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\def\navigator{{Dror Bar-Natan: Talks: Toronto-150514:}}

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\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
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\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\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
$\BigCirc\neq\lefttrefoil$.
}}}}

\def\Books{{\raisebox{2mm}{\parbox[t]{4in}{
{\red Some knot theory books.}
\par Colin C.~Adams, {\em The Knot Book, an Elementary Introduction to the
Mathematical Theory of Knots,} American Mathematical Society, 2004.
\par Meike Akveld and Andrew Jobbings, {\em Knots Unravelled, from Strings
to Mathematics,} Arbelos 2011.
\par J.~Scott Carter and Masahico Saito, {\em Knotted Surfaces and Their
Diagrams,} American Mathematical Society, 1997.
\par Peter Cromwell, {\em Knots and Links,} Cambridge University Press,
2004.
\par W.B.~Raymond Lickorish, {\em An Introduction to Knot Theory,} Springer
1997.
}}}}

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