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

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  \newline\null\hfill Slides w/ no handout/URL should be banned!
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\begin{multicols*}{2} %\raggedcolumns

{\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.

\vskip -3mm\rule{\linewidth}{1pt}\vspace{-1mm}

\parpic[r]{$\displaystyle v_{d,f}(K)=\sum_{Y\subset X(K),\ |Y|=d}f(Y)$}
{\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:

\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}

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

So there ought to be further poly-time invariants.

{\red Also.} $\bullet$ The diagonal above the Alexander diagonal in the Melvin-Morton-Rozansky \cite{MM,Ro} of the coloured Jones polynomial. $\bullet$ The 2-loop contribution to the Kontsevich integral.

\vskip -3mm\rule{\linewidth}{1pt}\vspace{-1mm}

{\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'':
    \[ \fbox{\parbox[t][1.8in]{0.75in}{\tiny
      TODO: a schematic picture of stitching
    }} \]
  }
}\hfill
\raisebox{\dimexpr\baselineskip-\height}{
  \includegraphics[width=\dimexpr\linewidth-1in]{../../../Projects/Killam-2017/AKT-1607.pdf}
}

\rule{\linewidth}{1pt}\vspace{-2mm}

\parpic[r]{
  \includegraphics[height=0.56in]{../../NCSU-1604/AmplifiedMobilityPlatform.jpg}
}
{\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.

\vskip -3mm\rule{\linewidth}{1pt}\vspace{-1mm}

\pichskip{0mm}\parpic[l]{
  \includegraphics[height=0.5in]{../IASLogo.png}
}
{\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}
$

\[ \begin{CD}
  \begin{array}{c|ccc}
    \omega & a & b & S \\
    \hline
    a & \alpha & \beta & \theta \\
    b & \gamma & \delta & \epsilon \\
    S & \phi & \psi & \Xi
  \end{array}
  @>{\displaystyle m^{ab}_c}>{\displaystyle t_a,t_b\to t_c\atop\displaystyle \mu\coloneqq 1-\beta}>
  \left(\!\begin{array}{c|cc}
    \mu\omega & c & S \\
    \hline
    c & \gamma+\alpha\delta/\mu & \epsilon+\delta\theta/\mu \\
    S & \phi+\alpha\psi/\mu & \Xi+\psi\theta/\mu
  \end{array}\!\right)
\end{CD} \]
\parpic[r]{\ \includegraphics[height=0.4in]{../../../Projects/Gallery/Dunfield.png}}
For long knots, $\omega$ is Alexander, and that's the fastest \text{Alexander} algorithm I know!
\hfill{\footnotesize Dunfield: 1000-crossing fast.}

\vskip -2mm\rule{\linewidth}{1pt}\vspace{-1mm}

%\parpic[r]{\resizebox{0.6\linewidth}{!}{\input{Av.pdf_t}}}
{\red Theorem} \cite{EtingofKazhdan:BialgebrasI, Haviv:DiagrammaticAnalogue, Enriquez:Quantization, Severa:BialgebrasRevisited}. There is a ``homomorphic expansion''
\[ \arraycolsep=0pt \renewcommand{\arraystretch}{1}
  \begin{array}{c}
    Z\colon\left\{\parbox{0.8in}{$n$-component $v/b$-tangles}\right\}
    \to \calA^v_n \coloneqq \\
  \parbox{0.45\linewidth}{\vskip 2mm
  \includegraphics[height=0.4in]{../../../Projects/Gallery/Etingof.jpg}%
  \hfill\includegraphics[height=0.4in]{../../../Projects/Gallery/Kazhdan.jpg}%
  \hfill\includegraphics[height=0.4in]{../../../Projects/Gallery/Haviv.jpg}%
  \hfill\includegraphics[height=0.4in]{../../../Projects/Gallery/Enriquez.jpg}%
  \hfill\includegraphics[height=0.4in]{../../../Projects/Gallery/Severa.jpg}
  \hfill} \\
  \parbox{0.45\linewidth}{\tiny Etingof \hfill Kazhdan \hfill Haviv \hfill Enriquez \hfill \v{S}evera \hfill}
  \end{array}
  \begin{array}{c}\resizebox{0.55\linewidth}{!}{\input{Av.pdf_t}}\end{array}
\]
(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 $\calA^v_n$ that have poly-sized descriptions; \hfill\text{\ldots specifically, limit the co-brackets.}

\vskip -2mm\rule{\linewidth}{1pt}\vspace{-1mm}

\parpic[r]{\fbox{\parbox[t][0.4in]{2.25in}{\tiny TODO: 1co and 2co}}}
{\red 1-co and 2-co,} aka $\TC$ and $\TC^2$, on right. The primitives that remain are:

\resizebox{\columnwidth}{!}{\subimport{../../../Projects/OneCo-1604/}{1coGraphs.pdf_t}}
\newline\null\hfill\text{\red\ldots manageable but still exponential!}

\vskip -2mm\rule{\linewidth}{1pt}\vspace{-1mm}

{\red The $2D$ relations} come from the relation with 2D Lie bialgebras:
\newline\null
\[ \begin{array}{cc}
  \fbox{\parbox[t][0.5in]{3in}{\tiny TODO: The 2D relations.}} &
  \begin{array}{c}
    \includegraphics[height=0.5in]{../../../Projects/Gallery/Jones-2.jpg} \\
    \text{\tiny Jones}
  \end{array}
\end{array} \]

\vskip -2mm\rule{\linewidth}{1pt}\vspace{-1mm}

{{\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}}

\vskip -6mm\vfill\rule{\linewidth}{1pt}%\vspace{-1mm}

\includegraphics[height=0.6in]{../../../Projects/Gallery/Kronecker.jpg}
\hfill
\parbox[b]{2.7in}{
  ``God created the knots; all else in
  \newline topology is the work of mortals.''
  \vskip 1.5mm
  {\footnotesize Leopold Kronecker (modified)\hfill\href{http://katlas.org}{katlas.org}}
}
\hfill
\includegraphics[height=0.6in]{../../../Projects/Gallery/The_Knot_Atlas.png}

\end{multicols*}

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