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\title{The Strongest Genuinely Computable Knot Invariant (Since) In 2024}

\author{Dror Bar-Natan}

\institute{\url{http://drorbn.net/ktc25}}

%\date{The Fields Institute, October 2021}

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\begin{document} \latintext\beamertemplatenavigationsymbolsempty

\begin{frame}
  %\titlepage
  \begin{center}
  \vfill
  \LARGE{\bf The Strongest Genuinely Computable Knot Invariant \renewcommand{\ULthickness}{1.5pt}\sout{Since} In 2024}
  \vfill
  \par\Large \href{https://knots-congress.github.io/}{The First International On-line Knot Theory Congress}
  \par February 1-5, 2025
  \vfill
  \par\normalsize Dror Bar-Natan
  \end{center}

  \vfill\normalsize

  {\bf\red 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.
\end{frame}

\begin{frame}
  These slides and all the code within are available at \url{http://drorbn.net/ktc25}
  \vskip 2mm (not making your slides available {\bf before} your talk is a sin).
  \vskip 10mm
  (I'll post the video there too)
  \vskip 10mm If you can, please turn your video on!
\end{frame}

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

\begin{frame}
  Strongest? Genuinely Computable?
\end{frame}

\begin{frame}
{\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})
}
\vfill
\resizebox{\linewidth}{!}{
  \begin{tabular}[t]{c|c|c|c|>{\columncolor{Yellow}}c|c|c}
  & knots & $(H,Kh)$ & $(\Delta,\rho_1)$ & $\Theta=(\Delta,\theta)$ & $(\Delta,\theta,\rho_2)$ & all together \\
  \hline
  reign & & 2005-22 & 2022-24 & 2024 & 2025- \\
  \hline\hline
  xing $\leq 10$ & 249 & 248 (1) & 249 (0) & 249 (0) & 249(0) & 249 (0) \\
  \hline
  xing $\leq 11$ & 801 & 771 (30) & 787 (14) & 798 (3) & 798 (3) & 798 (3) \\
  \hline
  xing $\leq 12$ & 2,977 & (214) & (95) & (19) & (10) & (10) \\
  \hline
  xing $\leq 13$ & 12,965 & (1,771) & (959) & (194) & (169) & (169) \\
  \hline
  xing $\leq 14$ & 59,937 & (10,788) & (6,253) & (1,118) & (982) & (981) \\
  \hline
  xing $\leq 15$ & 313,230 & (70,245) & (42,914) & (6,758) & (6,341) & (6,337) \\
\end{tabular}}
\end{frame}

\begin{frame}
  \parbox[b]{0.49\linewidth}{
  {\red\bf Genuinely Computable.} Here's $\Theta$ on a random 300 crossing knot (from \cite{DHOEBL:Random}). For almost every other knot invariant, that's science fiction.
  }
  \hfill
  \parbox[b]{0.49\linewidth}{
    \includegraphics[width=\linewidth]{../UBC-241004/DK300.pdf}
  }
\end{frame}

\begin{frame}
  \parbox[b]{0.49\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?
  }
  \hfill
  \parbox[b]{0.49\linewidth}{
    \includegraphics[width=\linewidth]{../UBC-241004/DK300.pdf}
  }
\end{frame}

\begin{frame}
Random knots (from \cite{DHOEBL:Random}) with 101--115 crossings:
\newline\includegraphics[width=\linewidth]{Beehive.png}
\end{frame}

\begin{frame}
The Rolfsen Table:
\vfill\includegraphics[width=\linewidth]{../UBC-241004/g250@.png}
\end{frame}

\begin{frame}
The torus knots $T_{13/2}$, $T_{17/3}$, $T_{13/5}$, and $T_{7/6}$:
\vfill\includegraphics[width=\linewidth]{TorusKnots.pdf}
\end{frame}

\begin{frame}
The torus knot $T_{22/7}$: \includegraphics[width=0.6\linewidth]{T227.pdf}
\end{frame}

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

\begin{frame}
  {\bf\red Convention.} $T$, $T_1$, and $T_2$ are indeterminates and $T_3\coloneqq T_1T_2$.
\end{frame}

\begin{frame}
  \parbox[b]{0.49\linewidth}{
  {\red\bf Preparation.} Draw an $n$-crossing knot $K$ as a diagram $D$ 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 $\varphi_k$.
  }
  \hfill \input{figs/KnotDiagram.pdf_t} \hfill\null
\end{frame}

\begin{frame}
\[ 
  \def\dtA{{\tiny image credits:}}
  \def\dtB{{\tiny \href{https://diamondtraffic.com/productcategory/Portable-Counters}{diamondtraffic.com}}}
  \def\DallEA{{\tiny image credits:}}
  \def\DallEB{{\tiny \href{https://labs.openai.com/}{Dall-E}}}
  \input{figs/TRulesTop.pdf_t}
\]
{\bf\red Model $T$ Traffic Rules.} Cars always drive
forward. When a car crosses over a sign-$s$ bridge it goes through with
(algebraic) probability $T^s\sim 1$, but falls off with probability
$1-T^s\sim 0$. At the very end, cars fall off and disappear.
On various edges {\em traffic counters} are placed.
See also~\cite{Jones:Hecke, LinTianWang:RandomWalk}.
\[ \input{figs/TRulesBottom.pdf_t} \]
\end{frame}

\begin{frame}
\[ \input{figs/TrafficFunction.pdf_t} \]
{\bf\red Definition.} The {\em traffic function} $G=(g_{\alpha\beta})$
(also, the {\em Green function} or the {\em two-point function}) is the
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). There are also model-$T_\nu$ traffic functions
$G_\nu=(g_{\nu\alpha\beta})$ for $\nu=1,2,3$.

{\bf\red Example.}
\[
  \def\kinkA{{$\sum_{p\geq 0}(1\!-\!T)^p=T^{-1}$}}
  \def\kinkG{{$G=\begin{pmatrix}1&T^{-1}&1\\0&T^{-1}&1\\0&0&1\end{pmatrix}$}}
  \input{figs/TrafficOnKink.pdf_t}
\]

\end{frame}

\begin{frame}
Given crossings $c=(s,i,j)$, $c_0=(s_0,i_0,j_0)$, and $c_1=(s_1,i_1,j_1)$, let
\begin{alignat*}{1}
  F_1(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. \\
  &\quad
    \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. \\
  &\quad
      + (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) \\
  &\quad
    \left. -\,(T_1^s-1) (T_2^s+1) (T_3^s-1) g_{1ji} g_{3ji} \right] \\
  F_2(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) \\
  F_3(\varphi_k,k) =\ &
     \varphi_k(g_{3kk}-1/2)
\end{alignat*}
\hfill(Computers don't care!)
\end{frame}

\begin{frame}
{\red\bf Main Theorem.}
The following is a knot invariant:\hfill{\footnotesize (the $\Delta_\nu$ are normalizations discussed later)}
\[ 
  \theta(D) \coloneqq \Delta_1\Delta_2\Delta_3
  \left(\sum_c F_1(c) + \sum_{c_0,c_1} F_2(c_0,c_1) + \sum_kF_3(\varphi_k,k)\right).
\]
\[ \input{figs/FeynmanDiagrams.pdf_t} \]
If these pictures remind you of Feynman diagrams, it's because they are Feynman
diagrams~\cite{IType}.
\end{frame}

\begin{frame}
{\bf\red Lemma 1.} The traffic function $g_{\alpha\beta}$ is a ``relative invariant'':
\[ \input{figs/RelativeInvariance.pdf_t} \]
{\footnotesize (There is some small print for R1 and R2 which change the numbering of the edges and sometimes collapse a pair of edges into one)}
\end{frame}

\begin{frame}
{\bf\red Proof.}
\[
  \def\messA{{$(1\!-\!T)^2\!+\!T(1\!-\!T)$}}
  \def\messB{{$(1\!-\!T)T$}}
  \def\messC{{$T(1\!-\!T)$}}
  \def\messD{{$1\!-\!T$}}
  \input{figs/RIProof.pdf_t}
\]
\end{frame}

\begin{frame}
{\bf\red Lemma 2.} With $k^+\coloneqq k+1$, the ``$g$-rules'' hold near a crossing $c=(s,i,j)$:
\[
  g_{j\beta} = g_{j^+\beta} + \delta_{j\beta}
  \quad g_{i\beta} = T^sg_{i^+\beta} + (1-T^s)g_{j^+\beta} + \delta_{i\beta}
  \quad g_{2n^+,\beta} = \delta_{2n^+,\beta}
\]
\[
  g_{\alpha i^+} = T^sg_{\alpha i} + \delta_{\alpha i^+}
  \quad g_{\alpha j^+} = g_{\alpha j} + (1-T^s)g_{\alpha i} + \delta_{\alpha j^+}
  \quad g_{\alpha,1} = \delta_{\alpha,1}
  %\quad g_{\alpha,2n^+} = 1
\]
\vskip 5mm
\[ \input{figs/GenericCrossing.pdf_t} \]
\end{frame}

\begin{frame}
{\bf\red Corollary 1.} $G$ is easily computable, for $AG=I$ ($=GA$), with $A$ the $(2n+1)\times(2n+1)$
identity matrix with additional contributions:
\parbox[b]{0.75\linewidth}{
\[
  c=(s,i,j) \mapsto \begin{array}{c|cccc}
    A & \text{col }i^+ & \text{col }j^+ \\
    \hline
    \text{row }i & -T^s & T^s-1 \\
    \text{row }j & 0 & -1
  \end{array}
\]
For the trefoil example, we have:
\[
A=\left(
\begin{array}{ccccccc}
 1 & \mbluem{-T} & 0 & 0 & \mbluem{T-1} & 0 & 0 \\
 0 & 1 & \mpinkm{-1} & 0 & 0 & \mpinkm{\ 0\ } & 0 \\
 0 & 0 & 1 & \myellowm{-T} & 0 & 0 & \myellowm{T-1} \\
 0 & \mbluem{\ 0\ } & 0 & 1 & \mbluem{-1} & 0 & 0 \\
 0 & 0 & \mpinkm{T-1} & 0 & 1 & \mpinkm{-T} & 0 \\
 0 & 0 & 0 & \myellowm{\ 0\ } & 0 & 1 & \myellowm{-1} \\
 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)
\]
}\hfill \input{figs/KnotDiagram.pdf_t} \hfill\null
\end{frame}

\begin{frame}
And so,
\[
G=\left(
\begin{array}{ccccccc}
 1 & T & 1 & T & 1 & T & 1 \\
 0 & 1 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T^2}{T^2-T+1} &
   1 \\
 0 & 0 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T^2}{T^2-T+1} &
   1 \\
 0 & 0 & \frac{1-T}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} &
   1 \\
 0 & 0 & \frac{1-T}{T^2-T+1} & -\frac{(T-1) T}{T^2-T+1} & \frac{1}{T^2-T+1} &
   \frac{T}{T^2-T+1} & 1 \\
 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)
\]
\end{frame}

\begin{frame}
{\bf\red Note.} The Alexander polynomial $\Delta$ is given by

\[ \Delta = T^{(-\varphi-w)/2}\det(A), \]
with
\[ \varphi = \sum_k \varphi_k, \qquad w = \sum_c s. \]
We also set $\Delta_\nu\coloneqq\Delta(T_\nu)$ for $\nu=1,2,3$. This defines and explain the normalization factors in the Main Theorem.
\end{frame}

\begin{frame}
{\bf\red Corollary 2.} Proving invariance is easy:
\vskip 5mm
\resizebox{\linewidth}{!}{\import{../Toronto-241030}{R3.pdftex_t}}
\end{frame}

\import{.}{Theta.tex}

\begin{frame}
\vfill
{\red\bf Questions, Conjectures, Expectations, Dreams.}
\end{frame}

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

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

\begin{frame}
{\bf\red 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}
\end{frame}

\begin{frame}
{\bf\red 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}.
\end{frame}

\begin{frame}\null
{\bf\red 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}.

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

\vfill{\bf\red 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 $K$.
\end{frame}

\begin{frame}
{\bf\red 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.
\end{frame}

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

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

\begin{frame}
  Thank You!
\end{frame}


{\red\bf References.}
\footnotesize

% Now get rid of all the colours
\setbeamercolor*{bibliography entry title}{fg=black}
\setbeamercolor*{bibliography entry author}{fg=black}
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\def\bysame{{---}}
\begin{thebibliography}{}

\bibitem[BN1]{DPG} D.~Bar-Natan,
  {\em Everything around $sl_{2+}^\epsilon$ is DoPeGDO. So what?,}
  talk in Da Nang, May 2019. Handout and video at \web{DPG}.

\bibitem[BN2]{IType} \bysame,
  {\em Knot Invariants from Finite Dimensional Integration,}
  talks in Beijing (July 2024, \web{icbs24}) and in Geneva (August 2024, \web{ge24}).

\bibitem[BV1]{APAI} \bysame, R.~van~der~Veen,
  {\em A Perturbed-Alexander Invariant,}
  Quantum Topology {\bf 15} (2024) 449--472, \web{APAI}.

\bibitem[BV2]{PG} \bysame, \bysame,
  {\em Perturbed Gaussian Generating Functions for Universal Knot Invariants,}
  \arXiv{2109.02057}.

\bibitem[DHOEBL]{DHOEBL:Random} N.~Dunfield, A.~Hirani, M.~Obeidin, A.~Ehrenberg,
  S.~Bhattacharyya, D.~Lei, and others,
  {\em Random Knots: A Preliminary Report,} lecture notes at \web{DHOEBL}. Also a
  data file at \web{DD}.

\bibitem[GK]{GaroufalidisKashaev:Multivariable}  S.~Garoufalidis, R.~Kashaev,
  {\em Multivariable Knot Polynomials from Braided Hopf Algebras with Automorphisms,}
  \arXiv{2311.11528}.

\bibitem[GL]{GaroufalidisLi:Patterns} \bysame, S.~Y.~Li,
  {\em Patterns of the $V_2$-polynomial of knots,}
  \arXiv{2409.03557}.

\bibitem[GR]{GaroufalidisRozansky:LoopExpansion} \bysame, L.~Rozansky,
  {\em The Loop Expansion of the Kontsevich Integral, the Null-Move, and $S$-Equivalence,}
  \arXiv{math.GT/0003187}.

\bibitem[Jo]{Jones:Hecke} V.~F.~R.~Jones,
  {\em Hecke Algebra Representations of Braid Groups and Link Polynomials,}
  Annals Math., {\bf 126} (1987) 335-388.

\bibitem[Kr]{Kricker:Lines} A.~Kricker,
  {\em The Lines of the Kontsevich Integral and Rozansky's Rationality Conjecture,}
  \arXiv{math/0005284}.

\bibitem[LTW]{LinTianWang:RandomWalk} X-S.~Lin, F.~Tian, Z.~Wang,
  {\em Burau Representation and Random Walk on String Links,}
  Pac.\ J.\ Math., {\bf 182-2} (1998) 289--302, \arXiv{q-alg/9605023}.

\bibitem[Oh]{Ohtsuki:TwoLoop} T.~Ohtsuki,
  {\em On the 2–loop Polynomial of Knots,}
  Geom.\ Top.\ {\bf 11} (2007) 1357--1475.

\bibitem[Ov]{Overbay:Thesis} A.~Overbay,
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\end{document}


