\documentclass[10pt,notitlepage]{article}
\usepackage[english,greek]{babel}
\usepackage{amsmath,graphicx,amssymb,datetime,multicol,stmaryrd,amscd,dbnsymb}
\usepackage{tensor}
\usepackage{txfonts}	% for the likes of \coloneqq.
\usepackage[usenames,dvipsnames]{xcolor}

% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}
\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor=blue
}

\def\navigator{{Dror Bar-Natan: Talks: Oberwolfach-1405:}}
\def\webdef{{{\greektext web}$\coloneqq$\url{http://www.math.toronto.edu/~drorbn/Talks/Oberwolfach-1405/}}}
\def\web#1{{\href{http://www.math.toronto.edu/drorbn/Talks/Oberwolfach-1405/#1}{{\greektext web}/#1}}}

\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:#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\Kbh{{\calK^{bh}}}
\def\lr{$\leftrightarrow$}
\def\ori{$\circlearrowleft$}

\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\calD{{\mathcal D}}
\def\calG{{\mathcal G}}
\def\calK{{\mathcal K}}
\def\calO{{\mathcal O}}
\def\frakg{{\mathfrak g}}

%%%

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Abstract.} \normalsize I will describe some very good formulas for a
{\em (matrix plus scalar)}-valued extension of the Alexander
polynnomial to tangles, then say that everything extends to virtual
tangles, then roughly to simply knotted balloons and hoops in 4D,
then the target space extends to {\em (\mbox{free} Lie algebras plus
cyclic words)}, and the result is a universal finite type of the knotted
objects in its domain. Taking a cue from the BF topological quantum field
theory, everything should extend (with some modifications) to arbitrary
codimension-2 knots in arbitrary dimension and in particular, to arbitrary
2-knots in 4D. But what is really going on is still a mystery.
}}}}

\def\MetaAssoc{{
  (meta-associativity: $m^{ab}_a\act m^{ac}_a=m^{bc}_b\act m^{ab}_a$)
}}

\def\WhyTangles{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 5 0in 1.25in 0in 1.25in 0in 3.95in 0in 3.95in 0in 1.9in
{\red Why Tangles?}
\newline $\bullet$ Finitely presented.
\newline $\bullet$ Divide and conquer proofs and computations.
\newline $\bullet$ ``Algebraic Knot Theory'': If $K$ is ribbon,
\[ Z(K)\in\{cl_2(Z)\colon cl_1(Z)=1\}. \]
(Genus and crossing number are also definable properties).
}}}}

\def\ThmOne{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Theorem 1.} $\exists!$ an invariant $\gamma\colon\{$pure
framed $S$-component tangles$\}\to R\times M_{S\times S}(R)$, where
$R=R_S=\bbZ((T_a)_{a\in S})$ is the ring of rational functions in $S$
variables, intertwining
\newline {\bf 1.}
$\begin{CD}
  \left(
    \begin{array}{c|c} \omega_1&S_1\\ \hline S_1&A_1 \end{array},
    \begin{array}{c|c} \omega_2&S_2\\ \hline S_2&A_2 \end{array}
  \right)
  @>\sqcup>>
  \begin{array}{c|cc}
    \omega_1\omega_2 & S_1 & S_2 \\
    \hline
    S_1 & A_1 & 0 \\
    S_2 & 0 & A_2
  \end{array}
\end{CD}$,
\newline {\bf 2.}
$\begin{CD}
  \begin{array}{c|ccc}
    \omega & a & b & S \\
    \hline
    a & \alpha & \beta & \theta \\
    b & \gamma & \delta & \epsilon \\
    S & \phi & \psi & \Xi
  \end{array}
  @>m^{ab}_c>\mu\coloneqq 1-\beta>
  \left(\!\begin{array}{c|cc}
    \mu\omega & c & S \\
    \hline
    c & \gamma+\alpha\delta/\mu & \epsilon+\delta\theta/\mu \\
    S & \phi+\alpha\psi/\mu & \Xi+\psi\theta/\mu
  \end{array}\!\right)_{T_a,T_b\to T_c}
\end{CD}$,
\newline and satisfying
$\left(
  \mid_a;\,
  \tensor[_a]{\overcrossing}{_b},\tensor[_b]{\undercrossing}{_a}
  \right)
  \overset{\gamma}{\longrightarrow}
  \left(
    \begin{array}{c|c} 1 & a \\ \hline a & 1 \\ \end{array};\,
    \begin{array}{c|cc} 
      1 & a & b \\
      \hline
      a & 1 & 1-T_a^{\pm 1} \\
      b & 0 & T_a^{\pm 1}
    \end{array}
  \right)
$.
}}}}

\def\InAddition{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 8 0in 2.75in 0in 2.75in 0in 2.75in 0in 2.75in 0in 2.75in 0in 2.75in 0in 2.75in 0in 3.95in
{\red In Addition,}
$\bullet$ This is really ``just'' a stitching formula for Burau/Gassner
\cite{LeDimet:Gassner, KirkLivingstonWang:Gassner,
CimasoniTuraev:LagrangianRepresentation}.
\newline $\bullet$ $L\mapsto\omega$ is Alexander, mod units.
\newline $\bullet$ $L\mapsto(\omega,A) \mapsto \omega\det'(A-I)/(1-T')$ is
the MVA, mod units.
\newline $\bullet$ The ``fastest'' Alexander algorithm.
\newline $\bullet$ There are also formulas for strand deletion, reversal,
and doubling.
\newline $\bullet$ Every step along the computation is the invariant
of something.
\newline $\bullet$ Extends to and more naturally defined on v/w-tangles.
\newline $\bullet$ Fits in one column, including propaganda \& implementation.
}}}}

\def\Implementation{{\raisebox{2.5mm}{\parbox[t]{2in}{
\parshape 2 0in 2in 0.1in 1.5in
{\red Implementation} key idea:
\newline $\left(\omega,A=(\alpha_{ab})\right)\leftrightarrow$
\newline $\left(\omega,\lambda=\sum\alpha_{ab}t_ah_b\right)$
}}}}

\def\Weaknesses{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Weaknesses,}
$\bullet$ $m^{ab}_c$ is non-linear.
\newline $\bullet$ The product $\omega A$ is always Laurent, but proving
this takes induction with exponentially many conditions.
}}}}

\def\ExampleNotes{{\raisebox{1mm}{\parbox[t]{4in}{
$\bullet$ $\delta$ injects u-knots into $\calK^{bh}$ (likely u-tangles
  too).
\newline$\bullet$ $\delta$ maps v-tangles to $\calK^{bh}$; the kernel
contains the above and {\red conjecturally} (Satoh), that's all.
\newline$\bullet$ Allowing punctures and cuts, $\delta$ is onto.
}}}}

\def\MetaGroupAction{{\raisebox{1.5mm}{\parbox[t]{1.4in}{
If $X$ is a space, $\pi_1(X)$ is a group,
$\pi_2(X)$ is an Abelian group, and $\pi_1$ acts on $\pi_2$.
\newline{\red Proposition.} The generators generate.
}}}}

\def\Ktm{{$K\sslash tm^{uv}_w$:}}
\def\Khm{{$K\sslash hm^{xy}_z$:}}
\def\Ktha{{$K\sslash tha^{ux}$:}}

%\def\GensGen{{\raisebox{2mm}{\parbox[t]{1.5in}{
%{\red Proposition.} The generators generate.
%}}}}

\def\ThmTwo{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Definition.} $l_{xu}$ is the linking number of hoop $x$ with balloon
$u$. For $x\in H$, $\sigma_x\coloneqq\prod_{u\in T}T_u^{l_{xu}}\in
R=R_T=\bbZ((T_a)_{a\in T})$, the ring of rational functions in $T$
variables.

{\red Theorem 2} \cite{Bar-NatanSelmani:MetaMonoids}.
$\exists!$ an invariant $\beta\colon w\Kbh(H;T) \to R\times M_{T\times
H}(R)$, intertwining
\newline {\bf 1.}
$\begin{CD}
  \left(
    \begin{array}{c|c} \omega_1&H_1\\ \hline T_1&A_1 \end{array},
    \begin{array}{c|c} \omega_2&H_2\\ \hline T_2&A_2 \end{array}
  \right)
  @>\sqcup>>
  \begin{array}{c|cc}
    \omega_1\omega_2 & H_1 & H_2 \\
    \hline
    T_1 & A_1 & 0 \\
    T_2 & 0 & A_2
  \end{array}
\end{CD}$,
\newline {\bf 2.}
$\begin{CD}
  \begin{array}{c|c}
    \omega & H \\
    \hline
    u & \alpha \\
    v & \beta \\
    T & \Xi
  \end{array}
  @>tm^{uv}_w>>
  \left(\!\begin{array}{c|c}
    \omega & H \\
    \hline
    w & \alpha+\beta \\
    T & \Xi
  \end{array}\!\right)_{T_u,T_v\to T_w}
\end{CD}$,
\newline {\bf 3.}
$\begin{CD}
  \begin{array}{c|ccc}
    \omega & x & y & H \\
    \hline
    T & \alpha & \beta & \Xi
  \end{array}
  @>{hm^{xy}_z}>>
  \begin{array}{c|cc}
    \omega & z & H \\
    \hline
    T & \alpha+\sigma_x\beta & \Xi
  \end{array}
\end{CD}$,
\newline {\bf 4.}
$\begin{CD}
  \begin{array}{c|cc}
    \omega & x & H \\
    \hline
    u & \alpha & \theta \\
    T & \phi & \Xi
  \end{array}
  @>{tha^{ux}}>\nu\coloneqq 1+\alpha>
  \begin{array}{c|cc}
    \nu\omega & x & H \\
    \hline
    u & \sigma_x\alpha/\nu & \sigma_x\theta/\nu \\
    T & \phi/\nu & \Xi-\phi\theta/\nu
  \end{array}
\end{CD}$,
\newline and satisfying
$\left(\epsilon_x;\epsilon_u;\rho^\pm_{ux}\right)
  \overset{\beta}{\longrightarrow}
  \left(
    \begin{array}{c|c} 1 & x \\ \hline & \end{array};
    \begin{array}{c|c} 1 & \\ \hline u & \end{array};
    \begin{array}{c|c} 1 & x \\ \hline u & T_u^{\pm 1}-1 \end{array}
  \right)
$.
{\red Proposition.} If $T$ is a u-tangle and $\beta(\delta T)=(\omega,A)$,
then $\gamma(T)=(\omega, \sigma-A)$, where $\sigma=\diag(\sigma_a)_{a\in
S}$. Under this, $m^{ab}_c\leftrightarrow tha^{ab}\act tm^{ab}_c\act
hm^{ab}_c$.
}}}}

\def\ThmThree{{\raisebox{2mm}{\parbox[t]{3.95in}{

{\red Theorem 3} \cite{WKO, KBH}.
$\exists!$ a homomorphic expansion, aka a homomorphic universal finite type
invariant $Z$ of w-knotted balloons and hoops. $\zeta\coloneqq\log Z$ takes
values in $\FL(T)^H\times\CW(T)$.
}}}}

\def\computable{{\raisebox{2.5mm}{\parbox[t]{4in}{
{\red $\zeta$ is computable!} $\zeta$ of the Borromean tangle,
to degree 5:
}}}}

\def\modbeta{{\raisebox{3.5mm}{\parbox[t]{4in}{
\parshape 6 0pt 2.0in 0pt 2.0in 0pt 2.0in 0pt 2.0in 0pt 2.0in 0pt 4in
{\red Proposition} \cite{KBH}. Modulo all relations that universally hold
for the 2D non-Abelian Lie algebra and after some changes-of-variable,
$\zeta$ reduces to $\beta$ and the KBH operations on $\zeta$ reduce to
the formulas in Theorem 2.
}}}}

\def\BigQ{{\raisebox{3.5mm}{\parbox[t]{4in}{
{\red A Big Question.} Does it all extend to arbitrary 2-knots (not
necessarily ``simple'')? To arbitrary codimension-2 knots?
}}}}

\def\BFBox{{\raisebox{2mm}{\parbox[t]{3.4in}{
{\red BF Following~\cite{CattaneoRossi:WilsonSurfaces}.}
$A\in\Omega^1(M=\bbR^4,\frakg)$, $B\in\Omega^2(M,\frakg^*)$,
\[ S(A,B)\coloneqq\int_M\langle B,F_A\rangle. \]
With $\kappa\colon(S=\bbR^2)\to M$, $\beta\in\Omega^0(S,\frakg)$,
$\alpha\in\Omega^1(S,\frakg^*)$, set
\[
  \calO(A,B,\kappa)\coloneqq\int\calD\beta\calD\alpha\exp\left(
    \frac{i}{\hbar}
      \int_S\left\langle\beta,d_{\kappa^*A}\alpha+\kappa^*B\right\rangle
  \right).
\]
}}}}

\def\FeynmanRules{{\raisebox{2mm}{\parbox[t]{4in}{
\parshape 4 0in 3.3in 0in 3.3in 0in 3.3in 0in 4in
{\red The BF Feynman Rules.} For an edge $e$, let $\Phi_e$ be its
direction, in $S^3$ or $S^1$. Let $\omega_3$ and $\omega_1$ be volume forms
on $S^3$ and $S_1$. Then 
\[ \def\neg{{\hspace{-1mm}}}
  Z_{BF}=\neg\sum_{\text{diagrams}\atop D}\neg
  \frac{[D]}{|\text{Aut}(D)|}
  \underbrace{\int_{\bbR^2}\neg\!\!\cdots\neg\int_{\bbR^2}}_{S\text{-vertices}}
  \underbrace{\int_{\bbR^4}\neg\!\!\cdots\neg\int_{\bbR^4}}_{M\text{-vertices}}
  \prod_{\text{red}\atop e\in D}\Phi_e^\ast\omega_3
  \prod_{\text{black}\atop e\in D}\Phi_e^\ast\omega_1
\]
(modulo some $STU$- and $IHX$-like relations).
}}}}

\def\BFIssues{{\raisebox{3.5mm}{\parbox[t]{4in}{
{\red Issues.} $\bullet$ Signs don't quite work out, and BF seems to
reproduce only ``half'' of the wheels invariant.
\newline$\bullet$ There are many more configuration space integrals than BF
Feynman diagrams and than just trees and wheels.
\newline$\bullet$ I don't know how to define ``finite type'' for arbitrary
2-knots.
}}}}

\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,}
  \web{WKO1}, \web{WKO2}, \arXiv{1405.1956}, \arXiv{1405.1955}.

\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[CR]{CattaneoRossi:WilsonSurfaces} A.~S.~Cattaneo and C.~A.~Rossi,
  {\em Wilson Surfaces and Higher Dimensional Knot Invariants,}
  Commun.\ in Math.\ Phys.\ {\bf 256-3} (2005) 513--537,
  \arXiv{math-ph/0210037}.

\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[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.

\end{thebibliography}
}}}}

\def\issues{{\raisebox{2mm}{\parbox[t]{4in}{
{\red Issues.}\begin{itemize}
\item 
\end{itemize}
}}}}

\paperwidth 8.5in
\paperheight 11in
\textwidth 8.5in
\textheight 11in
\oddsidemargin -1in
\evensidemargin \oddsidemargin
\topmargin -1in
\headheight 0in
\headsep 0in
\footskip 0in
\parindent 0in
\setlength{\topsep}{0pt}
\pagestyle{empty}

\begin{document} \latintext
\setlength{\jot}{0ex}
\setlength{\abovedisplayskip}{0.5ex}
\setlength{\belowdisplayskip}{0.5ex}
\setlength{\abovedisplayshortskip}{0ex}
\setlength{\belowdisplayshortskip}{0ex}

\begin{center}
\null\vfill \input{GoodFormulas1.pstex_t} \vfill\null
\newpage \null\vfill \input{GoodFormulas2.pstex_t} \vfill\null
\newpage \null\vfill \input{GoodFormulas3.pstex_t} \vfill\null
\end{center}

%\newpage\null\vspace{0.5in}
%\hspace{0.5in}\begin{minipage}{7.5in}
%\begin{multicols}{2}
%
%{\large\bf A GoodFormulas Discussion.}
%
%\end{multicols}
%\end{minipage}

\end{document}

\endinput