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\def\bbR{{\mathbb R}}

\def\red{\color{red}}

\def\navigator{{Dror Bar-Natan: Pensieve: Classes: 13-Aarhus Handouts @ \today, \ampmtime}}

\def\talkurl{{\url{http://www.math.toronto.edu/drorbn/Talks/Aarhus-1305/}}}

\def\uKTGz{{\text{uKTG$_0$}}}
\def\uKTGb{{\text{uKTG$^\bullet$}}}
\def\wTFr{{\text{wTF$^r$}}}
\def\wTFo{{\text{wTF$^o$}}}
\def\wTFe{{\text{wTF$^e$}}}
\def\CA{\operatorname{CA}}
\def\PA{\operatorname{PA}}

\def\uU{{\raisebox{2mm}{\parbox[t]{4in}{
{\red The $u$-Universe.} 
\hfill$\displaystyle
  \text{uKTG} =
  \PA\left\langle
    \text{uGens}
    \left|\text{uRels}\right|
    \text{uOps}
  \right\rangle_{1,1}
$\hfill\null
}}}}

\def\wW{{\raisebox{2mm}{\parbox[t]{4in}{
{\red The $w$-World.} 
\hfill$\displaystyle
  \text{wTF} =
  \CA\left\langle
    \text{wGens}
    \left|\text{wRels}\right|
    \text{uOps}
  \right\rangle
$\hfill\null
}}}}

\def\signs{{\raisebox{0mm}{\parbox[t]{8in}{
\parshape 8 0in 6.4in 0in 6.4in 0in 6.4in 0in 6.4in 0in 6.4in 0in 6.4in 0in 6.4in 0in 8in
If $x$ is an oriented $S^1$ and $u$ is an oriented $S^2$ in an oriented
$S^4$ (or $\bbR^4$) and the two are disjoint, their linking number
$l_{ux}$ is defined as follows. Pick a ball $B$ whose oriented boundary
is $u$ (using the ``outward pointing normal'' convention for orienting
boundaries), and which intersects $x$ in finitely many transversal
intersection points $p_i$. At any of these intersection points $p_i$,
the concatenation of the orientation of $B$ at $p_i$ (thought of a basis
to the tangent space of $B$ at $p_i$) with the tangent to $x$ at $p_i$ is
a basis of the tangent space of $S^4$ at $p_i$, and as such it may either
be positively oriented or negatively oriented.  Define $\sigma(p_i)=+1$
in the former case and $\sigma(p_i)=-1$ in the latter case. Finally,
let $l_{ux}:=\sum_i\sigma(p_i)$. It is a standard fact that $l_{ux}$
is an isotopy invariant of $(u,x)$.

An efficient thumb rule for deciding the linking-number signs for
a balloon $u$ and a hoop $x$ presented using our standard notation
is the ``right-hand rule'' of the figure on
the right, shown here without further
explanation.  The lovely figure is adopted from
[\href{http://en.wikipedia.org/wiki/Right-hand_rule}{Wikipedia:
Right-hand rule}].
}}}}

\def\tubeA{{\raisebox{0mm}{\parbox[t]{8in}{
v-Knots
are oriented knots
drawn on an oriented surface $\Sigma$
(meaning, ``embedded in
$\Sigma\times[-\epsilon,\epsilon]$''),
modulo ``stabilization'', which is the addition and/or removal of empty
handles (handles that do not intersect with the knot). We prefer an
equivalent, yet even more bare-bones approach. For us, a virtual knot is an
oriented knot $\gamma$ drawn on a ``virtual surface $\Sigma$ for
$\gamma$''. More precisely, $\Sigma$ is an oriented surface that may have
a boundary, $\gamma$ is drawn on $\Sigma$, and the pair $(\Sigma,\gamma)$
is taken modulo the following relations:
\begin{itemize}
\item Isotopies of $\gamma$ on $\Sigma$ (meaning, in
  $\Sigma\times[-\epsilon,\epsilon]$).
\item Tearing and puncturing parts of $\Sigma$ away from $\gamma$:
\end{itemize}
}}}}

\def\tubeB{{\raisebox{0mm}{\parbox[t]{8in}{
(We call $\Sigma$ a ``virtual surface'' because tearing and puncturing
imply that we only care about it in the immediate vicinity of $\gamma$).

We can now define a map $\delta_0$, defined on v-knots and taking values
in ribbon tori in $\bbR^4$: given $(\Sigma,\gamma)$, embedd $\Sigma$
arbitrarily in $\bbR^3_{xyz}\subset\bbR^4$. Note that the unit normal
bundle of $\Sigma$ in $\bbR^4$ is a trivial circle bundle and it has a
distiguished trivialization, constructed using its positive-$t$-direction
section and the orientation that gives each fiber a linking number
$+1$ with the base $\Sigma$.  We say that a normal vector to $\Sigma$
in $\bbR^4$ is ``near unit'' if its norm is between $1-\epsilon$ and
$1+\epsilon$. The near-unit normal bundle of $\Sigma$ has as fiber
an annulus that can be identified with $[-\epsilon,\epsilon]\times
S^1$ (identifying the radial direction $[1-\epsilon,1+\epsilon]$
with $[-\epsilon,\epsilon]$ in an orientation-preserving manner), and
hence the near-unit normal bundle of $\Sigma$ defines an embedding
of $\Sigma\times[-\epsilon,\epsilon]\times S^1$ into $\bbR^4$. On the
other hand, $\gamma$ is embedded in $\Sigma\times[-\epsilon,\epsilon]$ so
$\gamma\times S^1$ is embedded in $\Sigma\times[-\epsilon,\epsilon]\times
S^1$, and we can let $\delta_0(\Sigma,\gamma)$ be the composition
\[ \gamma\times S^1
  \hookrightarrow\Sigma\times[-\epsilon,\epsilon]\times S^1
  \hookrightarrow\bbR^4,
\]
which is a torus in $\bbR^4$, oriented using the given orientation of
$\gamma$ and the standard orientation of $S^1$.
}}}}

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