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

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

\def\navigator{{Dror Bar-Natan: Talks: CUMC-1307:}}

\def\webdef{{$\omega:=$\url{http://www.math.toronto.edu/~drorbn/Talks/CUMC-1307}}}
\def\w#1{{\href{http://www.math.toronto.edu/drorbn/Talks/CUMC-1307/#1}{$\omega$/#1}}}

\def\Abstract{{\raisebox{2mm}{\parbox[t]{4in}{
{\red Abstract.} Much as we can understand 3-dimensional objects by staring
at their pictures and x-ray images and slices in 2-dimensions, so can we
understand 4-dimensional objects by staring at their pictures and x-ray
images and slices in 3-dimensions, capitalizing on the fact that we
understand 3-dimensions pretty well. So we will spend some time staring at
and understanding various 2-dimensional views of a 3-dimensional elephant,
and then even more simply, various 2-dimensional views of some
3-dimensional knots. This achieved, we'll take the leap and visualize some
4-dimensional knots by their various traces in 3-dimensional space, and
this achieved, I will tell you about the simplest problem in 4-dimensional
knot theory whose solution I don't know.
}}}}

\def\NotT{{\raisebox{2mm}{\parbox[t]{0.875in}{\begin{center}
``The third dimension isn't $t$''
\end{center}}}}}

\def\Conjecture{{\raisebox{2mm}{\parbox[t]{4in}{
\parshape 5 0pt 3.0in 0pt 3.0in 0pt 3.0in 0pt 3.0in 0pt 4in
{\red Satoh's Conjecture.} (\w{Sat}) The ``kernel'' of the ``double
inflation'' map $\delta$, mapping ``long'' w-knot diagrams in the plane to
``long'' knotted 2D tubes in 4D, is precisely the moves R1--R3, VR1--VR3,
D and OC listed below.

In other words, two long w-knot diagrams represent via $\delta$ the
{\blue same} long 2D knotted tube in 4D iff they differ by a sequence
of the said moves.
\[ \text{First Iso.\ Thm: }
  \phi\colon G\to H\ \Rightarrow\ \im\phi\cong G/\ker(\phi)
\]
$\delta$ is a map from algebra to topology. So a thing in ``hard''
topology (``ribbon 2-knots'') is the same as a thing in ``easy'' algebra.
}}}}

\def\Reidemeister{{\raisebox{2mm}{\parbox[t]{3in}{
{\red Reidemeister' Theorem.} Two knot diagrams represent the same 3D knot
iff they differ by a sequence of ``Reidemester moves'':
}}}}

\def\wM{{\raisebox{2mm}{\parbox[t]{2.5in}{
{\red w-Moves.} Same R1, R2, R3 as above, and also:
}}}}

\def\CS{{\raisebox{2mm}{\parbox[t]{1.125in}{
Many of the images are by Carter and Carter-Saito, \w{CS}.
}}}}

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