\label{fig:Rolfsen}
\end{figure}

\begin{figure}
\includegraphics[width=0.19\linewidth]{figs/PP300.png}
\includegraphics[width=0.19\linewidth]{figs/PP301.png}
\includegraphics[width=0.19\linewidth]{figs/PP302.png}
\includegraphics[width=0.19\linewidth]{figs/PP303.png}
\includegraphics[width=0.19\linewidth]{figs/PP304.png}
\includegraphics[width=0.19\linewidth]{figs/PP305.png}
\includegraphics[width=0.19\linewidth]{figs/PP306.png}
\includegraphics[width=0.19\linewidth]{figs/PP307.png}
\includegraphics[width=0.19\linewidth]{figs/PP308.png}
\includegraphics[width=0.19\linewidth]{figs/PP309.png}
\includegraphics[width=0.19\linewidth]{figs/PP310.png}
\includegraphics[width=0.19\linewidth]{figs/PP311.png}
\includegraphics[width=0.19\linewidth]{figs/PP312.png}
\includegraphics[width=0.19\linewidth]{figs/PP313.png}
\includegraphics[width=0.19\linewidth]{figs/PP317.png}
\caption{$\theta$ (hexagonal QR code only) of the 15 largest knots that
  we have computed by September 16, 2024. They are all ``generic'' in as
  much as we know, and they all have $\geq 300$ crossings. The knots come from~\cite{DHOEBL:Random}.
}
\label{fig:300}
\end{figure}

\section{Formulas} \label{sec:Formulas}

\subsection{Old Formulas} \label{ssec:OldFormulas}

The setup leading to the definition of $\Theta$ is the same as the setup leading to the definition of the
invariant $\rho_1$ of~\cite{APAI}, and hence we copy a few relevant paragraphs from~\cite{APAI} nearly
varbatim, with only a few examples removed.

\parpic[r]{\scalebox{0.95}{\input{figs/SampleKnot.pdf_t}}}
Given an oriented $n$-crossing knot $K$, we draw it in the plane as a
long knot diagram $D$ in such a way that the two strands intersecting
at each crossing are pointed up (that's always possible because we
can always rotate crossings as needed), and so that at its beginning
and at its end the knot is oriented upward. We call such a diagram an
{\em upright knot diagram.} An example of an upright knot diagram
is shown on the right.

We then label each edge of the diagram with two integer labels: a 
running index $k$ which runs from 1 to $2n+1$, and a ``rotation number''
$\varphi_k$, the geometric rotation number of that edge (the signed
number of times the tangent to the edge is horizontal and heading right,
with cups counted with $+1$ signs and caps with $-1$; this number is well 
defined because at their ends, all edges are headed up). On the right the
running index runs from $1$ to $7$, and the rotation numbers for all edges
are $0$ (and hence are omitted) except for $\varphi_4$, which is $-1$.

{\em A Technicality.} Some Reidemeister moves create or lose an edge 
and to avoid the need for renumbering it is beneficial to also allow
labelling the edges with non-consecutive labels. Hence we allow that, and
write $\ip$ for the successor of the label $i$ along the knot, and $\ipp$
for the successor of $\ip$ (these are $i+1$ and $i+2$ if the labelling is
by consecutive integers). Also, ``$1$'' will always refer to the label of
the first edge, and ``$2n+1$'' will always refer to the label of the last.

We let $A$ be the $(2n+1)\times(2n+1)$ matrix of Laurent polynomials in 
the formal variable $T$ defined by
\[ A \coloneqq I - \sum_c \left( T^sE_{i,\ip} + (1-T^s)E_{i,\jp} + E_{j,\jp} \right), \]
where $I$ is the identity matrix and $E_{\alpha\beta}$ denotes the elementary
matrix with $1$ in row $\alpha$ and column $\beta$ and zeros elsewhere.
The summation is over the crossings $c=(s,i,j)$ of the diagram $D$, and once $c$
is chosen, $s$ denotes its sign and $i$ and $j$ denote the labels below
the crossing where the label $i$ belongs to the over-strand and $j$
to the under-strand.

Alternatively, $A=I + \sum_c A_c$, where $A_c$ is a matrix of zeros except for the blocks as follows:
\begin{equation} \label{eq:A}
  \begin{array}{c}\input{figs/Xings.pdf_t}\end{array}
  \qquad\longrightarrow\qquad
  \begin{array}{c|cccc}
    A_c &   \text{column }\ip  &  \text{column }\jp \\
    \hline
    \text{row }i & -T^s  & T^s-1 \\
    \text{row }j & 0  & -1
  \end{array}
\end{equation}

We note (as we did in~\cite{APAI}) that the determinant of $A$ is equal up to a unit
to the normalized Alexander polynomial $\Delta$ of $K$. In fact, we have that
\begin{equation} \label{eq:Delta} \Delta = T^{(-\varphi(D)-w(D))/2}\det(A), \end{equation}
where $\varphi(D)\coloneqq\sum_k\varphi_k$ is the total rotation number of $D$ and where $w(D)=\sum_cs_c$ is
the writhe of $D$, namely the sum of the signs $s_c$ of all the crossings $c$ in $D$.

We let $G=(g_{\alpha\beta})=A^{-1}$ and, thinking of it as a function
$g_{\alpha\beta}$ of a pair of edges $\alpha$ and $\beta$, we call
it the Green function of the diagram $D$. When inspired by physics
(e.g.~\cite{IType}) we sometimes call it ``the 2-point function'',
and when thinking of car traffic (e.g.~\cite{Toronto-241030}) we sometimes
call it ``the traffic function''.

We note that the computation of $G$ is the bottleneck in the computation of $\Theta$. It requires inverting a
$(2n+1)\times(2n+1)$ matrix whose entries that are (degree 1) Laurent polynomials in $T$. It's a daunting task
yet it takes polynomial time, it can be performed in practice even if $n$ is in the hundreds, and everything which
follows is easier.

\subsection{New Formulas} Let $T_1$ and $T_2$ be indeteminates and let
$T_3\coloneqq T_1T_2$. Let $\Delta_\nu\coloneqq\Delta\,/\,(T\to T_\nu)$
and $G_\nu=(g_{\nu\alpha\beta})\coloneqq G\,/\,(T\to T_\nu)$ be $\Delta$
and $G$ affected by the substitution $T\to T_\nu$, where $\nu=1,2,3$
(these are easy to compute once $\Delta$ and $G$ have been computed).

The formulas for $\theta$ depend on three fixed polynomials $R_{11}(c)$,
$R_{12}(c_0,c_1)$ and $\Gamma_1(\varphi,k)$ in the $g_{\nu\alpha\beta}$'s,
which we admit, are rather ugly. So we prefer to assert their existance
and postpone displaying them to a few paragraphs later.

\begin{theorem}[Proof in Section~\ref{sec:Proof}] With $c=(s,i,j)$, $c_0=(s_0,i_0,j_0)$, and
$c_1=(s_1,i_1,j_1)$ denoting crossings, there is a quadratic polynomial
$R_{11}(c)\in\bbQ(T_1,T_2)[g_{\nu\alpha\beta}:\alpha,\beta\in\{i,j\}]$
in the $g_{\nu\alpha\beta}$'s with coefficients in the ring of rational
functions in $T_1$ and $T_2$ and with $\alpha,\beta\in\{i,j\}$, and similarly a cubic
$R_{12}(c_0,c_1)\in\bbQ(T_1,T_2)[g_{\nu\alpha\beta}:\alpha,\beta\in\{i_0,j_0,i_1,j_1\}]$,
and a linear $\Gamma_1(\varphi,k)$ such that the following is a knot
invariant:
\begin{equation} \label{eq:Main}
  \theta(D) \coloneqq \Delta_1\Delta_2\Delta_3
  \left(\sum_c R_{11}(c) + \sum_{c_0,c_1} R_{12}(c_0,c_1) + \sum_k\Gamma_1(\varphi_k,k)\right).
\end{equation}
\end{theorem}

We note without detail that there is an alternative formula for $\theta$
in terms of perturbed Gaussian integration~\cite{IType}. In that language,
and using also the traffic motifs of~\cite{APAI, Toronto-241030}, the
three summands in~\eqref{eq:Main} become Feynman diagrams for processes
in which cars governed by parameter $T=T_1, T_2,$ or $T_3$ interact:

\[ \input{figs/FeynmanDiagrams.pdf_t} \]

In particular, the middle diagram which resembles the greek letter $\Theta$ gave the invariant its
name.

We note also that computationaly, the worst term in~\eqref{eq:Main}
is the middle one, and even it takes merely $\sim n^2$ operations in
the ring $\bbQ(T_1,T_2)$.

The polynomials $R_{11}(c)$,
$R_{12}(c_0,c_1)$ and $\Gamma_1(\varphi,k)$ are not unique, and we are not certain that we have the cleanest
possible formulas for them. As admitted, they are human-ugly. Yet from a computational perspective, having 18
terms (as is the case for $\Gamma_1(\varphi,k)$) isn't really a problem; computers don't care. Anyway, here are
the formulas:
\begin{multline*}
  R_{11}(c) \!=\! s
    \left[ 1/2 - g_{3ii} +  T_2^s g_{1ii} g_{2ji} - T_2^s g_{3jj} g_{2ji}
      - (T_2^s\!-\!1) g_{3ii} g_{2ji} \right. \\
    \left. + (T_3^s\!-\!1) g_{2ji} g_{3ji} - g_{1ii} g_{2jj} + 2 g_{3ii} g_{2jj}
      + g_{1ii} g_{3jj} - g_{2ii} g_{3jj} \right] \\
  + \frac{s}{T_2^s\!-\!1}
    \left[
      (T_1^s\!-\!1)T_2^s \left( g_{3jj} g_{1ji} - g_{2jj} g_{1ji} + T_2^s g_{1ji} g_{2ji} \right) \right. \\
      + (T_3^s\!-\!1) \left( g_{3ji} - T_2^s g_{1ii} g_{3ji} + g_{2ij} g_{3ji}
      + (T_2^s\!-\!2) g_{2jj} g_{3ji} \right) \\
    \left. - (T_1^s\!-\!1) (T_2^s\!+\!1) (T_3^s\!-\!1) g_{1ji} g_{3ji} \right]
\end{multline*}
\[
  R_{12}(c_0,c_1) \!=\!
    \frac{s_1 (T_1^{s_0}\!-\!1) (T_3^{s_1}\!-\!1) g_{1j_1i_0} g_{3j_0i_1}}{T_2^{s_1}\!-\!1}
    \left(T_2^{s_0} g_{2i_1i_0}+g_{2j_1j_0} - T_2^{s_0} g_{2j_1i_0}-g_{2i_1j_0} \right)
\]
\[ \Gamma_1(\varphi,k) = \varphi(-1/2+g_{3kk}) \]

\section{Implementation and Examples} A concise yet reasonably efficient implementation is worth a thousand
formulas. It completely removes ambiguities, it tests the theories, and it allows for experimentation. Hence
our next task is to implement. The section that follows was generated from a
Mathematica~\cite{Wolfram:Mathematica} notebook which is available at~\cite[Theta.nb]{Self}.

\input Implementation.tex

\section{Proof of Invariance} \label{sec:Proof}

\section{Strong and Meaningful}

\section{Conjectures and Dreams}
