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\def\navigator{{%
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}:
  \href{http://www.math.toronto.edu/~drorbn/Talks}{Talks}:
  \href{http://drorbn.net/mbw}{MoscowByWeb-1511}:
}}
\def\webdef{{Video and more at {\greektext web}$\coloneqq$\url{http://drorbn.net/mbw}}}
\def\w#1{{\href{http://drorbn.net/mbw/#1}{{\greektext web}/#1}}}

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

\def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:\linebreak[0]#1}}}

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\newcommand{\Aut}{\operatorname{Aut}}
\def\FG{\text{\it FG}}
\def\uB{{\mathit u\!B}}
\def\uT{{\mathit u\!T}}
\def\vB{{\mathit v\!B}}
\def\vT{{\mathit v\!T}}
\def\wB{{\mathit w\!B}}
\def\wT{{\mathit w\!T}}

%%%

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red Abstract.} The subject will be very close to Manturov's
representation of $\vB_n$ into $\Aut(\FG_{n+1})$ --- I'll describe
how I think about it in terms of a very simple minded map $\Zhe$ from
$n$-component v-tangles to $(n+1)$-component w-tangles. It is possible that
you all know this already. Possibly my talk will be very short --- it will
be as long as it is necessary to describe $\Zhe$ and say a few more words,
and if this is little, so be it.
}}}}

\def\AllYouNeed{{\raisebox{2mm}{\parbox[t]{2in}{
{\red All you need is $\Zhe$\ldots} $\bullet$ What is its domain? $\bullet$ What is its target?
$\bullet$ Why should one care?
}}}}

\def\VirtualKnots{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red Virtual Knots.} Virtual knots are the algebraic structure underlying the Reidemeister
presentation of ordinary knots, without the topology. Locally they are knot diagrams modulo
the Reidemeister relations; globally, who cares? So,
\[ \vT =
  \operatorname{CA}\left\langle
    \overcrossing,\undercrossing\colon {\gray R1,} R2, R3
  \right\rangle
  \qquad \operatorname{CA}=\text{``Circuit Algebra''}
\]
}}}}

\def\FlyingPogs{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red Flying Pogs} for $v2_1$ and for $8_{17}$:
}}}}

\def\No{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red No!} Note that also\hfill (with $\operatorname{PA}=$``Planar Algebra'')
\[ \vT =
  \operatorname{PA}\left\langle
    \overcrossing,\undercrossing,\virtualcrossing\colon {\gray R1,} R2, R3, VR1, VR2, VR3, M
  \right\rangle,
\]
but I have a prejudice, or a deeply held belief, that {\red this is morally wrong!}
}}}}

\def\BodenManturov{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red My moment of reckoning.} Manturov's $\mathit{VG}(K)$:\hfill
\cite{Manturov:InvariantsOfVirtualLinks, BodenEtAl:VirtualKnotGroups}
\[
  \tensor*[_x^z]{\text{\huge$\overcrossing$}}{_y^w} \raisebox{2mm}{$\to$}
    \begin{array}[b]{l} z=xyx^{-1} \\ w=x \end{array}
  \quad
  \tensor*[_y^w]{\text{\huge$\undercrossing$}}{_x^z} \raisebox{2mm}{$\to$}
    \begin{array}[b]{l} z=x^{-1}yx \\ w=x \end{array}
  \quad
  \tensor*[_x^z]{\text{\huge$\virtualcrossing$}}{_y^w} \raisebox{2mm}{$\to$}
    \begin{array}[b]{l} z=q^{-1}yq \\ w=qxq^{-1} \end{array}
\]

Manturov's $\mu\colon\vB_n\to\Aut(F(x_1,\ldots,x_n,q))$:\hfill 
\cite{Manturov:InvariantsOfVirtualLinks, BodenEtAl:VirtualKnotGroups}
\[
  \sigma_i=\overcrossing_i\mapsto\left\{{
    x_i\mapsto x_ix_{i+1}x_i^{-1} \atop x_{i+1}\mapsto x_i
  }\right.
  \qquad
  \tau_i=\virtualcrossing_i\mapsto\left\{{
    x_i\mapsto qx_{i+1}q^{-1} \atop x_{i+1}\mapsto q^{-1}x_iq
  }\right..
\]

{\red Easy resolution.} Setting $y_i\coloneqq q^ix_iq^{-i}$, we find that $\mu$ is equivalent to
\[
  \overcrossing_i\mapsto\left\{{
    y_i\mapsto y_iq^{-1}y_{i+1}qy_i^{-1} \atop y_{i+1}\mapsto qy_iq^{-1}
  }\right.
  \qquad
  \virtualcrossing_i\mapsto\left\{{
    y_i\mapsto y_{i+1} \atop y_{i+1}\mapsto y_i
  }\right.,
\]
and to me, virtual braids are anyways always pure. So really,
\[ \sigma_{ij}\mapsto\left\{{
    y_i\mapsto qy_iq^{-1} \atop y_j\mapsto y_i^{-1}q^{-1}y_jqy_i
  }\right..
\]
But why does it exist?\hfill{\red Especially, wherefore $\vB_n\to\wB_{n+1}$?}
}}}}

\def\wTangles{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red w-Tangles.} $\wT\coloneqq\vT/OC$ where ``Overcrossings Commute'' is:
}}}}

\def\descends{{\raisebox{0mm}{\parbox[t]{3.96in}{
$\pi_1$ is defined on $\wT$; Artin's representation $\phi$ is defined on $\wB_n$.
}}}}

\def\BackToZhe{{\raisebox{2mm}{\parbox[t]{3.96in}{
\parshape 9 0in 3.125in 0in 3.125in 0in 3.125in 0in 3.125in 0in 2.5in 0in 2.5in 0in 2.5in 0in 2.5in 0in 3.96in
{\red Back to $\Zhe$.} The ``crossing the crossings'' map
$\Zhe\colon\vT_n\to\wT_{n+1}$ is defined by the picture below. Equally
well, it is $\Zhe\colon\vB_n\to\wB_{n+1}$. Better, it is
$\Zhe\colon\vT_n\to(nv+1w)\!T$ or $\Zhe\colon\vB_n\to(nv+1w)\!B$.
\par{\red Claims.}
\begin{myenumerate}
\item $\Zhe$ is well defined.
\item On u-links, $\Zhe$ ``factors''.
\item $\Zhe$ does not respect $OC$.
\item $\Zhe$ recovers Manturov's $VG$ and $\mu$: $VG(K)=\pi_1(\Zhe(K))$,
  $\mu = \Zhe\circ\phi = \phi\act\Zhe$.
\end{myenumerate}
{\red Even better,} $\Zhe$ pulls back {\em any} invariant of 2-component
w-knots to an invariant of virtual knots. in particular, there is a
wheel-valued ``non-commutative'' invariant $\omega$ as in~\cite{KBH}
and \href{http://www.math.toronto.edu/~drorbn/Talks/Hamilton-1412/}{DBN:
Talks: Hamilton-1412} (next page).
\par{\red Likely,} the various ``2-variable Alexander polynomials'' for virtual knots arise in
this way.
}}}}

\def\ProofTwo{{\raisebox{2mm}{\parbox[t]{1.6in}{
{\red Proof of 2.} The net ``red flow'' into every face is $0$, so the red arrows can be paired.
They form cycles that can hover off the picture.
\par{\red No proof of 3.} Well, there simply is no proof that $OC$ is respected, and it's easy to
come up with counter-examples.
}}}}

\def\ProofFour{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red Proof of 4.} A simple verification, except my conventions are off\ldots
}}}}

\def\refs{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red References.}
\par\vspace{-2mm}
\renewcommand{\section}[2]{}%
\begin{thebibliography}{}
\setlength{\parskip}{0pt}
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\bibitem[BN]{KBH} D.~Bar-Natan,
  {\em
    \href{http://www.math.toronto.edu/~drorbn/papers/KBH/}{Balloons and
      Hoops and their Universal Finite Type Invariant, BF
      Theory, and an Ultimate Alexander Invariant,}
  }
  Acta Mathematica Vietnamica {\bf 40-2} (2015) 271--329, \arXiv{1308.1721}.

\bibitem[BGHNW]{BodenEtAl:VirtualKnotGroups} H.~U.~Boden, A.~I.~Gaudreau, E.~Harper,
  A.~J.~Nicas, and L.~White,
  {\em Virtual Knot Groups and Almost Classical Knots,}
  \arXiv{1506.01726}.

\bibitem[Ma]{Manturov:InvariantsOfVirtualLinks} V.~O.~Manturov,
  {\em On Invariants of Virtual Links,}
  Acta Applicandae Mathematica {\bf 72-3} (2002) 295--309.

\end{thebibliography}
}}}}

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