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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{Banff-2607}
\def\title{$\Theta$: $4\smiley4\frownie$}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\thanks{{OMG, thanks! (Again!)}}
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%%%

\def\Abstract{{\raisebox{1.6mm}{\parbox[t]{3.95in}{
\parshape 7 0in 3.3in 0in 3.3in 0in 3.3in 0in 3.3in 0in 3.3in 0in 3.3in 0in 3.95in
{\red\bf Abstract.} Recently, in September 2025, Roland van der Veen and myself released a
paper titled ``A Fast, Strong, Topologically Meaningful, and Fun Knot Invariant''
(that's $4\smiley$; see \cite{Theta}). More recently, in January 2025 in Les Diablerets (see
https://drorbn.net/ld26), I gave a talk in which I suggested 27 homework tasks related to
that same invariant $\Theta$ (plus a bonus task).
\qquad Today, after a brief review, I'll talk a bit more about just 4 of those tasks:

$\frownie$~Figure out the relationship of $\Theta$ with Chern
Simons theory.

$\frownie$~Complete the discussion of the relationship of $\Theta$
with $sl_3$.

\hangindent=5mm\hangafter=1
$\frownie$~Find formulas for $\Theta$ corresponding to other
presentations of the Alexander module.

\hangindent=5mm\hangafter=1
$\frownie$~Fully understand $\Theta$-like formulas and their
non-uniqueness (i.e., clean the mess).

\footnotesize {\bf\red Acknowledgement.} This work was supported by NSERC
grant RGPIN-2025-06718 and by the Chu Family Foundation (NYC).
}}}}

\def\A{{\raisebox{1.7mm}{\parbox[t]{3.95in}{
{\red $A$.} With $T$ an indeterminate, start from a presentation
matrix $A$ for the Alexander module of $K$, coming from the Wirtinger
presentation of $\pi_1(K)$: $A\coloneqq I_{2n+1}+\sum_c A_c$, where

\vskip -4mm
\parshape 1 0in 3.125in
\[
  \begin{array}{c}\input{figs/Xings.pdf_t}\end{array}
  \ \rightarrow\ 
  \begin{array}{c|cccc}
    A_c &   i+1  &  j+1 \\
    \hline
    i & -T^s  & T^s-1 \\
    j & 0  & -1
  \end{array}
\]
\[
A=\left(
\begin{array}{ccccccc}
 1 & \mbluem{-T} & 0 & 0 & \mbluem{T-1} & 0 & 0 \\
 0 & 1 & \mpinkm{-1} & 0 & 0 & \mpinkm{\ 0\ } & 0 \\
 0 & 0 & 1 & \myellowm{-T} & 0 & 0 & \myellowm{T-1} \\
 0 & \mbluem{\ 0\ } & 0 & 1 & \mbluem{-1} & 0 & 0 \\
 0 & 0 & \mpinkm{T-1} & 0 & 1 & \mpinkm{-T} & 0 \\
 0 & 0 & 0 & \myellowm{\ 0\ } & 0 & 1 & \myellowm{-1} \\
 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)
\]
}}}}

\def\G{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red $G$.} Let $G = (g_{\alpha\beta}) \coloneqq A^{-1}$, the ``two point function'':

$G=\left( \arraycolsep=3.5pt
\begin{array}{ccccccc}
 1 & T & 1 & T & 1 & T & 1 \\
 0 & 1 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} &
\frac{T^2}{T^2-T+1} &
   1 \\
 0 & 0 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} &
\frac{T^2}{T^2-T+1} &
   1 \\
 0 & 0 & \frac{1-T}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{1}{T^2-T+1} &
\frac{T}{T^2-T+1} &
   1 \\
 0 & 0 & \frac{1-T}{T^2-T+1} & -\frac{(T-1) T}{T^2-T+1} & \frac{1}{T^2-T+1} &
   \frac{T}{T^2-T+1} & 1 \\
 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)$

Let $T_1$ and $T_2$ be new indeteminates, let $T_3=T_1T_2$, and
let $G_\nu=(g_{\nu\alpha\beta})$ be $G$ with $T\to T_\nu$, for $\nu=1,2,3$.
}}}}

\def\thet{{\raisebox{3mm}{\parbox[t]{2.25in}{
\[
  {\red\theta} \sim \Delta_1\Delta_2\Delta_3\sum_{c_0,c_1}g_{1i_0i_1}g_{2i_0i_1}g_{3i_1i_0}
  + \text{l.o.}
\]
\[ {\red\Theta} = (\Delta,\theta)\in\bbZ[T^{\pm1}]\times\bbZ[T_1^{\pm1},T_2^{\pm1}] \]
}}}}

\def\random{{\raisebox{1mm}{\parbox[t]{1.8in}{
A random 300 xing knot from \cite{DHOEBL:Random}. For most invariants,
300 is science fiction.
}}}}

\def\Strong{{\raisebox{2mm}{\parbox[t]{4in}{
{\bf\purple Strong.} $\Theta$ vs.\ a slew of other reasonably-computable invariants
(deficits shown):

\!\resizebox{4.02in}{!}{\def\s{$\sim$}
\begin{tabular}{c|c|c|c|c|c|c}
\hline
$n$&				$\leq 10$&	$\leq 11$&	$\leq 12$&	$\leq 13$&	$\leq 14$&	$\leq 15$ \\ \hline
knots&				249&		801&		2,977&		12,965&		59,937&		313,230 \\ \hline
$\Delta$&			(38)&		(250)&		(1,204)&	(7,326)&	(39,741)&	(236,326) \\ \hline
$\sigma_{LT}$&			(108)&		(356)&		(1,525)&	(7,736)&	(40,101)&	(230,592) \\ \hline
$J$&				(7)&		(70)&		(482)&		(3,434)&	(21,250)&	(138,591) \\ \hline
$\Kh$&				(6)&		(65)&		(452)&		(3,226)&	(19,754)&	(127,261) \\ \hline
$H$&				(2)&		(31)&		(222)&		(1,839)&	(11,251)&	(73,892) \\ \hline
$\Vol$&				(\s6)&		(\s25)&		(\s113)&	(\s1,012)&	(\s6,353)&	(\s43,607) \\ \hline
$(\Kh,H,\Vol)$&			(\s0)&		(\s14)&		(\s84)&		(\s911)&	(\s5,917)&	(\s41,434) \\ \hline
$(\Delta,\rho_1)$&		(0)&		(14)&		(95)&		(959)&		(6,253)&	(42,914) \\ \hline
$(\Delta,\rho_1,\rho_2)$&	(0)&		(14)&		(84)&		(911)&		(5,926)&	(41,469) \\ \hline
$(\rho_1,\rho_2,\Kh,H,\Vol)$&	(0)&		(\s14)&		(\s84)&		(\s911)&	(\s5,916)&	(\s41,432) \\ \hline
\rowcolor{yellow}
$\Theta$&			(0)&		(3)&		(19)&		(194)&		(1,118)&	(6,758) \\ \hline
$(\Theta,\rho_2)$&		(0)&		(3)&		(10)&		(169)&		(982)&		(6,341) \\ \hline
$(\Theta,\sigma_{LT})$&		(0)&		(3)&		(19)&		(194)&		(1,118)&	(6,758) \\ \hline
$(\Theta,\Kh)$&			(0)&		(3)&		(18)&		(185)&		(1,062)&	(6,555) \\ \hline
$(\Theta,H)$&			(0)&		(3)&		(18)&		(185)&		(1,064)&	(6,563) \\ \hline
$(\Theta,\Vol)$&		(0)&		(\s3)&		(\s10)&		(\s169)&	(\s973)&	(\s6,308) \\ \hline
$(\Theta,\rho_2,\Kh,H,\Vol)$&	(0)&		(\s3)& 		(\s10)&		(\s169)&	(\s972)&	(\s6,304) \\ \hline
\end{tabular}}
}}}}

\def\TopMean{{\raisebox{2mm}{\parbox[t]{4in}{
{\bf\purple Topologically Meaningful.} $\theta$ is near $\Delta$ and
we dream that anything $\Delta$ can do, $\theta$ does too (sometimes
better). The following two conjectures are verified for knots with $\leq 13$ crossings:
\par{\bf\red Conjecture 1.} $\deg_{T_1}\theta(K)\leq 2g(K)$.
\par{\bf\red Conjecture 2.} If $K$ is a fibered knot and $d$ is the degree of $\Delta(K)$ (the highest power of
$T$), then the coefficient of $T_2^{2d}$ in $\theta(K)$, which is a
polynomial in $T_1$, is an integer multiple of $T_1^d\Delta(K)|_{T\to T_1}$.
\par{\bf\red Dream.} $\theta$ has something to say about ribbon knots.
}}}}

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{\normalsize\red\bf References.}

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\begin{center}
\includegraphics[height=\textheight, page=1]{../../Projects/Theta/Theta_16up_2.pdf}
\newpage\includegraphics[height=\textheight, page=2]{../../Projects/Theta/Theta_16up_2.pdf}
\end{center}

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