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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{Greece-1607}
\def\title{The Brute and the Hidden Paradise}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
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\def\ad{\operatorname{ad}}
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\def\calG{{\mathcal G}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}
\def\calO{{\mathcal O}}
\def\calP{{\mathcal P}}
\def\calS{{\mathcal S}}
\def\calT{{\mathcal T}}
\def\calU{{\mathcal U}}
\def\frakg{{\mathfrak g}}
\def\tilE{\tilde{E}}

%%%

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Abstract.} There is expected to be a hidden paradise of poly-time 
computable knot polynomials lying just beyond the Alexander polynomial. I
will describe my brute attempts to gain entry.
}}}}

\def\Expected{{\raisebox{4.5mm}{\begin{minipage}[t]{3.95in}
\parpic[r]{$\displaystyle v_{d,f}(K)=\sum_{Y\subset X(K),\ |Y|=d}f(Y)$}
\picskip{2}
{\red Why ``expected''?} Gauss diagram formulas~\cite{PV,GPV} show that
finite-type invariants are all poly-time, and tempt to conjecture that
there are no others. But Alexander shows it nonsense:

\vspace{-2mm}

\definecolor{lightgreen}{RGB}{127,255,127}
\definecolor{lightergreen}{RGB}{191,255,191}
\definecolor{lightestgreen}{RGB}{223,255,223}
\begin{center}\begin{tabular}{r|ccc>{\columncolor{lightgreen}}c>{\columncolor{lightergreen}}c>{\columncolor{lightestgreen}}cccc}
  $d$ &                             2 & 3 & 4 &         5 & 6 & 7 & 8  &
$\cdots$ \\
  \hline
  \it known invts$^\ast$ in $O(n^d)$ &  1 & 1 & $\infty$ &  3 & 4 & 8 & 11
& $\cdots$
\end{tabular}\end{center}
\vskip -3mm

This is an unreasonable picture!
\hfill$^\ast${\footnotesize\it Fresh, numerical, no cheating.}

So there ought to be further poly-time invariants.

\parshape 3 0in 3.6in 0in 3.6in 0in 3.95in
{\red Also.} $\bullet$ The line above the Alexander line in the
Melvin-Morton \cite{MM,Ro} expansion of the coloured Jones polynomial.
$\bullet$ The 2-loop contribution to the Kontsevich integral.
\end{minipage}}}}

\def\Paradise{{\raisebox{2mm}{\begin{minipage}[t]{3.95in}
{\red Why ``paradise''?} Foremost answer: {\red\sl OBVIOUSLY.}
{\footnotesize Cf.\ proving (incomputable $A$)$=$(incomputable $B$), or
categorifying (incomputable $C$).}
\parbox[t]{0.8in}{\raggedright
  \web{K17}:
  \newline{\footnotesize (extend to tangles, perhaps detect non-slice
ribbon knots)
  }
  \vskip 5mm
  {\red Moral.} Need ``stitching'':
}\hfill
\raisebox{\dimexpr\baselineskip-\height}{
  \includegraphics[width=\dimexpr\linewidth-1in]{../../Projects/Killam-2017/AKT-1607.pdf}
}
\end{minipage}}}}

\def\Brute{{\raisebox{2mm}{\begin{minipage}[t]{3.125in}
{\red Why ``brute''?} Cause it's the only thing I know, for now. There may
be better ways in, and it's fair to hope that sooner or later they will be
found.
\end{minipage}}}}

\def\Gold{{\raisebox{5mm}{\begin{minipage}[t]{3.95in}
\pichskip{0mm}\parpic[l]{
  \includegraphics[height=0.5in]{IASLogo.png}
} \picskip{3}
{\red The Gold Standard} is set by the formulas
\cite{Bar-NatanSelmani:MetaMonoids, KBH} for Alexander. An $S$-component
tangle $T$ has $\Gamma(T) \in R_S\times M_{S\times S}(R_S) =
\left\{\begin{array}{c|c}\omega&S\\\hline S&A\end{array}\right\}$ with
$R_S\coloneqq\bbZ(\{t_a\colon a\in S\})$:

$\displaystyle
  \left(\tensor[_a]{\overcrossing}{_b},\tensor[_b]{\undercrossing}{_a}\right)
  \to
  \begin{array}{c|cc} 1 & a & b \\ \hline a & 1 & 1-t_a^{\pm 1} \\ b & 0 &
t_a^{\pm 1} \end{array}
$
\hfill$\displaystyle
  T_1\sqcup T_2
  \to
  \begin{array}{c|cc} \omega_1\omega_2 & S_1 & S_2 \\ \hline S_1 & A_1 & 0
\\ S_2 & 0 & A_2 \end{array}
$
\vskip -1mm
\[ \begin{CD}
  \begin{array}{c|ccc}
    \omega & a & b & S \\
    \hline
    a & \alpha & \beta & \theta \\
    b & \gamma & \delta & \epsilon \\
    S & \phi & \psi & \Xi
  \end{array}
%  @>{\displaystyle m^{ab}_c}>{\displaystyle t_a,t_b\to t_c\atop\displaystyle \mu\coloneqq 1-\beta}>
  @>{\displaystyle m^{ab}_c}>{\displaystyle t_a,t_b\to t_c}>
  \left(\!\begin{array}{c|cc}
%    \mu\omega & c & S \\
%    \hline
%    c & \gamma+\alpha\delta/\mu & \epsilon+\delta\theta/\mu \\
%    S & \phi+\alpha\psi/\mu & \Xi+\psi\theta/\mu
    (1-\beta)\omega & c & S \\
    \hline
    c & \gamma+\frac{\alpha\delta}{1-\beta} & \epsilon+\frac{\delta\theta}{1-\beta} \\
    S & \phi+\frac{\alpha\psi}{1-\beta} & \Xi+\frac{\psi\theta}{1-\beta}
  \end{array}\!\right)
\end{CD} \]
\end{minipage}}}}

\def\Dunfield{{\raisebox{0mm}{\parbox[t]{2.85in}{
For long knots, $\omega$ is Alexander, and that's the fastest Alexander
algorithm I know!
\newline\null\hfill\text{\footnotesize Dunfield: 1000-crossing fast.}
}}}}

\def\EKTheorem{{\raisebox{2mm}{\begin{minipage}[t]{3.95in}
{\red Theorem} \cite{EtingofKazhdan:BialgebrasI,
Haviv:DiagrammaticAnalogue, Enriquez:Quantization,
Severa:BialgebrasRevisited}. There is a ``homomorphic expansion''
\[ \arraycolsep=0pt \renewcommand{\arraystretch}{1}
  \begin{array}{c}
    Z\colon\left\{\parbox{0.85in}{$S$-component ($v/b$-)tangles}\right\}
    \to \calA^v_S \coloneqq \\
  \parbox{0.45\linewidth}{\vskip 2mm
  \includegraphics[height=0.4in]{../../Projects/Gallery/Etingof.jpg}%
  \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Kazhdan.jpg}%
  \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Haviv.jpg}%
  \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Enriquez.jpg}%
  \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Severa.jpg}
  \hfill} \\
  \parbox{0.45\linewidth}{\tiny Etingof \hfill Kazhdan \hfill Haviv \hfill
Enriquez \hfill \v{S}evera \hfill}
  \end{array}
  \begin{array}{c}\resizebox{0.55\linewidth}{!}{\subimport{TeXAttempt/}{Av.pdf_t}}\end{array}
\]
(it is enough to know $Z$ on $\overcrossing$ and have disjoint union and
stitching formulas)\hfill\text{\red\ldots exponential and too hard!}

{\red Idea.} Look for ``ideal'' quotients of $\calA^v_S$ that have
poly-sized descriptions; \hfill\text{\ldots specifically, limit the
co-brackets.}
\end{minipage}}}}

\def\OneCoTwoCo{{\raisebox{2mm}{\parbox[t]{1.6in}{
{\red 1-co and 2-co,} aka $\TC$ and $\TC^2$, on the right. The primitives that
remain are:
}}}}

\def\TwoD{{\raisebox{2mm}{\parbox[t]{3.1in}{
{\red The $2D$ relations} come from the relation with 2D Lie bialgebras:
}}}}

\def\TwoTwoDefs{{\raisebox{0mm}{\parbox[t]{3.95in}{
We let $\calA^{2,2}$ be $\calA^v$ modulo 2-co and $2D$, and $z^{2,2}$ be
the projection of $\log Z$ to $\calP^{2,2}\coloneqq\pi\calP^v$, where
$\calP^v$ are the primitives of $\calA^v$.

{\red Main Claim.} $z^{2,2}$ is poly-time computable.
}}}}

\def\MainPoint{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Main Point.} $\calP^{2,2}$ is poly-size, so how hard can it be?
Indeed, as a module over $\bbQ\llbracket b_i\rrbracket$, $\calP^{2,2}$ is at
most
\vskip 23mm
{\red Claim.} $R_{jk}=e^{a_{jk}}e^{\rho_{jk}}$ is a solution of the
Yang-Baxter / R3 equation $R_{12}R_{13}R_{23}= R_{23}R_{13}R_{12}$ in
$\exp\calP^{2,2}$, with $\rho_{jk} \coloneqq$
\[
  \psi(b_j)\left(-c_k + \frac{c_ka_{jk}}{b_j} - \frac{\delta
    a_{jk}a_{jk}}{b_j^2}\right)
  + \frac{\phi(b_j)\psi(b_k)}{b_k\phi(b_k)}\left(c_ka_{kk} - \frac{\delta
    a_{jk}a_{kk}}{b_j}\right),
\]
and with $\phi(x)\coloneqq e^{-x}-1 = -x+x^2/2-\dots$, and
$\psi(x)\coloneqq\left((x+2)e^{-x}-2+x\right)/(2x) = x^2/12-x^3/24+\dots$.
(This already gives some new (v-)braid group representations, as below).
}}}}

\def\SnG{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Problem.} How do we multiply in $\exp(\calP^{2,2})$? How do we
stitch? BCH is a theoretical dream. Instead, use ``scatter and glow'' and
``feedback loops'':

\vskip 2mm
\parshape 1 0in 1.75in \footnotesize
The Euler trick:\newline 
With $Ef\coloneqq(\deg f)f$ get $Ee^x=xe^x$ and
$E(e^xe^ye^z)=$\newline
\null\hfill $xe^xe^ye^z + e^xye^ye^z + e^xe^yze^z.$
}}}}

\def\LocalAlgebra{{\raisebox{2mm}{\parbox[t]{3.375in}{
{\red Local Algebra} (with van der Veen) Much can be reformulated as
(non-standard) ``quantum algebra'' for the 4D Lie algebra $\frakg=\langle
b,c,u,w\rangle$ over $\bbQ[\epsilon]/(\epsilon^2=0)$, with $b$
central and $[w,c]=w$, $[c,u]=u$, and $[u,w]=b-2\epsilon c$. The key:
$a_{ij}=(b_i-\epsilon c_i)c_j+u_iw_j$ in $\calU(\frakg)^{\otimes\{i,j\}}$.
}}}}

\def\cellscale{0.625}

\def\NewRepsA{{\raisebox{2mm}{\parbox[t]{4in}{
{\red Some (new) representationss of the (v-)braid groups.}\hfill\text{\web{Reps}}

\hfill{\red Burau (old)}
\vskip -4mm\includegraphics[scale=\cellscale]{Snips/Burau.pdf}

\hfill{\red \ldots testing R3}
\vskip -4mm\includegraphics[scale=\cellscale]{Snips/BurauR3.pdf}

\hfill{\red Gassner (old)}
\vskip -4mm\includegraphics[scale=\cellscale]{Snips/Gassner.pdf}

\hfill{\red \ldots Overcrossings Commute (OC):}
\includegraphics[scale=\cellscale]{Snips/GassnerOC.pdf}

\hfill{\red \ldots Undercrossings Commute (UC):}
\includegraphics[scale=\cellscale]{Snips/GassnerUC.pdf}

\hfill{\red Gassner Plus (new?)}
\includegraphics[scale=\cellscale]{Snips/GassnerPlus.pdf}

\hfill{\red \ldots R3 (left)}
\vskip -4mm\includegraphics[scale=\cellscale]{Snips/GPR3l.pdf}

\hfill{\red \ldots R3 (rest)}
\vskip -4mm\includegraphics[scale=\cellscale]{Snips/GPR3.pdf}

\hfill{\red \ldots OC}
\vskip -4mm\includegraphics[scale=\cellscale]{Snips/GPOC.pdf}

{\bf Question.} Does Gassner Plus factor through Gassner?

\vskip 3mm
\hfill{\red Turbo-Gassner (new!)}
\vskip -4mm\includegraphics[scale=\cellscale]{Snips/TG.pdf}

Satisfies R3\ldots

}}}}

\def\NewRepsB{{\raisebox{0mm}{\parbox[t]{4in}{

\hfill{\red \ldots OC}
\vskip -4mm\includegraphics[scale=\cellscale]{Snips/TGOC.pdf}

\hfill{\red Turbo-Burau (new!)}
\vskip -4mm\includegraphics[scale=\cellscale]{Snips/TB.pdf}

\hfill{\red \ldots OC}
\vskip -4mm\includegraphics[scale=\cellscale]{Snips/TBOC.pdf}

}}}}

\def\FlowerSurgery{{\raisebox{2mm}{\parbox[t]{1.2in}{\small
{\red Flower Surgery Theorem.} A knot is ribbon iff it is the result
of $n$-petal flower surgery (from thin petals to wide petals) on an
$n$-componenet unlink, for some $n$.
}}}}

\def\Help{{\raisebox{2mm}{\parbox[t]{2.4in}{\small
{\red Help Needed!} Disorganized videos of talks in a private
seminar are at \web{PP}.
}}}}

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

\bibitem[BN]{KBH} D.~Bar-Natan,
  {\em Balloons and Hoops and their Universal Finite Type Invariant, BF
    Theory, and an Ultimate Alexander Invariant,}
  \web{KBH}, \arXiv{1308.1721}.

\bibitem[BND]{WKO} D.~Bar-Natan and Z.~Dancso,
  {\em Finite Type Invariants of W-Knotted Objects I, II, IV,}
  \web{WKO1}, \web{WKO2}, \web{WKO4}, \arXiv{1405.1956}, \arXiv{1405.1955},
  \arXiv{1511.05624}.

\bibitem[BNG]{Bar-NatanGaroufalidis:MMR} D.~Bar-Natan and S.~Garoufalidis,
  {\em On the Melvin-Morton-Rozansky conjecture,}
  Invent.\ Math.\ {\bf 125} (1996) 103--133.

\bibitem[BNS]{Bar-NatanSelmani:MetaMonoids} D.~Bar-Natan and S.~Selmani,
  {\em Meta-Monoids, Meta-Bicrossed Products, and the Alexander
    Polynomial,}
  J.\ of Knot Theory and its Ramifications {\bf 22-10} (2013),
  \arXiv{1302.5689}.

%\bibitem[CT]{CimasoniTuraev:LagrangianRepresentation} D.~Cimasoni and
%V.~Turaev,
%  {\em A Lagrangian Representation of Tangles,}
%  Topology {\bf 44} (2005) 747--767, \arXiv{math.GT/0406269}.

\bibitem[En]{Enriquez:Quantization} B.~Enriquez,
  {\em A Cohomological Construction of Quantization Functors of Lie
    Bialgebras,}
  Adv.\ in Math.\ {\bf 197-2} (2005) 430–-479,
  \arXiv{math/0212325}.

\bibitem[EK]{EtingofKazhdan:BialgebrasI} P.~Etingof and D.~Kazhdan,
  {\em Quantization of Lie Bialgebras, I,}
  Selecta Mathematica {\bf 2} (1996) 1--41, \arXiv{q-alg/9506005}.

%\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[GPV]{GPV} M.~Goussarov, M.~Polyak, and O.~Viro,
  {\em Finite type invariants of classical and virtual knots,}
  Topology {\bf 39} (2000) 1045--1068, \arXiv{math.GT/9810073}.

\bibitem[Ha]{Haviv:DiagrammaticAnalogue} A.~Haviv,
  {\em Towards a diagrammatic analogue of the Reshetikhin-Turaev link
invariants,}
  Hebrew University PhD thesis, Sep.\ 2002, \arXiv{math.QA/0211031}.

%\bibitem[KLW]{KirkLivingstonWang:Gassner} P.~Kirk, C.~Livingston, and Z.~Wang,
%  {\em The Gassner Representation for String Links,}
%  Comm.\ Cont.\ Math.\ {\bf 3} (2001) 87--136, \arXiv{math/9806035}.

%\bibitem[LD]{LeDimet:Gassner} J.~Y.~Le Dimet,
%  {\em Enlacements d'Intervalles et Repr\'esentation de Gassner,}
%  Comment.\ Math.\ Helv.\ {\bf 67} (1992) 306--315.

\bibitem[MM]{MM} P.~M.~Melvin and H.~R.~Morton,
  {\em The coloured Jones function,}
  Commun.\ Math.\ Phys.\ {\bf 169} (1995) 501--520.

\bibitem[PV]{PV} M.~Polyak and O.~Viro,
  {\em  Gauss Diagram Formulas for Vassiliev Invariants,}
  Inter.\ Math.\ Res.\ Notices {\bf 11} (1994) 445--453.

\bibitem[Ro]{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[Se]{Severa:BialgebrasRevisited} P.~\v{S}evera,
  {\em Quantization of Lie Bialgebras Revisited,}
  Sel.\ Math., NS, to appear, \arXiv{1401.6164}.

\end{thebibliography}
}}}}

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