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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{UBC-241004}
\def\title{The Strongest Genuinely Computable Knot Invariant in 2024}

\def\navigator{{
  \href{https://www.mathematics.utoronto.ca/}{University of Toronto}:
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\thanks{{Thanks for allowing me in UBC!}}
\def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/ubc24}{http://drorbn.net/ubc24}}}
\def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}}
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%%%

\def\Abstract{{\raisebox{1.2mm}{\parbox[t]{3.95in}{
\parshape 6 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.95in
{\red\bf Abstract.} ``Genuinely computable'' means we have computed it
for random knots with over 300 crossings. ``Strongest'' means it separates
prime knots with up to 15 crossings better than the less-computable
HOMFLY-PT and Khovanov homology taken together. And hey, it's also meaningful
and fun.

Continues Rozansky, Garoufalidis, Kricker, and Ohtsuki, joint with van der Veen.

{\bf\red Acknowledgement.} This work was supported by NSERC
grant RGPIN-2018-04350 and by the Chu Family Foundation (NYC).
}}}}

\def\Strongest{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Strongest.} Testing $\Theta=(\Delta,\theta)$ on prime knots
up to mirrors and reversals, counting the number of distinct values
(with deficits in parenthesis):
\hfill{\footnotesize
  ($\rho_1$: \cite{Ro, Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis, APAI})
}

\centering\small\begin{tabular}[t]{c|c|c|c|c|c}
  & knots & $(H,Kh)$ & $(\Delta,\rho_1)$ & $\Theta=(\Delta,\theta)$ & together \\
  \hline
  reign & & 2005-22 & 2022-24 & 2024- \\
  \hline\hline
  xing $\leq 10$ & 249 & 248 (1) & 249 (0) & 249 (0) & 249 (0) \\
  \hline
  xing $\leq 11$ & 801 & 771 (30) & 787 (14) & 798 (3) & 798 (3) \\
  \hline
  xing $\leq 12$ & 2,977 & (214) & (95) & (19) & (18) \\
  \hline
  xing $\leq 13$ & 12,965 & (1,771) & (959) & (194) & (185) \\
  \hline
  xing $\leq 14$ & 59,937 & (10,788) & (6,253) & (1,118) & (1,062) \\
  \hline
  xing $\leq 15$ & 313,230 & (70,245) & (42,914) & (6,758) & (6,555) \\
\end{tabular}
}}}}

\def\GC{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 1 0in 2.3in
{\red\bf Genuinely Computable.} Here's $\Theta$ on a random 300 crossing
knot (from \cite{DHOEBL:Random}). For almost every other
invariant, that's science fiction.

\parshape 6 0in 2.3in 0in 2.3in 0in 2.3in 0in 2.3in 0in 2.3in 0in \linewidth
{\red\bf Fun.} There's so much more to see in 2D pictures
than in 1D ones! Yet almost nothing
of the patterns you see we know how to prove. We'll have fun with that
over the next few years. Would you join?

{\bf\red Meaningful.} $\Theta$ gives a genus bound (unproven yet with confidence).
We hope (with reason) it says something about ribbon knots.

{\bf\red The Bad(?).} $\Theta$ art is more glass blowing than pottery.
}}}}

\def\Jones{{\raisebox{2mm}{\parbox[t]{2.95in}{
{\red\bf Jones:}
\vskip 1mm
\par Formulas stay;
\par stories change with time.
}}}}

\def\Arules{{$\displaystyle
  \begin{array}{c|cccc}
    A &   \text{col }i\!+\!1  &  \text{col }j\!+\!1 \\
    \hline
    \text{row }i &  -T^s  & T^s-1 \\
    \text{row }j &  0  & -1
  \end{array}
$}}

\def\FormulasA{{\raisebox{2mm}{\parbox[t]{3.125in}{
{\red\bf Formulas.} Draw an $n$-crossing knot $K$ as on the right: all crossings face up, and the edges
are marked with a running index ${k\in\{1,\ldots,2n+1\}}$ and with rotation numbers $\varphi_k$. Let $A$
be the
$(2n+1)\times(2n+1)$ matrix constructed by starting with the identity matrix $I$, and adding a $2\times 2$
block for each crossing:
}}}}

\def\FormulasB{{\raisebox{0mm}{\parbox[t]{3.95in}{
Let $G=(g_{\alpha\beta})=A^{-1}$. For the trefoil example, it is:
\par $\displaystyle
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),
$
\par $\displaystyle\quad
G=\left(
\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)
$
}}}}

\def\FormulasC{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\bf\red Note.} The Alexander polynomial $\Delta$ is given by
\newline\null
  \hfill$\Delta = T^{(-\varphi-w)/2}\det(A)$,
  \hfill with $\varphi = \sum_k \varphi_k$, $w = \sum_c s$.
  \hfill\null
\newline
%\[ \Delta = T^{(-\varphi-w)/2}\det(A), \qquad
%  \text{with }\varphi = \sum_k \varphi_k,\ w \!=\! \sum_c s.
%\]
{\red\bf Classical Topologists:} This is boring. Yawn.
}}}}

\def\Interpretation{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 5 0in 3.3in 0in 3.3in 0in 3.3in 0in 3.3in 0in 3.95in
{\bf\red Cars, Interchanges, and Traffic Counters.} Cars always drive
forward. When a car crosses over a bridge it goes through with (algebraic)
probability $T^s\sim 1$, but falls off with probability $1-T^s\sim
0^\ast$. At the very end, cars fall off and disappear. See
also~\cite{Jones:Hecke, LinTianWang:RandomWalk}.
}}}}

\def\Foot{{\raisebox{0mm}{\parbox[t]{3.95in}{\footnotesize
\ $^\ast$ In algebra $x\sim 0$ if for every $y$ in the ideal generated by $x$, $1-y$ is invertible.
}}}}

\def\dtA{{\tiny image credits:}}
\def\dtB{{\tiny \href{https://diamondtraffic.com/productcategory/Portable-Counters}{diamondtraffic.com}}}

\def\DallEA{{\tiny image credits:}}
\def\DallEB{{\tiny \href{https://labs.openai.com/}{Dall-E}}}

\def\gab{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 5 0in 2.8in 0in 2.8in 0in 2.8in 0in 2.8in 0in \linewidth
{\bf\red Theorem.} The Green function $g_{\alpha\beta}$ is the \text{reading}
of a traffic counter at $\beta$, if car traffic is injected at $\alpha$
(if $\alpha=\beta$, the counter is {\em after} the injection point).
\par{\bf\red Example.}
\vskip 0.625in

\parshape 5 0in 3.4in 0in 3.4in 0in 3.4in 0in 3.4in 0in \linewidth
{\bf\red Proof.} Near a crossing $c$ with sign $s$, incoming upper edge~$i$ and incoming lower edge $j$,
both sides satisfy the {\em $g$-rules}:
\[ g_{i\beta} = \delta_{i\beta}+T^sg_{i+1,\beta}+(1-T^s)g_{j+1,\beta},
  \quad g_{j\beta} = \delta_{j\beta}+g_{j+1,\beta},
\]
and always, $g_{\alpha,2n+1} = 1$: use common sense and $AG=I\ (=GA)$.
\par{\bf\red Bonus.} Near $c$, both sides satisfy the further {\em $g$-rules}:
\[ g_{\alpha i} = T^{-s}(g_{\alpha,i+1}-\delta_{\alpha,i+1}),
  \quad g_{\alpha j} = g_{\alpha,j+1} - (1-T^s)g_{\alpha i} - \delta_{\alpha,j+1}.
\]
}}}}

\def\kinkA{{$\sum_{p\geq 0}(1\!-\!T)^p=T^{-1}$}}
\def\kinkG{{$G=\begin{pmatrix}1&T^{-1}&1\\0&T^{-1}&1\\0&0&1\end{pmatrix}$}}

\def\InvarianceA{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\bf\red Invariance of $\Theta$.} We start with the hardest, Reidemeister 3:
}}}}

\def\InvarianceB{{\raisebox{0mm}{\parbox[t]{3.95in}{
$\Rightarrow$ Overall traffic patterns are unaffected by Reid3!
\newline $\Rightarrow$ Green's $g_{\alpha\beta}$ is unchanged by Reid3,
provided the cars injection site $\alpha$ and the traffic counters $\beta$
are away.

\parshape 6 0in 2.5in 0in 2.5in 0in 2.5in 0in 2.5in 0in 2.5in 0in \linewidth
$\Rightarrow$ Only the contribution from the $R_1$ and $\theta$
terms within the Reid3 move matters, and using $g$-rules the relevant
$g_{\alpha\beta}$'s can be pushed outside of the Reid3 area:
\import{.}{Invariance.tex}
The other Reidemeister moves are treated in a similar manner.\qed
}}}}

\def\messA{{$(1\!-\!T)^2\!+\!T(1\!-\!T)$}}
\def\messB{{$(1\!-\!T)T$}}
\def\messC{{$T(1\!-\!T)$}}
\def\messD{{$1\!-\!T$}}
\def\i{{$i$}} \def\j{{$j$}} \def\k{{$k$}} \def\m{{$m$}} \def\n{{$n$}} \def\s{{$s$}}
\def\ip{{$i^+$}} \def\jp{{$j^+$}} \def\kp{{$k^+$}}
\def\ipp{{$i^{+\!+}$}} \def\jpp{{$j^{+\!+}$}} \def\kpp{{$k^{+\!+}$}}

\def\Conjectures{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Questions, Conjectures, Expectations, Dreams.}

{\bf Question 1.} What's the relationship between $\Theta$ and the Garoufalidis-Kashaev
invariants \cite{GaroufalidisKashaev:Multivariable, GaroufalidisLi:Patterns}?

{\bf Conjecture 2.} On classical (non-virtual) knots, $\theta$ always has hexagonal 
($D_6$) symmetry.

{\bf Conjecture 3.} $\theta$ is the $\epsilon^1$ contribution to
the ``solvable approximation'' of the $sl_3$ universal invariant,
obtained by running the quantization machinery on the double
$\calD(\frakb,b,\eps\delta)$, where $\frakb$ is the Borel subalgebra of
$sl_3$, $b$ is the bracket of $\frakb$, and $\delta$ the cobracket.
See \cite{PG, DPG, Schaveling:Thesis}

{\bf Conjecture 4.} $\theta$ is equal to the ``two-loop
contribution to the Kontsevich Integral'', as studied by
Garoufalidis, Rozansky, Kricker, and in great detail by Ohtsuki
\cite{GaroufalidisRozansky:LoopExpansion, Ro, Rozansky:Burau,
Rozansky:U1RCC, Kricker:Lines, Ohtsuki:TwoLoop}.

{\bf Fact 5.} $\theta$ has a perturbed Gaussian integral formula,
with integration carried out over over a space $6E$, consisting of 6
copies of the space of edges of a knot diagram $D$. See \cite{IType}.

{\bf Conjecture 6.} For any knot $K$, its genus $g(K)$ is bounded by the $T_1$-degree of
$\theta$: $g(K) < \lceil\deg_{T_1}\theta(K)\rceil$.

{\bf Conjecture 7.} $\theta(K)$ has another perturbed Gaussian integral
formula, with integration carried out over over the space $6H_1$,
consisting of 6 copies of $H_1(\Sigma)$, where $\Sigma$ is a Seifert
surface for $K$.

{\bf Expectation 8.} There are many further invariants like $\theta$, given by Green function
formulas and/or Gaussian integration formulas. One or two of them may be stronger than $\theta$
and as computable.

{\bf Dream 9.} These invariants can be explained by something less foreign than semisimple
Lie algebras.

{\bf Dream 10.} $\theta$ will have something to say about ribbon knots.
}}}}

\def\refs{{\raisebox{5.5mm}{\parbox[t]{3.95in}{
%{\red\bf References.}
{\footnotesize
\def\bysame{{---}}
%\par\vspace{-3mm}
\renewcommand{\section}[2]{}%
\begin{thebibliography}{}
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.3ex}

\input refs.tex

\end{thebibliography}}
}}}}

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\begin{document} \latintext
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\null\vfill\input{SGC24T1.pdftex_t}\vfill\null
\end{center}

\newgeometry{textwidth=8in,textheight=10.5in}

\begin{multicols*}{2}

{\red\bf New Stuff.}
Now let $T_1$ and $T_2$ be indeterminates and let $T_3=T_1T_2$. For $\nu=1,2,3$ let $\Delta_\nu$ and
$G_\nu = (g_{\nu\alpha\beta})$ be $\Delta$ and $G$ subject to the substitution $T\to T_\nu$.
Define
\[
  \theta(K) \coloneqq \Delta_1\Delta_2\Delta_3\left(\sum_c R_1(c) + \sum_{c_0,c_1} \theta(c_0,c_1)
  + \sum_k\Gamma_1(\varphi_k,k)\right),
\]
where the first summation is over crossings $c=(s,i,j)$, the second is over pairs of crossings
$(c_0=(s_0,i_0,j_0),c_1=(s_1,i_1,j_1))$, and the third is over edges $k$, and where
\begin{multline*}
  R_1(c) \!\coloneqq\! 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*}
\begin{multline*}
  \theta(c_0,c_1) \!\coloneqq\! \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} \\
    \cdot \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)
\end{multline*}
\[ \Gamma_1(\varphi,k) \coloneqq \varphi(-1/2+g_{3kk}) \]

{\red\bf Theorem.} $\theta$ and hence $\Theta$ are knot invariants.

\rule{\linewidth}{1pt}\vspace{0mm}

\input Theta.tex

\end{multicols*}

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\begin{center}
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\newpage \newgeometry{textwidth=8in,textheight=10.5in}

The torus knot $T_{22/7}$:\hfill(many more at \web{TK})
\[ \includegraphics[width=0.9\linewidth]{T227.pdf} \]

Random knots from \cite{DHOEBL:Random}, with 50-73 crossings:\hfill(many more at \web{DK})

\[ \includegraphics[width=0.95\linewidth]{Gallery50-73.png} \]

\end{document}

\endinput

