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

\def\bbF{{\mathbb F}}
\def\bbR{{\mathbb R}}
\def\calK{{\mathcal K}}

\def\navigator{{Dror Bar-Natan: Talks: HUJI-140101:}}

\def\w#1{{\href{http://www.math.toronto.edu/drorbn/Talks/HUJI-140101/#1}{$\omega$/#1}}}
\def\webdef{{$\omega\coloneqq$\url{http://www.math.toronto.edu/~drorbn/Talks/HUJI-140101}}}
\def\webnote{{Handout and links at \w{}}}

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.9in}{
{\red Abstract.} I will describe a few 2-dimensional knots in 4 dimensional
space in detail, then tell you how to make many more, then tell you that I
don't really understand my way of making them, yet I can tell at least some
of them apart in a colourful way.
}}}}

\def\hookright{{$S^1\hookrightarrow\bbR^3_{xyz}$}}
\def\xyt{{in $\bbR^3_{xyt}$}}

\def\PA{{\raisebox{0mm}{\parbox[t]{2.2in}{\small
\parshape 5 0in 1.1in 0in 1.1in 0in 1.1in 0in 1.1in 0in 2.2in
``Planar Algebra'': The objects are ``tiles'' that can be composed in
arbitrary planar ways to make bigger tiles, which can then be composed
even further\ldots.
}}}}

\def\Conjecture{{\raisebox{2mm}{\parbox[t]{4in}{
\parshape 4 0in 3.125in 0in 3.125in 0in 3.125in 0in 4in
{\red Satoh's Conjecture.} (\w{Sat}) The ``kernel'' of the double
inflation map $\delta$, mapping w-knot diagrams in the plane to knotted
2D tubes and spheres in 4D, is precisely the moves R2-3, VR1-3, M,
CP and OC listed above.  In other words, two w-knot diagrams represent
via $\delta$ the same 2D knot in 4D iff they differ by a sequence of
the said moves.
\[ \blue
  \text{First Isomorphism Thm: }
  \delta\colon G\to H\ \Rightarrow\ \operatorname{im}\delta\cong G/\ker(\delta)
\]
$\delta$ is a map from algebra to topology. So a thing in ``hard''
topology ($\operatorname{im}\delta$) is the same as a thing in ``easy''
algebra ($w\calK$).
}}}}

\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; $\lambda(K)\coloneqq |\{\text{3-colourings}\}|$.

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

{\red Exercise.} Show that the set of colourings of $K$ is a vector space
over $\bbF_3$ hence $\lambda(K)$ is always a power of $3$.
}}}}

\def\Reidemeister{{\raisebox{2mm}{\parbox[t]{3.125in}{
{\red Reidemeister's Theorem.}
%Two knot diagrams represent the same u-knot
%iff they differ by a sequence of ``Reidemester moves'':
}}}}

\def\extend{{\raisebox{2mm}{\parbox[t]{2.5in}{
{\red Extend} $\lambda$ to $w\calK$ by declaring that arcs ``don't see''
v-xings, and that caps are always ``kosher''. Then
$\lambda(\multimapdotboth)=3\neq 9=\lambda(\text{CS 2-knot})$, so assuming
Conjecture, the CS 2-knot is indeed knotted.
}}}}

\def\expansion{{\raisebox{2mm}{\parbox[t]{4in}{
{\red Expansions.} Given a ``ring'' $K$ and an ideal $I\subset K$, set
\[ A\coloneqq I^0/I^1\oplus I^1/I^2\oplus I^2/I^3\oplus\cdots. \]
A homomorphic expansion is a multiplicative $Z\colon
K\to A$ such that if $\gamma\in I^m$, then $Z(\gamma) =
(0,0,\ldots,0,\gamma/I^{m+1},\ast,\ast,\ldots)$.

{\red Example.} Let $K=C^\infty(\bbR^n)$ be smooth functions on $\bbR^n$,
and $I\coloneqq\{f\in K\colon f(0)=0\}$. Then $I^m=\{f\colon f\text{ vanishes as
}|x|^m\}$ and $I^m/I^{m+1}$ is $\{\text{homogeneous polynomials of degree
}m\}$ and $A$ is the set of power series. So $Z$ is ``a Taylor expansion''.

Hence Taylor expansions are vastly general; even {\red knots can be
Taylor expanded!}
}}}}

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