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\def\navigator{{Video, links, and more @ Dror Bar-Natan: Talks: Hamilton-1412:}}
\def\webdef{{{\greektext web}$\coloneqq$\url{http://www.math.toronto.edu/~drorbn/Talks/Hamilton-1412}}}
\def\w#1{{\href{http://www.math.toronto.edu/drorbn/Talks/Hamilton-1412/#1}{{\greektext web}/#1}}}

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\def\dgreen{\color{ForestGreen}}
\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\CW{\text{\it CW}}
\def\FA{\text{\it FA}}
\def\FL{\text{\it FL}}

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

\def\vK{{\mathit v\mathcal K}}
\def\wK{{\mathit w\mathcal K}}

%%%

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red Abstract.} I will describe a {\dgreen computable, non-commutative}
invariant of tangles with values in wheels, almost generalize it to some
balloons, and then tell you why I care. Spoilers: tangles are you know
what, wheels are linear combinations of cyclic words in some alphabet,
balloons are 2-knots, and one reason I care is because quantum field
theory predicts more than I can actually get (but also less).
}}}}

\def\CompNonComm{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red Why I like {\dgreen ``non-commutative''}?} With $\FA(x_i)$ the free
associative non-commutative algebra,
\[ \dim\bbQ[x,y]_d\sim d \ll 2^d \sim \dim\FA(x,y)_d. \]
{\red Why I like {\dgreen ``computable''}?}
\newline $\bullet$ Because I'm weird.
\hfill $\bullet$ Note that $\pi_1$ isn't computable.
}}}}

\def\Prelims{{\raisebox{2mm}{\parbox[t]{3.96in}{
\parshape 4 0in 2.2in 0in 2.2in 0in 2.2in 0in 3.96in
{\red Preliminaries from Algebra.} $\FL(x_i)$ denotes the free
Lie algebra in $(x_i)$; $\FL(x_i)=($binary trees with AS vertices
and coloured leafs$)/($IHX relations$)$. There an obvious map
$\FA(\FL(x_i))\to\FA(x_i)$ defined by $[a,b]\to ab-ba$, which in itself,
is IHX.
}}}}

\def\prelims{{\raisebox{0mm}{\parbox[t]{3.96in}{
$\CW(x_i)$ denotes the vector space of cyclic words in $(x_i)$:
$\CW(x_i)=\FA(x_i)/(x_iw=wx_i)$. There an obvious map
$\CW(\FL(x_i))\to\CW(x_i)$. In fact, connected uni-trivalent 2-in-1-out
graphs with univalents with colours in $\{1,\ldots,n\}$, modulo AS and
IHX, is precisely $\CW(x_i)$:
}}}}

\def\important{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red Most important.} $\displaystyle e^x=\sum\frac{x^d}{d!}$ and
$e^{x+y}=e^xe^y$.
}}}}

\def\sumklm{$\displaystyle\sum_{k,l,m\geq 0}\frac{(+)^k(-)^l(+)^m}{k!l!m!}$}
\def\sumkm{$\displaystyle\sum_{k,m\geq 0}\frac{(+)^k(-)^m}{k!m!}$}

\def\Theorem{{\raisebox{3mm}{\parbox[t]{3.96in}{
{\red Theorem.} $\omega$, the connected part of the procedure below, is an
invariant of $S$-component tangles with values in $\CW(S)$:
}}}}

\def\computable{{\raisebox{2mm}{\parbox[t]{2.3in}{
{\red $\omega$ is practically computable!} For the Borromean tangle, to
degree 5, the result is:\hfill(see~\cite{KBH})
}}}}

\def\further{{\raisebox{-3mm}{\parbox[t]{3.96in}{
\parbox{3.3in}{\begin{myitemize}
\item $\omega$ is really the second part of a (trees,wheels)-valued
invariant $\zeta=(\lambda,\omega)$. The tree part $\lambda$ is just a
repackaging of the Milnor $\mu$-invariants.
\end{myitemize}}
\begin{myitemize}
\item On u-tangles, $\zeta$ is equivalent to the trees\&wheels part of the
Kontsevich integral, except it is computable and is defined with no need
for a choice of parenthesization.
\item On long/round u-knots, $\omega$ is equivalent to the Alexander
polynomial.
\item The multivariable Alexander polynomial (and Levine's factorization
thereof~\cite{Levine:Factorization}) is contained in the Abelianization of
$\zeta$~\cite{Bar-NatanSelmani:MetaMonoids}.
\item $\omega$ vanishes on braids.
\item Related to / extends
Farber's~\cite{Farber:NoncommutativeRationalFunctions}?
\item Should be summed and categorified.
\item Extends to v and descends to w:
\end{myitemize}
\hfill meaning, $\zeta$ satisfies \hfill\hfill $\omega$ also satisfies
\hfill\hfill so $\omega$'s ``true domain'' is
\hfill\null
}}}}

\def\does{{$\text{Does $\omega$ extend}\atop\text{to balloons?}$}}

\def\ff{{\raisebox{2.25mm}{\parbox[t]{3.96in}{
\begin{myitemize}
\item Agrees with BN-Dancso~\cite{WKO1,WKO2} and with~\cite{KBH}.
\item $\zeta$, $\omega$ are universal finite type invariants.
\item Using $\Zhe\colon\vK_n\to\wK_{n+1}$, defines a strong invariant
of v-tangles / long v-knots.\hspace{1in}($\Zhe$~in~\LaTeX:~\w{zhe})
\end{myitemize}
}}}}

\def\BigOpen{{\raisebox{2.5mm}{\parbox[t]{3.96in}{
{\red A Big Open Problem.} $\delta$ maps w-knots onto simple 2-knots. To
what extent is it a bijection? What other relations are required? In other
words, {\bf find a simple description of simple 2-knots}.
}}}}

\def\BigQ{{\raisebox{3.5mm}{\parbox[t]{3.96in}{
{\red 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]{3.96in}{
\parshape 4 0in 3.3in 0in 3.3in 0in 3.3in 0in 3.96in
{\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 IHX-like relations).\hfill See also~\cite{Watanabe:CSI}
}}}}

\def\BFIssues{{\raisebox{3.5mm}{\parbox[t]{3.96in}{
{\red Issues.} $\bullet$ Signs don't quite work out, and BF seems to
reproduce only ``half'' of the wheels invariant on simple 2-knots.
\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 / analyze ``finite type''
for general 2-knots.
\newline$\bullet$ I don't know how to reduce $Z_{BF}$ to
combinatorics / algebra.
}}}}

\def\Riddles{{\raisebox{0mm}{\parbox[t]{3.96in}{
{\bf Riddles}, in case you are bored.
\begin{myitemize}
\item Can you find uncountably many distinct subsets $\{A_\alpha\}$
of $\bbZ$ such that whenever $\alpha\neq\beta$ either $A_\alpha\subset
A_\beta$ or $A_\beta\subset A_\alpha$?
\item Can you find uncountably many distinct subsets $\{B_\alpha\}$
of $\bbZ$ such that whenever $\alpha\neq\beta$ the intersection
$B_\alpha\cap B_\beta$ is finite?
\end{myitemize}
}}}}

\def\refs{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red References.} \small
\par\vspace{-2mm}
\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,}
  \w{KBH}, \arXiv{1308.1721}.

\bibitem[BND1]{WKO1} D.~Bar-Natan and Z.~Dancso,
  {\em Finite Type Invariants of W-Knotted Objects I: W-Knots and the
  Alexander Polynomial,}
  \w{WKO1}, \arXiv{1405.1956}.

\bibitem[BND2]{WKO2} D.~Bar-Natan and Z.~Dancso,
  {\em Finite Type Invariants of W-Knotted Objects II: Tangles and
  the Kashiwara-Vergne Problem,}
  \w{WKO2}, \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[CS]{CS} J.~S.~Carter and M.~Saito,
  {\em Knotted surfaces and their diagrams,}
  Math.\ Surv.\ and Mono.\ {\bf 55}, Amer.\ Math.\ Soc., Providence 1998.

\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[Fa]{Farber:NoncommutativeRationalFunctions} M.~Farber,
  {\em Noncommutative Rational Functions and Boundary Links,}
  Math.\ Ann.\ {\bf 293} (1992) 543--568.

\bibitem[Le]{Levine:Factorization} J.~Levine,
  {\em A Factorization of the Conway Polynomial,}
  Comment.\ Math.\ Helv.\ {\bf 74} (1999) 27--53, \arXiv{q-alg/9711007}.

\bibitem[Wa]{Watanabe:CSI} T.~Watanabe,
  {\em Configuration Space Integrals for Long $n$-Knots, the Alexander
    Polynomial and Knot Space Cohomology,}
  Alg.\ and Geom.\ Top.\ {\bf 7} (2007) 47--92, \arXiv{math/0609742}.

\end{thebibliography}
}}}}

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