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{\LARGE{\bf (Proposed) Agenda}}\hfill ? \hfill
\parbox[b]{2.275in}{\tiny
   Dropbox and \hfill $|$ \hfill modified \today, \ampmtime
   \newline\null\hfill\url{http://drorbn.net/AcademicPensieve/People/ThurstonD/}
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\begin{multicols*}{2}

{\bf (Proposed) Agenda.} Using \AA rhus-like techniques, construct a map $Z\colon\calT_{\text{\it\!vous}}\to\calA_{\text{\it vous}}$, where $\calT_{\text{\it\!vous}}$ is the space of VOUS-tangles: Virtual tangles with only Over or Under strands, some labeled as Surgery strands, with a non-singular linking matrix between the surgery strands, modulo acyclic Reidemeister 2 moves and Kirby slide relations, and where $\calA_{\text{\it vous}}$ is some space of arrow diagrams modulo appropriate relations. The construction will either fix the definitions of $\calT_{\text{\it\!vous}}$ and $\calA_{\text{\it vous}}$ or will allow some flexibility that will be fixed so that the following will hold true:

\begin{enumerate}

\item $\calT_{\text{\it\!vous}}$ should have a clearer topological interpretation, perhaps in terms of Heegaard diagrams.

\item $\calA_{\text{\it vous}}$ should pair with some kind of Lie bialgebras.

\item $\calA_{\text{\it vous}}$ should be the associated graded of $\calT_{\text{\it\!vous}}$ and $Z$ should be an expansion.

\item Ordinary tangles $\calT_{\text{\it\!ord}}$ and ordinary virtual tangles $\calT_{\text{\it\!v-ord}}$ should map into $\calT_{\text{\it\!vous}}$, and when viewed on $\calT_{\text{\it\!(v-)ord}}$, the invariant $Z$ should explain the Drinfel'd double construction.

\end{enumerate}

It may be better to first construct a $Z$ and only later worry about the numbered properties. Yet property 4 has stand-alone topological content which may be very interesting: $\calT_{\text{\it\!vous}}$ is a space with an $R3$-free presentation and which contains $\calT_{\text{\it\!(v-)ord}}$, at least nearly faithfully. What does it mean? To what extent does it make $R3$ superfluous in knot theory?

As for constructing $Z$, the first step should be a $Z\colon\calT_{\text{\it\!vou}}\to\calA_{\text{\it vou}}$ (no surgery), which would have a prescribed behaviour on strand-doubling.

\end{multicols*}

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