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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\blue{\color{blue}}
\def\red{\color{red}}
\def\cbox#1#2{{\setlength{\fboxsep}{0pt}\colorbox{#1}{#2}}}

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

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\def\Ad{\operatorname{Ad}}
\def\CW{\text{\it CW}}
\def\diag{\operatorname{diag}}
\def\FL{\text{\it FL}}
\def\lr{$\leftrightarrow$}
\def\ori{$\circlearrowleft$}
\def\PvT{{\mathit P\!v\!T}}
\def\remove{\!\setminus\!}
\def\tr{\operatorname{tr}}
\def\vT{{\mathit v\!T}}

\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\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{GPV,PV}. If $g$ is a Gauss diagram and
$F$ an unsigned Gauss diagram, $\displaystyle \langle F,g\rangle_{\text{GPV}}
\coloneqq \sum_{y\subseteq g} (-1)^y\delta(F,\bar{y})$:
}}}}

\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)
$.
}}}}

\def\ThmTwo{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Theorem 2.} With $k=1$ and $F_A$ defined by
\[ F_A(\overset{s}{\longrightarrow},\gamma) = \left.
  \frac{s(\gamma_{22}\gamma_{33}-\gamma_{23}\gamma_{32})}
    {\gamma_{33}+\gamma_{13}\gamma_{32}-\gamma_{12}\gamma_{33}}
  \right|_{T_a\to T}, \]
\[ F_A(\overset{s}{\longleftarrow},\gamma) = \left.
  \frac{s(\gamma_{13}\gamma_{32}-\gamma_{12}\gamma_{33})}
    {\gamma_{32}-\gamma_{23}\gamma_{32}+\gamma_{22}\gamma_{33}}
  \right|_{T_a\to T}, \]
\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{0.75in}{\scriptsize\raggedright
  Jones, Melvin, Morton, Rozansky, Garoufalidis
}}

\def\mmr{{\raisebox{0mm}{\parbox[t]{3.95in}{
  \ldots and they the ``higher diagonals'' in the MMR expansion of $JJ$:
}}}}

\def\vTPA{{$\vT\coloneqq$PA}}

\def\grI{{\raisebox{0mm}{\parbox[t]{3.95in}{
Let $\calI\coloneqq\langle\slashoverback-\crossing\rangle$. Then
$\calA^v\coloneqq\prod I^n/I^{n+1}=$``universal
$\calU(D\frakg)^{\otimes S}$''$=$

\vskip 0.8in

\parshape 1 0in 2.7in
{\red Likely Theorem.} \cite{EtingofKazhdan:BialgebrasI,
Enriquez:Quantization}
There exists a homomorphic expansion (universal finite type invariant)
$Z\colon\vT\to\calA^v$. \hfil{\footnotesize (issues~suppressed)}

{\red Too hard!} Look for ``meta-monoid'' quotients.
}}}}

\def\fineprint{{\raisebox{0mm}{\parbox[t]{3.9in}{
Fine print: No sources no sinks, AS vertices, internally acyclic,
$\deg=(\#\text{vertices})/2$.
}}}}

\def\Aw{$\calA^w\cong\calU(\FL(S)^S\ltimes\CW(S))$}

\def\ThmThree{{\raisebox{2mm}{\parbox[t]{3.25in}{
{\red Theorem 3} \cite{WKO}.
$\exists!$ a homomorphic expansion, aka a homomorphic universal finite type
invariant $Z^w$ of pure w-tangles. $z^w\coloneqq\log Z^w$ takes
values in $\FL(S)^S\times\CW(S)$.
}}}}

\def\computable{{\raisebox{2.5mm}{\parbox[t]{3.95in}{
{\red $z$ is computable~\cite{KBH}.} But the computations involve coloured
trees ($\FL$) and coloured wheels
($\CW$) \ldots\hfill\ldots{\red Too hard!}
}}}}

\def\modbeta{{\raisebox{3.5mm}{\parbox[t]{2in}{
%\parshape 6 0pt 2.0in 0pt 2.0in 0pt 2.0in 0pt 2.0in 0pt 2.0in 0pt 3.95in
{\red Proposition} \cite{KBH}. Modulo all relations that universally hold
for the 2D non-Abelian Lie algebra and after some changes-of-variable,
$z^w$ reduces to $z_0$.
}}}}

\def\Lamb{$\Lambda$}

\def\AbstractContext{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 4 0in 2.08in 0in 2.08in 0in 3in 0in 3.95in
{\red Definition.} (Compare~\cite{Bar-NatanSelmani:MetaMonoids, KBH})
A meta-monoid is a functor $M\colon$(finite sets,
injections)$\to$(sets) (think ``$M(S)$ is quantum $G^S$'', for $G$ a group)
along with natural operations $\ast\colon M(S_1)\times M(S_2)\to
M(S_1\sqcup
S_2)$ whenever $S_1\cap S_2=\emptyset$ and $m^{ab}_c\colon
M(S)\to M((S\remove\{a,b\})\sqcup\{c\})$ whenever $a\neq b\in S$ and
$c\notin S\remove\{a,b\}$, such that
\[ \text{meta-associativity:}\quad
  m^{ab}_x\act m^{xc}_y = m^{bc}_x\act m^{ax}_y \]
\[ \text{meta-locality:}\quad
  m^{ab}_c\act m^{de}_f = m^{de}_f\act m^{ab}_c \]
and, with $\epsilon_b=M(S\hookrightarrow S\sqcup\{b\})$,
\[ \text{meta-unit:}\quad
  \epsilon_b\act m^{ab}_a = Id = \epsilon_b\act m^{ba}_a.
\]

{\red Claim.} Pure virtual tangles $\PvT$ form a meta-monoid.

{\red Theorem.} $S\mapsto\Gamma_0(S)$ is a meta-monoid and
$z_0\colon\PvT\to\Gamma_0$ is a morphism of meta-monoids.

{\red Theorem.} There exists an extension of $\Gamma_0$ to a
bigger meta-monoid $\Gamma_{01}(S) = \Gamma_0(S)\times\Gamma_1(S)$, along
with
an extension of $z_0$ to $z_{01}\colon\PvT\to\Gamma_{01}$, with
\[ \Gamma_1(S) =
  R_S\oplus V\oplus V^{\otimes 2}\oplus V^{\otimes 3}\oplus
\calS^2(V)^{\otimes 2}
  \qquad(\text{with }V\coloneqq R_S\langle S\rangle).
\]
{\red Furthermore,} upon reducing to a single variable everything is
polynomial size and polynomial time.

{\red Furthermore,} $\Gamma_{01}$ is given using a
``meta-2-cocycle $\rho^{ab}_c$ over $\Gamma_0$'': In
addition to $m^{ab}_c\to m^{ab}_{0c}$, there are $R_S$-linear
$m^{ab}_{1c}\colon\Gamma_1(S\sqcup\{a,b\})\to\Gamma_1(S\sqcup\{c\})$,
a meta-right-action
$\alpha^{ab}\colon\Gamma_1(S)\times\Gamma_0(S)\to\Gamma_1(S)$
$R_S$-linear in the first variable, and a
first order differential operator (over $R_S$)
$\rho^{ab}_c\colon\Gamma_0(S\sqcup\{a,b\})\to\Gamma_1(S\sqcup\{c\})$
such that
\[ (\zeta_0,\zeta_1)\act m^{ab}_c
  = \left(
    \zeta_0\act m^{ab}_{0c},
    (\zeta_1,\zeta_0)\act\alpha^{ab}\act m^{ab}_{1c}
      + \zeta_0\act\rho^{ab}_c
  \right)
\]

{\red What's done?} The braid part, with still-ugly formulas.

{\red What's missing?} A lot of concept- and detail-sensitive work towards
$m^{ab}_{1c}$, $\alpha^{ab}$, and $\rho^{ab}_c$. The ``ribbon element''.
}}}}

\def\refs{{\raisebox{2mm}{\parbox[t]{3.95in}{\small
{\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[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[PV]{PV} M.~Polyak and O.~Viro,
  {\em  Gauss Diagram Formulas for Vassiliev Invariants,}
  Inter.\ Math.\ Res.\ Notices {\bf 11} (1994) 445--453.

\end{thebibliography}
}}}}

\def\legalities{{\raisebox{0mm}{\parbox[t]{3.95in}{
It should be a {\red legal requirement} that the slides of slide-based
talks be linked from the conference web site {\em before} the actual talks.
}}}}

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