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

\def\bbR{{\mathbb R}}
\def\lr{\leftrightarrow}

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

\def\navigator{{Dror Bar-Natan: Talks: Cornell-150925:}}

\def\webdef{{$\omega:=$\url{http://drorbn.net/C15}}}
\def\w#1{{\href{http://drorbn.net/C15/#1}{$\omega$/#1}}}

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.875in}{
{\red Abstract.} Much as we can understand 3-dimensional objects by staring
at their pictures and x-ray images and slices in 2\nobreak-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
if we'll still have time, we'll prove that these knots are really knotted.
}}}}

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

\def\formally{{\raisebox{1mm}{\parbox[t]{2.5in}{
Formally, ``a differentiable embedding of $S^1$ in $\bbR^3$ modulo
differentiable deformations of such''.
}}}}

\def\Formally{{\raisebox{.05mm}{\parbox[t]{2.5in}{
Formally, ``a differentiable embedding of $S^2$ in $\bbR^4$ modulo
differentiable deformations of such''.
}}}}

\def\Reidemeister{{\raisebox{2mm}{\parbox[t]{3in}{
{\red Reidemeister's Theorem.} (a) Every knot has a ``broken curve
diagram'', made only of curves and ``crossings'' like $\slashoverback$.
(b) Two knot diagrams represent the same 3D knot
iff they differ by a sequence of ``Reidemester moves'':
}}}}

\def\ThreeColourings{{\raisebox{2mm}{\parbox[t]{4in}{
\parshape 5 0in 2.5in 0in 2.5in 0in 2.5in 0in 2.5in 0in 4in
{\red 3-Colourings.} Colour the arcs of a broken arc diagram in {\red
R}{\green G}{\blue B} so that every crossing is either mono-chromatic or
tri-chromatic. Let $\lambda(K)$ be the number of such 3-colourings that $K$
has.

{\red Example.} $\lambda(\BigCirc)=3$ while $\lambda(\lefttrefoil)=9$; so
$\BigCirc\neq\lefttrefoil$.

{\red Riddle.} Is $\lambda(K)$ always a power of $3$?

{\red Proof sketch.} It is enough to show that for each Reidemeister move,
there is an end-colours-preserving bijection between the colourings of the
two sides. E.g.:
}}}}

\def\RosI{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Theorem.} Every 2-knot can be represented by a ``broken surface
diagram'' made of the following basic ingredients,
}}}}

\def\RosII{{\raisebox{1mm}{\parbox[t]{3.95in}{
\ldots and any two representations of the same knot differ by a sequence of
the following {\red ``Roseman moves''}:
}}}}

\def\Stronger{{\raisebox{2mm}{\parbox[t]{3.875in}{
{\red A Stronger Invariant.} There is an assigment of groups to
knots / 2-knots as follows. Put an arrow ``under'' every un-broken curve /
surface in a broken curve / surface diagram and label it with the name
of a group generator. Then mod out by relations as below.
}}}}

\def\pione{{\raisebox{2mm}{\parbox[t]{3.875in}{
{\red Facts.} The resulting ``Fundamental group'' $\pi_1(K)$ of a knot /
2-knot $K$ is a very strong but not very computable invariant of $K$.
Though it has computable projections; e.g., for any finite $G$, count the
homomorphisms from $\pi_1(K)$ to $G$.

{\red Exercise.} Show that
$|\operatorname{Hom}(\pi_1(K)\to S_3)| = \lambda(K)+3$.
}}}}

\def\Conjecture{{\raisebox{2mm}{\parbox[t]{3.875in}{
\parshape 7 0pt 1.9in 0pt 1.9in 0pt 2.9in 0pt 2.9in 0pt 2.9in 0pt 2.9in 0pt 3.875in
{\red Satoh's Conjecture.} (Satoh, {\em Virtual Knot Presentations of
Ribbon Torus-Knots,} J.~Knot Theory and its Ramifications {\bf 9} (2000)
531--542).  Two long w-knot diagrams represent via the map $\delta$ the
same simple long 2D knotted tube in 4D iff they differ by a sequence of
R-moves as above and the ``w-moves'' VR1--VR3, D and OC listed below:
}}}}

\def\Books{{\raisebox{2mm}{\parbox[t]{3.875in}{
{\red Some knot theory books.}
\par$\bullet$ Colin C.~Adams, {\em The Knot Book, an Elementary
Introduction to the Mathematical Theory of Knots,} American Mathematical
Society, 2004.
\par$\bullet$ Meike Akveld and Andrew Jobbings, {\em Knots Unravelled,
from Strings to Mathematics,} Arbelos 2011.
\par$\bullet$ J.~Scott Carter and Masahico Saito, {\em Knotted Surfaces
and Their Diagrams,} American Mathematical Society, 1997.
\par$\bullet$ Peter Cromwell, {\em Knots and Links,} Cambridge University
Press, 2004.
\par$\bullet$ W.B.~Raymond Lickorish, {\em An Introduction to Knot
Theory,} Springer 1997.
}}}}

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