We construct $\sleps$, a certain ``lossless approximation'' of $sl_2$,
and show that ``everything that matters'' around its universal enveloping
algebra and its quantization, namely the products, the co-products, the
$R$-matrix, and other essential ingredients can be described in terms
of a certain category \dpg\ of ``{\bf Do\fb}cile {\bf Pe\fb}rturbed
{\bf G\fb}aussian differential operators''.

\par Those essential ingredients are what one needs in order to construct
powerful knot invariants with good algebraic properties. Also, as we
show, \dpg\ is ``easy'' in the sense of computational complexity. Hence
we get (and implement and compute) powerful poly-time-computatble knot
invariants with favourable algebraic properties. Hooray!

\par Similar constructions ought to exist for all semi-simple Lie algebras,
but we do not pursue this here.