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\def\blue{\color{blue}}
\def\red{\color{red}}

\newcommand{\im}{\operatorname{im}}

\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\calA{{\mathcal A}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}

\def\ori{$\circlearrowleft$}

\def\aAS{{\overrightarrow{AS}}}
\def\Abh{{\calA^{bh}}}
\def\aft{{\overrightarrow{4T}}}
\def\aIHX{{\overrightarrow{IHX}}}
\def\aSTU{{\overrightarrow{STU}}}
\def\Atbh{{\tilde\calA^{bh}}}
\def\CW{\text{\it CW}}
\def\FL{\text{\it FL}}
\def\Kbh{{\calK^{bh}}}

\def\lr{$\leftrightarrow$}

\def\navigator{{Dror Bar-Natan: Talks: Geneva-131024:}}

\def\webdef{{$\omega:=$\url{http://www.math.toronto.edu/~drorbn/Talks/Geneva-131024}}}
\def\w#1{{\href{http://www.math.toronto.edu/drorbn/Talks/Geneva-131024/#1}{$\omega$/#1}}}

\def\Abstract{{\raisebox{2mm}{\parbox[t]{4in}{
{\red Abstract.} On my September 17 Geneva talk (\w{sep}) I described
a certain trees-and-wheels-valued invariant $\zeta$ of ribbon knotted loops and
2-spheres in 4-space, and my October 8 Geneva talk (\w{oct}) describes
its reduction to the Alexander polynomial. Today I will explain how that
same invariant arises completely naturally within the theory of finite
type invariants of ribbon knotted loops and 2-spheres in 4-space.
}}}}

\def\goal{{\raisebox{0mm}{\parbox[t]{4in}{
My goal is to tell you why such an invariant is expected, yet not to derive
the computable formulas.
}}}}

\def\expansions{{\raisebox{0mm}{\parbox[t]{4in}{
Let $\calI^n:=\langle\text{pictures with $\geq n$
semi-virts}\rangle\subset\Kbh$.

We seek an ``expansion''
\[ Z\colon\Kbh\to\operatorname{gr}\Kbh
  = \widehat{\bigoplus}\,\calI^n/\calI^{n+1}
  =: \Abh
\]
satisfying ``property U'': if $\gamma\in\calI^n$, then
\[ Z(\gamma)=(0,\ldots,0,\gamma/\calI^{n+1},\ast,\ast,\ldots). \]
}}}}

\def\why{{\raisebox{2mm}{\parbox[t]{4in}{
{\red Why?} $\bullet$ Just because, and this is vastly more general.

$\bullet$ $\left(\Kbh/\calI^{n+1}\right)^{\!\star}$ is
``finite-type/polynomial invariants''.

$\bullet$ The Taylor example: Take $\calK=C^\infty(\bbR^n)$,
$\calI=\left\{f\in\calK\colon f(0)=0\right\}$. Then $\calI^n=\left\{f\colon
f\text{ vanishes like }|x|^n\right\}$ so $\calI^n/\calI^{n+1}$ is
homogeneous polynomials of degree $n$ and $Z$ is a ``Taylor expansion''!
(So Taylor expansions are vastly more general than you'd think).
}}}}

\def\AExpansionDiagram{$\xymatrix{
  & \Atbh \ar@<-2pt>[d]_\pi \\
  \Kbh \ar[ur]^{\tilde{Z}} \ar[r]_<>(0.4)Z
  & \Abh \ar@<-2pt>[u]_{\operatorname{gr}\tilde{Z}}
}$}

\def\AExpansion{{\raisebox{3mm}{\parbox[t]{2.8in}{
{\red Plan.} We'll construct a graded $\Atbh$, a surjective graded
$\pi\colon\Atbh\to\Abh$, and a filtered $\tilde{Z}\colon\Kbh\to\Abh$
so that $\pi\sslash\operatorname{gr}\tilde{Z}=\text{\it
Id}$ (property U: if $\deg D=n$,
$\tilde{Z}(\pi(D)) = \pi(D)+(\text{deg}\geq n)$). Hence
$\bullet$~$\pi$ is an isomorphism.  $\bullet$~$Z:=\tilde{Z}\sslash\pi$
is an expansion.
}}}}

\def\exercise{{\raisebox{3mm}{\parbox[t]{0.75in}{
{\color{red}Exercise.} Prove property U.
}}}}

\def\BRC{{\raisebox{3mm}{\parbox[t]{4in}{
{\color{red}Corollaries.} (1) Related to Lie algebras! (2) Only trees and
wheels persist.
}}}}

\def\MM{{\raisebox{2.5mm}{\parbox[t]{4in}{
{\color{red}Theorem.} $\Abh$ is a bi-algebra. The space of its primitives
is $\FL(T)^H\times\CW(T)$, and $\zeta=\log Z$.
}}}}

\def\computable{{\raisebox{2mm}{\parbox[t]{4in}{
{\red $\zeta$ is computable!} $\zeta$ of the Borromean tangle,
to degree 5:
}}}}

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