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\def\navigator{{Dror Bar-Natan: Talks: Fields-1411:}}
\def\webdef{{\url{http://www.math.toronto.edu/~drorbn/Talks/Fields-1411/}}}

\def\blue{\color{blue}}
\def\dgreen{\color{ForestGreen}}
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\def\purple{\color{purple}}
\def\red{\color{red}}
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\def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:#1}}}

\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\calK{{\mathcal K}}
\def\calO{{\mathcal O}}
\def\frakg{{\mathfrak g}}
\def\gr{{\operatorname{gr}}}

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%%%

\def\Abstract{{\raisebox{1.5mm}{\parbox[t]{3.96in}{
\parshape 6 0in 2.85in 0in 2.85in 0in 2.85in 0in 2.85in 0in 2.85in 0in 3.96in
{\red Abstract.} \normalsize I will describe my former student's Jonathan
Zung work on finite type invariants of ``doodles'', plane curves modulo the
second Reidemeister move but not modulo the third. We use a definition of
``finite type'' different from Arnold's and more along the lines of
Goussarov's ``Interdependent Modifications'', and come to a conjectural
combinatorial description of the set of all such invariants. We then
describe how to construct many such invariants (though perhaps not all)
using a certain class of 2-dimensional ``configuration space integrals''.
\hfill{\purple An unfinished project!}
}}}}

\def\PriorArt{{\raisebox{1.5mm}{\parbox[t]{3.96in}{
\parshape 5 0in 2.75in 0in 2.75in 0in 2.75in 0in 2.75in 0in 3.96in
{\red Prior Art.} Arnold \cite{Arnold:PlaneCurves} first studied doodles
within his study of plane curves and the ``strangeness'' ${\mathit
St}$ invariant.  Vassiliev \cite{Vassiliev:TriplePointFree,
Vassiliev:Ornaments} defined finite type invariants in a different way,
and Merkov \cite{Me} proved that they separate
doodles.
}}}}

\def\dnd{{doodles and {\red detours} (dnd's)}}
\def\GFT{{\raisebox{2mm}{\parbox[t]{4in}{
{\red Def.} $V$ is of type $n$ if it vanishes on $\calK_{n+1}$.
\hfill$(\calK_0/\calK_{n+1})^\star \leftrightsquigarrow
  \calK_n/\calK_{n+1}$
}}}}

\def\Goals{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red Goals.}
$\bullet$ Describe $\calA_n\coloneqq\calK_n/\calK_{n+1}$ using
diagrams/relations.
$\bullet$ Get many or all finite type invariants of doodles using
configurations space integrals.
$\bullet$ Do these come from a TQFT?
$\bullet$ See if $\calA_n$ has a ``Lie theoretic'' (tensors/relations)
meaning.
$\bullet$ See if/how Arnold's ${\mathit St}$ and the Merkov invariants
integrate in.
}}}}

\def\SummaryDiagram{{\raisebox{1.5mm}{\parbox[t]{3.96in}{
{\red Summary Diagram.}
\vskip -10mm
\[ \xymatrix{
  & \calK \ar[dr]^-{Z} \ar@{=>}[d] \\
  \calD^c/MC \eqqcolon \calA^c \ar@{->>}[r]^-{\pi} \ar@/_1.5pc/[rr]^-{T:\ 1-1?}
  & \gr\calK \ar[r]^-{\gr Z}
  & \calA^t \coloneqq \calD^t/{\mathit FDR}
} \]
{\red $T$:}
}}}}

\def\UpperBound{{\raisebox{3.5mm}{\parbox[t]{3.25in}{
\begin{center}
\red Chord Diagrams and an Upper Bound on $\calK_n/\calK_{n+1}$
%\newline$\calA_n=\calK_n/\calK_{n+1}$
\end{center}
}}}}

\def\Rayman{{\raisebox{2.5mm}{\parbox[t]{3.96in}{
{\red The Rayman Principle.} In $\calK_n/\calK_{n+1}$,
}}}}

\def\Subdivision{{\raisebox{2.5mm}{\parbox[t]{3.96in}{
{\red The Subdivision Relations.} In $\calK_n/\calK_{n+1}$,
}}}}

\def\Kquo{{$\calK_n/\calK_{n+1}$}}

\def\cycsum{{$\displaystyle\sum_{\text{cyclic}\atop\text{perms}}\pm$}}

\def\LowerBound{{\raisebox{3mm}{\parbox[t]{4in}{
\begin{center}
\red Feynman Diagrams and a  Lower Bound on $(\calK_0/\calK_{n+1})^\ast$.
\end{center}
}}}}

\def\FeynmanDiagrams{{\raisebox{2.5mm}{\parbox[t]{4in}{
\parshape 10 0in 2in 0in 2in 0in 2in 0in 2in 0in 2in 0in 2in 0in 2in 0in 2in 0in 2in 0in 4in
{\red Feynman Diagrams.} A {\blue blue} ``skeleton line'' at the bottom. A
{\magenta magenta} ``arrow diagram'' (directed pairing of skeleton points)
on top, with a {\magenta magenta} dot at the middle of each arrow. A
{\dgreen green} directed graph on top, with 2-in 1-out antisymmetric
{\dgreen green} vertices, with arbitrary number of {\dgreen green} edges
starting at the {\magenta magenta} dots, and with some {\dgreen green}
edges terminating at distinct {\blue blue} skeleton points. The {\em
degree} is the total valency of the {\magenta magenta} dots.
}}}}

\def\cfi{{$\displaystyle
  \sumint_{C(D)}\Phi_D^\ast\left(\omega_1^{\wedge 3}\right)$}}

\def\vol{{$\omega_1:\text{ vol.}(S^1)$}}

\def\PartitionFunction{{\raisebox{2.5mm}{\parbox[t]{4in}{
{\red The ``Partition Function'' $Z$.}
\[ K\mapsto Z(K)\coloneqq
  \sum_{\text{Feynman}\atop\text{diagrams}}
    \sumint_{C(D)}\Phi_D^\ast\left(\omega_1^{\wedge e(D)}\right)
  \in\calA^t\coloneqq\langle D\rangle/(\text{$\partial$-relations}).
\]
{\red Theorem (90\%).} $Z$ is an invariant of doodles.
\par {\red $\partial$-relations.} STU, IHX, Foot Swap (FS), Arrow Exchange
(AE), and Combinatorial R2 (CR2):
}}}}

\def\sumterms{{$\displaystyle \sum_{2^n\text{ terms}}\pm$}}

\def\Unfinished{{\raisebox{1.5mm}{\parbox[t]{3.96in}{
{\purple An unfinished project!}
\begin{myitemize}
\item Nothing is written up.
\item We don't know if $T$ is injective (meaning, if our
upper and lower bounds agree).
\item We don't know if all of $\calA^t$ is necessary --- it is
very possible that it is enough to restrict to the {\dgreen green-less}
part of $\calA^t$ --- to ``Gauss Diagram Formulas''.
\item We haven't clarified the relationship with Merkov's~\cite{Me}.
\item A few further configuration space integrals can be
written beyond those that we have used. We don't know what to do with
those, if anything.
\item We don't know the relationship, if any, with algebra.
\item We don't know the relationship, if any, with quantum
field theory.
\item We don't know how to do similar things with 2-knots.
\end{myitemize}
}}}}

\def\Notes{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red Notes.}\begin{itemize}
\item Where does Arnold's strangeness fit in?
\item Perhaps I should put in the Merkov constructions?
\item Nearby objects: ``virtual doodles'', doodles with dots, planar graphs
of various kinds, flat braids.
\end{itemize}
}}}}

\def\Virtual{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red Jonathan's Comment.}
It seems that the configuration space integrals we defined are more
naturally invariants of virtual doodles. Virtual doodles are doodles with
some ordinary crossings and some virtual crossings, with the relation
that triple points having three virtual crossings are allowed. (Caution:
virtual doodles are not Gauss diagrams modulo Reidemeister 2.)

The integrals we defined are invariants for virtual doodles, if we use
the rule that the Gauss diagram skeleton is not allowed to use virtual
crossings.

What kinds of chords do our integrals detect? They detect ``semi-virtuals
with outer rings''.

(Conjectured) Punchline: Relations on Feynman diagrams correspond with
relations on chord diagrams.  This is just a matter of carefully checking
the analogues of the relations we already knew. What makes this work
here and not in the original theory is that we have degree 2 chords,
the semi-virtuals.

What would be nice is a clean formulation of finite type for virtual
doodles yielding chords which are ``semi-virtuals with outer rings''.
}}}}

\input refs.tex

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{\red Issues.}\begin{itemize}
\item 
\end{itemize}
}}}}

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