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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{Nara-2308}
\def\title{Cars, Interchanges, Traffic Counters, and some Pretty Darned Good Knot Invariants}

\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 inviting me to Nara!}}
\def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/na23}{http://drorbn.net/na23}}}
\def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}}
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\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 5 0in 2.1in 0in 2.1in 0in 2.1in 0in 2.1in 0in 3.95in
{\red\bf Abstract.} 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.
\hfill\text{{\red\bf Common misconception.} ``Dominated'' $\nRightarrow$ ``lesser''.}
}}}}

\def\lr{$\leftrightarrow$}

\def\WeSeek{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf We seek} strong, fast, homomorphic knot and tangle invariants.
\par {\bf Strong.} Having a small ``kernel''.
\par {\bf Fast.} Computable even for large knots (best: poly time).
}}}}

\def\Homomorphic{{\raisebox{0mm}{\parbox[t]{1.95in}{
{\bf Homomorphic.} Extends to tangles and behaves under tangle operations; especially gluings and
doublings:
}}}}

\def\WhyHomomorphic{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Why care for ``Homomorphic''?} {\bf Theorem.} A knot $K$ is
{\em ribbon} iff there exists a $2n$-component tangle $T$ with skeleton
as below such that $\tau(T)=K$ and  where $\delta(T)=U$ is the {\em
untangle}:
}}}}

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

\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\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\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
\[ \Delta = T^{(-\varphi-w)/2}\det(A), \qquad
  \text{with }\varphi = \sum_k \varphi_k,\ w \!=\! \sum_c s.
\]
{\red\bf Classical Topologists:} This is boring. Yawn.
}}}}

\def\FormulasD{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\bf\red Formulas, continued.} Finally, set
\[ R_1(c) \coloneqq s \left(
  g_{ji} \left(g_{j+1,j}+g_{j,j+1}-g_{ij}\right)
  -g_{ii} \left(g_{j,j+1}-1\right)
  -1/2
\right) \]
\[ \rho_1 \coloneqq \Delta^2\left(\sum_c R_1(c) - \sum_k\varphi_k\left(g_{kk}-1/2\right)\right). \]
In our example $\rho_1 = -T^2 +2T - 2 + 2T^{-1} - T^{-2}$.
\par{\red\bf Theorem.} $\rho_1$ is a knot invariant.\hfill Proof: later.
\par{\red\bf Classical Topologists:} Whiskey Tango Foxtrot?
}}}}

\def\Interpretation{{\raisebox{2mm}{\parbox[t]{4.95in}{
\parshape 5 0in 2.3in 0in 2.3in 0in 2.3in 0in 2.3in 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 $\rho_1$.} 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 $R_1$ 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{../Oaxaca-2210}{Invariance.tex}
}}}}

\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\ip{{$i^+$}} \def\jp{{$j^+$}} \def\kp{{$k^+$}}
\def\ipp{{$i^{+\!+}$}} \def\jpp{{$j^{+\!+}$}} \def\kpp{{$k^{+\!+}$}}

\def\TopologyHat{{\raisebox{2mm}{\parbox[t]{3.2in}{
{\bf\red Wearing my Topology hat} the formula for $R_1$, and even the
idea to look for $R_1$, remain a complete mystery to me.
}}}}

\def\QuantumAlgebraHat{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 4 0in 3.45in 0in 3.45in 0in 3.45in 0in \linewidth
{\bf\red Wearing my Quantum Algebra hat,} I spy a Heisenberg algebra
$\bbH=A\langle p,x\rangle/([p,x]=1)$:
\[ \text{cars}\leftrightarrow p \qquad \text{traffic counters}\leftrightarrow x \]

{\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 $q=\bbe^{\hbar\eps})$:
\[ [b,a]=0, \quad [b,x]=\eps x,\quad [b,y]=-\eps y, \]
\[ [a,x]=x, \quad [a,y]=-y, \quad xy-qyx=\frac{1-\bbe^{-\hbar(b+\eps a)}}{\hbar}. \]
Now $QU$ has an $R$-matrix solving Yang-Baxter (meaning Reid3),
\[ R=\sum_{m,n\geq 0}\frac{y^nb^m\otimes (\hbar a)^m(\hbar x)^n}{m![n]_q!},
  \quad([n]_q!\text{ is a ``quantum factorial''})
\]
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
\[ y\to -tp-\eps\cdot xp^2,\quad b\to t+\eps\cdot xp,\quad a\to xp,\quad x\to x, \]
(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}.

\parshape 1 0in 3.25in
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}.
}}}}

\def\Insanity{{\raisebox{0mm}{\parbox[t]{3.95in}{
If this all reads like {\bf\red insanity} to you, it should (and
you haven't seen half of it). Simple things should have simple
explanations.

Hence, {\bf\red Homework.} Explain $\rho_1$ with no reference to
quantum voodoo and find it a topology home (large enough to house
generalizations!). Make explicit the homomorphic properties of
$\rho_1$. Use them to do topology!

{\bf\red P.S.} As a friend of $\Delta$, $\rho_1$ gives a genus bound, sometimes better than $\Delta$'s. How
much further does this friendship extend?
}}}}

\def\Acknowledgement{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\bf\red Acknowledgement.} This work was supported by NSERC
grant RGPIN-2018-04350 and by the Chu Family Foundation (NYC).
}}}}

\def\refs{{\raisebox{6mm}{\parbox[t]{3.95in}{
%{\red\bf References.}
{\scriptsize
%\par\vspace{-3mm}
\renewcommand{\section}[2]{}%
\begin{thebibliography}{}
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.3ex}

\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[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[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}.

\parshape 1 11pt 3.0in

\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}}
}}}}

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\begin{multicols}{2}

{\bf\red A Small-Print Page on $\rho_d$, $d>1$.}

{\bf\red Definition.} $\langle f(z_i),h(\zeta_i)\rangle_{\{z_i\}} \coloneqq
\left.f(\partial_{\zeta_i})h\right|_{\zeta_i=0}$, so
$\langle p^2x^2,\bbe^{g\pi\xi}\rangle=2g^2$.

{\bf\red Baby Theorem.} There exist (non unique) power series
$r^\pm(p_1,p_2,x_1,x_2) = \sum_d \eps^d r^\pm_d(p_1,p_2,x_1,x_2) \in
\bbQ[T^{\pm 1},p_1,p_2,x_1,x_2]\llbracket\eps\rrbracket$ with $\deg
r^\pm_d\leq 2d+2$ (``docile'') such that the power series
$Z^b=\sum\rho^b_d\eps^d\coloneqq$
\[
  \left\langle \exp\left(\sum_c r^s(p_i,p_j,x_i,x_j)\right),
    \exp\left(\sum_{\alpha,\beta}g_{\alpha\beta}\pi_\alpha\xi_\beta\right)
  \right\rangle_{\{p_\alpha,x_\beta\}}
\]
is a bnot invariant. Beyond the once-and-for-all computation of $g_{\alpha\beta}$ (a matrix inversion),
$Z^b$ is computable in $O(n^d)$ operations in the ring $\bbQ[T^{\pm 1}]$.

(Bnots are knot diagrams modulo the braid-like Reidemeister moves, but not the cyclic ones).

{\bf\red Theorem.} There also exist docile
power series $\gamma^\varphi(\bar{p},\bar{x})
= \sum_d \eps^d \gamma^\varphi_d\in\bbQ[T^{\pm
1},\bar{p},\bar{x}]\llbracket\eps\rrbracket$ such that the power series
$Z=\sum\rho_d\eps^d\coloneqq$
\begin{multline*}
  \left\langle
    \exp\left(
      \sum_c r^s(p_i,p_j,x_i,x_j)
      + \sum_k \gamma^{\varphi_k}(\bar{p}_k,\bar{x}_k)
    \right),
  \right. \\
  \left.
    \exp\left(
      \sum_{\alpha,\beta}g_{\alpha\beta}(\pi_\alpha+\bar{\pi}_\alpha)(\xi_\beta+\bar{\xi}_\beta)
      + \sum_\alpha \pi_\alpha\bar{\xi}_\alpha
    \right)
  \right\rangle_{\{p_\alpha,\bar{p}_\alpha,x_\beta,\bar{x}_\beta\}}
\end{multline*}
is a knot invariant, as easily computable as $Z^b$.

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\input{Rhod.tex}

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