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\def\thistalk{Nara-2308}
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\begin{document}
\newdimen\captionwidth\captionwidth=\hsize

\title{Cars, Interchanges, Traffic Counters, and some Pretty Darned Good Knot Invariants}

\author{Dror~Bar-Natan, University of Toronto}
%\address{
%  Department of Mathematics\\
%  University of Toronto\\
%  Toronto Ontario M5S 2E4\\
%  Canada
%}
\email{drorbn@math.toronto.edu}
\urladdr{http://www.math.toronto.edu/~drorbn}

\begin{abstract}
A condensed summary of a talk I gave in Nara on August 13, 2023: Reporting on joint work with Roland van der Veen, I'll
tell you some stories about $\rho_1$, an easy to define, strong, fast
to compute, homomorphic, and well-connected knot invariant. $\rho_1$
was first studied by Rozansky and Overbay \cite{Ro, Rozansky:Burau,
Rozansky:U1RCC, Overbay:Thesis} and Ohtsuki \cite{Ohtsuki:TwoLoop}, it has far-reaching generalizations,
it is dominated by the coloured Jones polynomial, and I wish I understood it.
\end{abstract}

\date{\today}

\maketitle

\parpic[r]{\includegraphics[width=0.5\linewidth]{Slides/01.png}}
My talk's title and abstract were the same as the title and abstract of this summary. The talk used slides, and in this summary, they are shown on the right.

\vskip 7mm
\parpic[r]{\includegraphics[width=0.25\linewidth]{Slides/02.png}}
Those slides were all excerpts from a handout, which is attached at the end of this document. It is where the true content lies! It is also available on the web site of this talk, which is displayed on the next slide.

As an aside, I really believe in this way of giving talks, with slides and a handout. Slides are to save time and to allow for more elaborate figures. But slide talks without a handout are awful! Content disappears before it's been digested. A handout with identical content to the slides solves the problem -- you can always look back to recall (and ahead, to decide how hard you want to fight sleep). But then the best way to make sure that the handout and the slides are fully synced is to have the slides simply be zoomed-in parts of the handout, and that's precisely what I do.

But it's a waste of so much paper, I hear you say. Yes, I say, but it's completely trivial relative to our travel to hear each other talk. Save where it matters. Where it's useful, spend.

\parpic[r]{\includegraphics[width=2.5in]{Slides/03.png}}
Okay, it's all online, at \webdef. There's also a paper, at \web{APAI}.

\vskip 5mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/04.png}}
Thanks, NSERC and Arthur Chu!

\vskip 5mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/05.png}}
We seek strong, fast, and homomorphic invariants. Strong and fast are clear enough. Especially, we care for fast because of the likes of the GST48 knot~\cite{GompfScharlemannThompson:Counterexample} and the Piccirillo knot~\cite{Piccirillo:NotSlice}. Polynomial time is best!

\vskip 15mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/06.png}}
We then explained ``homomorphic''. It means, ``extends to tangles and is well behaved under tangle gluing and strand doubling''.

\vskip 0mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/07.png}}
We care for ``homomorphic'' because using tangles and tangle operations we can define interesting classes of knots, and thus invariants that are homomorphic with respect to these operations may be able to tell us something about these classes. See \web{AKT}.

\vskip 0mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/08.png}}
But enough with philosophy! I learned from Vaughan Jones that theories change with time, yet formulas stay. So let's start with formulas!

\vskip 5mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/09.png}}
To compute our knot invariant $\rho_1$, we cut it to a long knot and place it in the plane so that at all vertices, all edges are ``flowing up''. We then label each edge with serial number and with its rotation number $\varphi_k$.

\vskip 15mm
\parpic[r]{\includegraphics[width=3.5in]{Slides/10.png}}
We make a $(2n+1)\times(2n+1)$ matrix $A$ by starting with the identity matrix and adding a $2\times 2$ block for each crossing, as shown on the right. We let $G=(g_{\alpha\beta})$ be the inverse of $A$.

\vskip 17mm
\needspace{49mm} % 48mm is not enough.
\parpic[r]{\includegraphics[width=2.5in]{Slides/11.png}}
If we start from the trefoil knot diagram displayed before, the resulting $A$ is shown on the right.

\vskip 60mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/12.png}}
And now the corresponding $G$, the ``Green Function'', is shown.

\vskip 20mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/13.png}}
We noted that $\det(A)$ is (up to a normalization) the good old Alexander polynomial. If you are a classical topologist, you should yawn and perhaps fall asleep right now, for so far everything is very old material.

\vskip 0mm
\parpic[r]{\includegraphics[width=0.8in]{Slides/14.png}}
The $2\times 2$ matrices are the Burau matrices. The matrix $A$ is a presentation matrix of the Alexander module, derived by applying Fox calculus to the Wirtinger presentation. Even $G$ is not a great surprise; it is related to the ``Blanchfield Pairing''. All of these people are old timers, so much so that their pictures are in black and white.

\vskip 43mm
\parpic[r]{\includegraphics[width=3.5in]{Slides/15.png}}
All the news is in just one slide, the one on the right! We defined $R_1(c)$ and $\rho_1$, explained why $\rho_1$ is easy to compute (as easy as the Alexander polynomial), and asserted that it is invariant (to be proven below). If you are a classical topologist, these formulas should come as a complete surprise to you.

\vskip 0mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/16.png}}
These days I take what I learned from Vaughan Jones a step further. I care for programs even more than I care for formulas.

\vskip 3mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/17.png}}
We load some libraries that play a mild role: just tables of knots, and some older invariants for comparison, and a program to compute rotation numbers (something we could have done by hand).

\vskip 15mm
\parpic[r]{\includegraphics[width=3in]{Slides/18.png}}
Next is the main part of the program. It is almost one-to-one the same as the formulas for $\rho_1$, and if there's ever a disagreement, the program is to be trusted better because it's been tested extensively. Note that the program outputs the ordered pair $Z=(\Delta,\rho_1)$, because $\Delta$ is computed anyway within the computation of $\rho_1$, and we consider it as a part of $\rho_1$.

\vskip 28mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/19.png}}
We run the program on all knots with up to $6$ crossings.

\vskip 42mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/20.png}}
The program is fast! Here is the GST48 knot once again,\ldots

\vskip 33mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/21.png}}
and it takes only about 170 seconds to compute its $\rho_1$.

\vskip 55mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/22.png}}
$Z=(\Delta,\rho_1)$ is strong! It seems that it is stronger than HOMFLY-PT and Khovanov homology taken together.

\vskip 38mm
\parpic[r]{\includegraphics[width=3.5in]{Slides/23.png}}
On to interpretations, we discussed the traffic rules for cars on a knot diagram. All car crashes we discuss are gentle and no harm is ever caused to the occupants of our cars.

\vskip 12mm
\parpic[r]{\includegraphics[width=2.1in]{Slides/24.png}}
A relevant scene with Lightning McQueen, enacted by Roland's kids.

\vskip 20mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/25.png}}
We claim that the matrix $G$ of before is the traffic matrix for a knot diagram. Yet first we illustrate the traffic matrix using a very simple knot diagram (a single kink), and some simple-minded geometric summation.

\vskip 2mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/26.png}}
We then used ``$g$-rules'' to prove the claim. These are rules that tell us how to move the traffic injection sites and the traffic counting sites, and they will also be useful below, within the actual proof of invariance.

\vskip 3mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/27.png}}
On to the invariance under the hardest of the Reidemeister moves, Reid3. We first establish that traffic away from the Reid3 site is not affected by the move. This is essentially the invariance of the Burau representation.

\vskip 4mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/28.png}}
It follows that we only need to understand the contribution of the $R_1(c)$ terms from the crossings within the Reid3 area.

\vskip 8mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/29.png}}
We could have done it by hand, but we are lazy and we have good computer skills. So we type in the $g$-rules, the three $R_1$ contributions for the left hand side of Reid3 and the three $R_1$ contributions for the right hand side. We then apply the $g$-rules to move the traffic injection sites and the traffic counting sites to outside of the Reid3-move area, to where they are unchanged by the move. Comparing \verb$lhs$ with \verb$rhs$, the computer says \verb$True$, which means that $\rho_1$ is invariant under Reid3.

\vskip 4mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/30.png}}
As a second example we verify invariance under Reid1. Most of the work had already been done, because we computed already the ``traffic matrix'' of a kink. What remains is a little calculation (without forgetting the rotation-number correction!). We do that calculation on the right, using a hybrid of computer and human power (very little of each). A few further moves need to be shown --- they are discussed at \web{APAI}. This concludes the invariance proof for $\rho_1$.

\vskip 0mm
\parpic[r]{\includegraphics[width=4in]{Slides/31.png}}
This slide ought to be shown bigger. Wearing my topology hat, I genuinely, honestly, don't know what's going on.

\vskip 0mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/32.png}}
Unfortunately, at this point we had to rush towards the end, and be brief. Wearing my quantum algebra hat, the first thing to note is that there is a whiff of a Heisenberg relation in car traffic --- a difference of 1 between the traffic counting before and after the place where traffic is injected, and that may remind us of the Heisenberg commutation relation, $[p,x]=1$.

I have gone through the remaining few slides way too quickly. Here I will let them speak for themselves. The main things to learn from these reproduced slides are the references cited in them, and the comments in red.

\setlength{\columnseprule}{1pt}
\begin{multicols}{2}

{\bf\red Where did it come from?} Consider $\frakg_\eps\coloneqq
sl_{2+}^\eps\coloneqq L\langle y,b,a,x\rangle$ with relations
\[ [b,x]=\eps x, \quad [b,y]=-\eps y, \quad [b,a]=0, \]
\[ [a,x]=x, \quad [a,y]=-y, \quad [x,y]=b+\eps a. \]
At invertible $\eps$, it is isomorphic to
$sl_2$ plus a central factor, and it can be quantized \`a la
Drinfel'd~\cite{Drinfeld:QuantumGroups} much like $sl_2$ to get an
algebra $QU=A\langle y,b,a,x\rangle$ subject to (with $q=\bbe^{\hbar\eps})$:
\[ [b,a]=0, \quad [b,x]=\eps x,\quad [b,y]=-\eps y, \]
\[ [a,x]=x, \quad [a,y]=-y, \]
\[ xy-qyx=\frac{1-\bbe^{-\hbar(b+\eps a)}}{\hbar}. \]
Now $QU$ has an $R$-matrix solving Yang-Baxter (meaning Reid3),
\begin{multline*}
  R=\sum_{m,n\geq 0}\frac{y^nb^m\otimes (\hbar a)^m(\hbar x)^n}{m![n]_q!}, \\
  ([n]_q!\text{ is a ``quantum factorial''})
\end{multline*}
and so it has an associated ``universal quantum invariant'' \`a la
\text{Lawrence} and Ohtsuki~\cite{Lawrence:UniversalUsingQG,
Ohtsuki:QuantumInvariants}, $Z_\eps(K)\in QU$.

Now $QU\cong\calU(\frakg_\eps)$ (only as algebras!) and
$\calU(\frakg_\eps)$ represents into $\bbH$ via
\[ y\to -tp-\eps\cdot xp^2,\quad b\to t+\eps\cdot xp, \]
\[ a\to xp,\quad x\to x, \]
(abstractly, $\frakg_\eps$ acts on its Verma module
\[ \calU(\frakg_\eps)/(\calU(\frakg_\eps)\langle y,a,b-\eps a-t\rangle)\cong\bbQ[x] \]
by differential operators, namely via $\bbH$),
so $R$ can be pushed to $\calR\in\bbH\otimes\bbH$.

Everything still makes sense at $\eps=0$ and can be expanded near
$\eps=0$ resulting with $\calR=\calR_0(1+\eps \calR_1+\cdots)$, with
$\calR_0=\bbe^{t(xp\otimes 1-x\otimes p)}$ and $\calR_1$ a quartic
polynomial in $p$ and $x$. So $p$'s and $x$'s get created along
$K$ and need to be pushed around to a standard location (``normal
ordering''). This is done using
\begin{align*}
  (p\otimes 1)\calR_0 &= \calR_0(T(p\otimes 1)+(1-T)(1\otimes p)), \\
  (1\otimes p)\calR_0 &= \calR_0(1\otimes p),
\end{align*}
and when the dust settles, we get our formulas for $\rho_1$. But $QU$
is a quasi-triangular Hopf algebra, and hence {\red\bf $\rho_1$ is
homomorphic}. Read more at~\cite{PP1, PG} and hear more at \web{SolvApp},
\web{Dogma}, \web{DoPeGDO}, \web{FDA}, \web{AQDW}.

\parpic[r]{\parbox{0.65in}{
  \includegraphics[width=0.6in]{../../Projects/Gallery/Schaveling.jpg}
  \newline\footnotesize Schaveling
}}
Also, we can (and know how to) look at higher powers of $\eps$ and we can
(and more or less know how to) replace $sl_2$ by arbitrary semi-simple
Lie algebra (e.g.,~\cite{Schaveling:Thesis}). So {\red\bf $\rho_1$
is not alone!}

These constructions are very similar to Rozansky-Overbay \cite{Ro,
Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis} and hence to the ``loop
expansion'' of the Kontsevich integral and the coloured Jones polynomial~\cite{Ohtsuki:TwoLoop}.

\end{multicols}

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\vskip 0mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/39.png}}
We re-iterated that an invariant as simple as $\rho_1$ must have a simple explanation, hopefully, within topology. Our current understanding of $\rho_1$ within quantum algebra is simply way too complicated.

\vskip 0mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/40.png}}
We also remind that in some sense, $\rho_1$ is a ``friend'' of the Alexander polynomial $\Delta$, and that $\Delta$ is perhaps the most topologically-meaningful knot invariant. Like $\Delta$, $\rho_1$ gives a genus bound. Does it also give a ribbon criteria like the Fox-Milnor condition for $\Delta$?

\vskip 0mm
\parpic[r]{\includegraphics[width=2.5in]{Slides/41.png}}
At the end, we merely flashed our theorem regarding $\rho_d$, which generalizes $\rho_1$ when $d\geq 1$, and our implementation thereof. For $d\geq 2$, $\rho_d$ is more complicated than $\rho_1$, yet it retains some things in common with $\rho_1$: Once more the key is the matrix $G=(g_{\alpha\beta})$. To compute $\rho_1$ we carry out a 1-fold summation over the features of the knot (crossings and rotations), of polynomials of degree $\leq 2$ in the $g_{\alpha\beta}$'s. To compute $\rho_d$ we carry out a $d$-fold summation over the features of the knot, of polynomials of degree $\leq 2d$ in the $g_{\alpha\beta}$'s. Multiple summations are of course more costly than single summations, yet the computation of $\rho_2$ remains of polynomial time and for small $d$ it is completely practical.

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\vskip 20mm
\begin{thebibliography}{LTW}

\bibitem[BV1]{PP1} D.~Bar-Natan and R.~van~der~Veen,
  {\em A Polynomial Time Knot\newline Polynomial,}
  Proc.\ Amer.\ Math.\ Soc.\ {\bf 147} (2019) 377--397, \arXiv{1708.04853}.

\bibitem[BV2]{PG} D.~Bar-Natan and R.~van~der~Veen,
  {\em Perturbed Gaussian Generating Functions for Universal Knot Invariants,}
  \arXiv{2109.02057}.

\bibitem[Dr]{Drinfeld:QuantumGroups} V.~G.~Drinfel'd,
  {\em Quantum Groups,}
  Proc.\ Int.\ Cong.\ Math., 798--820, Berkeley, 1986.

\bibitem[GST]{GompfScharlemannThompson:Counterexample} R.~E.~Gompf,
  M.~Scharlemann, and A.~Thompson,
  {\em Fibered Knots and Potential Counterexamples to the Property 2R and
    Slice-Ribbon Conjectures,}
  Geom.\ and Top.\ {\bf 14} (2010) 2305--2347, \arXiv{1103.1601}.

\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[La]{Lawrence:UniversalUsingQG} R.~J.~Lawrence,
  {\em Universal Link Invariants using Quantum Groups,}
  Proc.\ XVII Int.\ Conf.\ on Diff.\ Geom.\ Methods in Theor.\ Phys., Chester,
  England, August 1988. World Scientific (1989) 55--63.

\bibitem[LTW]{LinTianWang:RandomWalk} X-S.~Lin, F.~Tian, and 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[Oh1]{Ohtsuki:QuantumInvariants} T.~Ohtsuki,
  {\em Quantum Invariants,}
  Series on Knots and Everything {\bf 29}, World Scientific 2002.

\bibitem[Oh2]{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,
  {\em Perturbative Expansion of the Colored Jones Polynomial,}
  Ph.D.\ thesis, University of North Carolina, August 2013, \web{Ov}.

\bibitem[Pi]{Piccirillo:NotSlice} L.~Piccirillo, 
  {\em The Conway knot is not slice,}
  Ann.\ of Math.\ (2) {\bf 191-2} (2020) 581--591, \arXiv{1808.02923}.

\bibitem[Ro1]{Ro} L.~Rozansky,
  {\em A Contribution of the Trivial Flat Connection to the Jones
  Polynomial and Witten's Invariant of 3D Manifolds, I,}
  Comm.\ Math.\ Phys.\ {\bf 175-2} (1996) 275--296, \arXiv{hep-th/9401061}.

\bibitem[Ro2]{Rozansky:Burau} L.~Rozansky,
  {\em The Universal $R$-Matrix, Burau Representation and the Melvin-Morton
    Expansion of the Colored Jones Polynomial,}
  Adv.\ Math.\ {\bf 134-1} (1998) 1--31, \arXiv{q-alg/9604005}.

\bibitem[Ro3]{Rozansky:U1RCC} L.~Rozansky,
  {\em A Universal $U(1)$-RCC Invariant of Links and Rationality Conjecture,}
  \arXiv{math/0201139}.

\bibitem[Sch]{Schaveling:Thesis} S.~Schaveling,
  {\em Expansions of Quantum Group Invariants,}
  Ph.D.\ thesis, Universiteit Leiden, September 2020, \web{Scha}.

\end{thebibliography}

\AtEndDocument{\includepdf[pages={-}]{Cars.pdf}}

\end{document}
\endinput

