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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{NCSU-1604}
\def\title{Gauss-Gassner Invariants}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\webdef{{{\greektext web}$\coloneqq$\url{http://drorbn.net/\thistalk/}}}
\def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}}
\def\titleA{{\title, What?}}
\def\titleB{{\title, Wherefore?}}
\def\titleC{{\title, C}}

\def\todo#1{\text{\Huge #1}}

\def\blue{\color{blue}}
\def\red{\color{red}}
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\def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:\linebreak[0]#1}}}

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\def\ad{\operatorname{ad}}
\def\Ad{\operatorname{Ad}}
\def\aft{$\overrightarrow{\text{4T}}$}
\def\CW{\text{\it CW}}
\def\diag{\operatorname{diag}}
\def\FL{\text{\it FL}}
\def\lr{$\leftrightarrow$}
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\def\PvT{{\mathit P\!v\!T}}
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\def\tr{\operatorname{tr}}
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\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\calA{{\mathcal A}}
\def\calD{{\mathcal D}}
\def\calG{{\mathcal G}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}
\def\calO{{\mathcal O}}
\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.} In a ``degree $d$ Gauss diagram formula'' one produces a
number by summing over all possibilities of paying very close attention
to $d$ crossings in some $n$-crossing knot diagram while observing the
rest of the diagram only very loosely, minding only its skeleton. The
result is always poly-time computable as only $\binom{n}{d}$ states
need to be considered.  An under-explained paper by Goussarov, Polyak,
and Viro~\cite{GPV} shows that every type
$d$ knot invariant has a formula of this kind. Yet only finitely many
integer invariants can be computed in this manner within any specific
polynomial time bound.

I suggest to do the same as~\cite{GPV},
except replacing ``the skeleton'' with ``the Gassner invariant'', which
is still poly-time. One poly-time invariant that arises in this way is
the Alexander polynomial (in itself it is infinitely many numerical
invariants) and I believe (and have evidence to support my belief)
that there are more.
}}}}

\def\QUILT{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red The QUILT Target.} QUick Invariants of Large Tangles, for little had
been found since Alexander (and if they're there, how can we not know all
about them?), and for $\{\text{ribbon}\}\neq\{\text{slice}\}$:
}}}}

\def\Ta{$\calT_{2n}$}
\def\Tb{$U\in\calT_n$}
\def\Tc{ribbon $K\in\calT_1$}
\def\Aa{$\calA_{2n}$}
\def\Ab{$1\in\calA_n$}
\def\Ac{$z(K)\in\calA_1$}

\def\GST{\parbox{0.5in}{\tiny
  Gompf, Scharlemann, Thompson \cite{GompfScharlemannThompson:Counterexample}
}}

\def\GPVBox{{\raisebox{2mm}{\parbox[t]{2.75in}{
{\red Gauss Diagram Formulas} \cite{PV,GPV}. If $g$ is a Gauss diagram and
$F$ an unsigned Gauss diagram, $\displaystyle \langle F,g\rangle_{\text{PV}}
\coloneqq \sum_{y\subseteq g} (-1)^y\delta(F,\bar{y})$:
}}}}

\def\GPVThm{{\raisebox{2mm}{\parbox[t]{1.3in}{
{\red Under-Explaind Theorem}~\cite{GPV}. Every finite type invariant
arises in this way.
}}}}

\def\MoreBox{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 5 0in 2.25in 0in 2.25in 0in 2.25in 0in 2.25in 0in 3.95in
{\red Gauss-Gassner Invariants.} Want more? Increase your environmental
awareness! Instead of nearly-forgetting $y^c$, compute its Burau/Gassner
invariant (note that $y^c$ is a tangle in a Swiss cheese; more easily, a
virtual tangle):
\[ GG_{k,F}(g)
  = \sum_{y\subseteq g,\,|y|\leq k}\bar{F}\left(y,z(y^c)\right)
  = \sum_{y\subseteq g,\,|y|\leq k}F\left(y,z(g\text{ cut near }y)\right),
\]
where $k$ is fixed and $F(y,\gamma)$ is a function of a list of arrows
$y$ and a square matrix $\gamma$ of side $|y|+1\leq k+1$.
}}}}

\def\ThmOne{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 5 0in 3.45in 0in 3.45in 0in 3.45in 0in 3.45in 0in 3.95in 
{\red Theorem 1.} $\exists!$ an invariant $z\colon\{$pure framed
$S$-component tangles$\}\to\Gamma(S)\coloneqq M_{S\times S}(R_S)$,
where $R_S=\bbZ((T_a)_{a\in S})$ is the ring of rational functions in
$S$ variables, intertwining
\newline
$\begin{CD}
  \left(
    \begin{array}{c|c} &S_1\\ \hline S_1&A_1 \end{array},
    \begin{array}{c|c} &S_2\\ \hline S_2&A_2 \end{array}
  \right)
  @>\displaystyle\sqcup>>
  \begin{array}{c|cc}
    & S_1 & S_2 \\
    \hline
    S_1 & A_1 & 0 \\
    S_2 & 0 & A_2
  \end{array}
\end{CD}$,
\newline
$\begin{CD}
  \begin{array}{c|ccc}
    & a & b & S \\
    \hline
    a & \alpha & \beta & \theta \\
    b & \gamma & \delta & \epsilon \\
    S & \phi & \psi & \Xi
  \end{array}
  @>{\displaystyle m^{ab}_c}>{\displaystyle T_a,T_b\to
T_c\atop\displaystyle \mu\coloneqq 1-\beta}>
  \left(\!\begin{array}{c|cc}
    & c & S \\
    \hline
    c & \gamma+\alpha\delta/\mu & \epsilon+\delta\theta/\mu \\
    S & \phi+\alpha\psi/\mu & \Xi+\psi\theta/\mu
  \end{array}\!\right)
\end{CD}$,
\newline and satisfying
$\left(
  \mid_a;\,
  \tensor[_a]{\overcrossing}{_b},\tensor[_b]{\undercrossing}{_a}
  \right)
  \overset{\displaystyle z}{\longrightarrow}
  \left(
    \begin{array}{c|c}  & a \\ \hline a & 1 \\ \end{array};\,
    \begin{array}{c|cc}
       & a & b \\
      \hline
      a & 1 & 1-T_a^{\pm 1} \\
      b & 0 & T_a^{\pm 1}
    \end{array}
  \right)
$.
See also~\cite{LeDimet:Gassner, KirkLivingstonWang:Gassner, CimasoniTuraev:LagrangianRepresentation,
Bar-NatanSelmani:MetaMonoids}.
}}}}

\def\ThmTwo{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Theorem 2.} With $k=1$ and $F_A$ defined by
\[ F_A(\overset{s}{\longrightarrow},\gamma) = \left.
  s\frac{\gamma_{22}\gamma_{33}-\gamma_{23}\gamma_{32}}
    {\gamma_{33}+\gamma_{13}\gamma_{32}-\gamma_{12}\gamma_{33}}
  \right|_{T_a\to T}, \]
\[ F_A(\overset{s}{\longleftarrow},\gamma) = \left.
  s\frac{\gamma_{13}\gamma_{32}-\gamma_{12}\gamma_{33}}
    {\gamma_{32}-\gamma_{23}\gamma_{32}+\gamma_{22}\gamma_{33}}
  \right|_{T_a\to T}, \]
\newline $GG_{1,F_A}(K)$ is a regular isotopy invariant. Unfortunately,
for every knot $K$, $GG_{1,F_A}(K) - T\frac{d}{dT}\log A(K)(T) \in \bbZ$,
where $A(K)$ is the Alexander polynomial of $K$.
}}}}

\def\Expect{{\raisebox{2mm}{\parbox[t]{3.4in}{
{\red Expectation.} Higher Gauss-Gassner invariants exist \ldots
\newline\scriptsize (though right now I can reach for them only wearing my
exoskeleton)
}}}}

\def\JMMRG{\parbox{0.85in}{\scriptsize\raggedright
  Jones, Melvin, Morton, Rozansky, Garoufalidis
}}

\def\mmr{{\raisebox{0mm}{\parbox[t]{3.95in}{
\ldots and they are the ``higher diagonals'' in the MMR expansion of the coloured Jones polynomial $J_\lambda$.

{\red Theorem} (\cite{Bar-NatanGaroufalidis:MMR}, conjectured~\cite{MM}, elucidated~\cite{Ro}). Let
$J_d(K)$ be the coloured Jones polynomial of $K$, in
the $d$-dimensional representation of $sl(2)$. Writing
\[ \left.
  \frac{(q^{1/2}-q^{-1/2})J_d(K)}{q^{d/2}-q^{-d/2}}
  \right|_{q=e^\hbar} =
  \sum_{j,m\geq 0} a_{jm}(K)d^j\hbar^m,
\]

\parpic[r]{\input{figs/MMR.pstex_t}}
``below diagonal'' coefficients vanish, $a_{jm}(K)=0$ if $j>m$, and
``on diagonal'' coefficients give the inverse of the
Alexander polynomial:
$\left(\sum_{m=0}^\infty a_{mm}(K)\hbar^m\right)\cdot A(K)(e^\hbar)=1$.
}}}}


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\begin{center}
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\parbox[b]{6in}{{\small \navigator} {\Large\bf\red Gauss-Gassner Invariants, Wherefore?}
 }
\hfill\parbox[b]{2in}{\tiny
  \null\hfill\webdef
  \newline\null\hfill Slides w/ no URL should be banned!
}
\vskip -3mm
\rule{\textwidth}{1pt}
\vspace{-8mm}

\begin{multicols}{2} \raggedcolumns

{\red Warning.} Conventions on this page change randomly from line to line.

\vskip -3mm\rule{\columnwidth}{1pt}

{\red $Z^{w/2}$.} The GGA story is about $Z^{w/2}\colon\calK\to\calA^{w/2}$, defined on arrows $a$ by $\pm a\mapsto\exp(\pm a)$:
\newline\resizebox{\columnwidth}{!}{
  \def\sumklm{$\displaystyle\longrightarrow\sum_{k,l,m\geq 0}\frac{(+)^k(-)^l(+)^m}{k!l!m!}$}
  \input{figs/ZDef.pstex_t}
}
Where the target space $\calA^{w/2}$ is the space of unsigned arrow diagrams modulo
\newline\resizebox{\columnwidth}{!}{\input{figs/w2Rels.pstex_t}}

($Z^{w/2}$ is a reduction of the much-studied $Z^w$~\cite{WKO,KBH}).

\vskip -3mm\rule{\columnwidth}{1pt}

{\red The Euler Trick.} How best do non-commutative
algebra with exponentials? Logarithms are from hell as
$e^fe^g=e^{\operatorname{bch}(f,g)}$, but Euler's from heaven: Let $E$ be the derivation
$Ef\coloneqq(\deg f)f$ ($=xf'$, in $\bbQ\llbracket x\rrbracket$) and let
$\tilE Z\coloneqq Z^{-1}EZ$ ($=x(\log Z)'$ in same). If $\deg x=1$
then $\tilE e^x=x$ and if $F=e^f$ and $G=e^g$, then $\tilE(FG)$ is
\[ (FG)^{-1}((EF)G+F(EG)) = G^{-1}(\tilE F)G+\tilE G = e^{-\ad g}(\tilE F)+\tilE G. \]

\vskip -3mm\rule{\columnwidth}{1pt}

{\red Scatter and Glow.} Apply $\tilE$ to $Z(K)$. $EZ$ is shown:
\newline\resizebox{\columnwidth}{!}{\input{figs/EZ.pstex_t}}

\vskip -3mm\rule{\columnwidth}{1pt}

\parpic[r]{\input{figs/HorArrDia.pstex_t}}
{\red Tail scattering.} The algebra $\bbQ\llbracket b_i\rrbracket\langle a_{ij}\rangle$
modulo $[a_{ij},a_{kl}] = 0$ (loc), $[a_{ij},a_{ik}] = 0$ (TC),
and $[a_{ik},a_{jk}] = -[a_{ij},a_{jk}] = b_ja_{ik}-b_ia_{jk}$ (CH and \aft),
acts on $V = {\bbQ\llbracket b_i\rrbracket\langle{x_i}=a_{i\infty}\rangle}$ by $[a_{ij},x_i]=0$,
$[a_{ij},x_j] = b_ix_j-b_jx_i$. Hence $e^{\ad a_{ij}}x_i=x_i$, $e^{\ad a_{ij}}x_j =
e^{b_i}x_j+\frac{b_j}{b_i}(1-e^{b_i})x_i$. Renaming $\bar{x}_i=x_i/b_i$, ${T_i}=e^{b_i}$, get $[e^{\ad
a_{ij}}]_{\bar{x}_i,\bar{x}_j} = \begin{pmatrix} 1 & 1-T_i \\ 0 & T_i \end{pmatrix}$. Alternatively,
\picskip{0}
\[ \hspace{-40mm}\input{figs/xydefs.pstex_t} \]

\vskip -3mm\rule{\columnwidth}{1pt}

{\red Linear Control Theory.}
\needspace{2cm}
\parpic[r]{\input{figs/LCT.pstex_t}}\picskip{3}
If $\begin{pmatrix}y\\y_n\end{pmatrix} =
\begin{pmatrix}\Xi&\phi\\\theta&\alpha\end{pmatrix}
\begin{pmatrix}x\\x_n\end{pmatrix}$, and we further impose $x_n=y_n$,
then $y=Bx$ where $\displaystyle B=\Xi+\frac{\phi\theta}{1-\alpha}$.
This fully explains the Gassner formulas and the GGA formula!

\vskip -3mm\rule{\columnwidth}{1pt}

All that remains now is to replace TC by something more interesting: with $\epsilon^2=0$,
\[ [a_{ij},a_{ik}] = \epsilon(c_ja_{ik}-c_ka_{ij}). \]
Many further changes are also necessary, and the algebra is a lot more complicated and revolves around ``quantization
of Lie bialgebras''~\cite{EtingofKazhdan:BialgebrasI, Enriquez:Quantization}. But the spirit is right.

\vskip -3mm\rule{\columnwidth}{1pt}

%\parpic[r]{\includegraphics[width=0.5in]{../../Projects/Gallery/BarNatan-2_400.eps}}
{\red References.}
\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[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}.

\end{thebibliography}

\end{multicols}

\end{document}

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