**Abstract. **My subject is a Cartesian product. It runs in
three parallel columns - the __u__ column, for __u__sual knots,
the __v__ column, for __v__irtual knots, and the __w__ column,
for __w__elded, or __w__eakly virtual, or __w__armup knots. Each
class of knots has a topological meaning and a "finite type" theory,
which leads to some combinatorics, somewhat different combinatorics
in each case. In each column the resulting combinatorics ends up
describing tensors within a different "low algebra" universe - the
universe of metrized Lie algebras for __u__, the richer universe of
Lie bialgebras for __v__, and for __w__, the wider and therefore
less refined universe of general Lie algebras. In each column there is
a "fundamental theorem" to be proven (or conjectured), and the means,
in each column, is a different piece of "high algebra": associators and
quasi-Hopf algebras in one, deformation quantization à la Etingof and
Kazhdan in the second, and in the third, the Kashiwara-Vergne theory
of convolutions on Lie groups. Finally, __u__ maps to __v__
and __v__ maps to __w__ at topology level, and the relationship
persists and deepens the further down the columns one goes.

The 12 boxes in this product each deserves its own talk, and the few that
are not yet fully understood deserve a few further years of research. Thus my
talk will only give the flavour of a few of the boxes that I understand, and
only hint at my expectations for the contents the (2,4) box, the one I
understand the least and the one I wish to understand the most.