===== recycled on Tue May  9 12:03:44 EDT 2017 by drorbn on Ubuntu-1404 ======


{\red 1-Smidgen $sl_2$} Let {\red $\frakg_1$}
be the 4-dimensional Lie algebra $\frakg_1=\langle h,e',l,f\rangle$ over
the ring $R=\bbQ[\epsilon]/(\epsilon^2=0)$, with $h$ central and with
$[f,l]=f$,
$[e',l]=-e'$, and $[e',f]=h-2\epsilon l$. Over $\bbQ$,
$\frakg_1$ is a {\red solvable approximation of $sl_2$}: $\frakg_1 \supset
\langle h,e',f,\epsilon h,\epsilon e',\epsilon l,\epsilon f\rangle \supset
\langle h,\epsilon h,\epsilon e',\epsilon l,\epsilon f\rangle \supset 0$.
Pragmatics: declare $\deg(h,e',l,f,\epsilon)=(1,1,0,0,1)$ and set
$t\coloneqq \bbe^h$ and $e\coloneqq (t-1)e'/h$.

===== recycled on Tue May  9 12:03:49 EDT 2017 by drorbn on Ubuntu-1404 ======


{\red Ordering Symbols.} $\bbO\left(\text{\it poly}\mid\text{\it
specs}\right)$ plants the
variables of {\it poly} in $\hat\calS(\oplus_i\frakg)$ along
$\hat\calU(\frakg)$ according to {\it specs}. E.g.,
\[ \bbO\left(e_1\bbe^{e_3}l_1^3l_2f_3^9\,|\,f_3l_1e_1e_3l_2\right)
  = f^9l^3e\bbe^{e}l \in \hat\calU(\frakg).
\]
This enables the description of elements of $\hat\calU(\frakg)$ using
commutative
polynomials / power series. In $\frakg_1$, no need to specify $h$~/~$t$.

===== recycled on Tue May  9 12:03:50 EDT 2017 by drorbn on Ubuntu-1404 ======


{\red Theorem.} The map (as defined below)
\[ \langle w\,\|\,\omega;L;Q;P \rangle \mapsto
  \bbO\left(\omega^{-1}\bbe^{L\log
t+\omega^{-1}Q}(1+\epsilon\omega^{-4}P)\colon\,w\right)
  \in\hat\calU(\frakg_1)
\]
is well defined modulo the sorting rules. It maps the initial preparation
to a product of ``$R$-matrices''
and ``cuap values'' satisfying the usual moves for Morse knots (R3, etc.).
(And hence the result is a
``quantum invariant'', except computed very differently; no representation
theory!).