This is the first in a series of papers studying w-knotted objects
(w-knots, w-braids, w-tangles, etc.), which make a class of knotted
objects which is {w}ider but {w}eaker than their usual counterparts.
The group of w-braids was studied (as "{w}elded braids") by
Fenn-Rimanyi-Rourke and was shown to be isomorphic to the McCool group of
"basis-conjugating" automorphisms of a free group Fn. Brendle-Hatcher,
tracing back to Goldsmith, have shown this group to be a group of movies
of flying rings in R3. Satoh studied several classes of w-knotted objects
(as "{w}eakly-virtual") and has shown them to be closely related to
certain classes of knotted surfaces in R4. So w-knotted objects are
algebraically and topologically interesting.
Here we study finite type invariants of w-knotted objects. Following
Berceanu-Papadima, we construct homomorphic universal finite type
invariants ("expansions") of w-braids and of w-tangles. We find that the
universal finite type invariant of w-knots is essentially the Alexander
polynomial.
We find that the spaces Aw of "arrow diagrams" for w-knotted objects
are related to not-necessarily-metrized Lie algebras. Many questions
concerning w-knotted objects turn out to be equivalent to questions
about Lie algebras. Most notably we find that a homomorphic expansion
of w-knotted foams is essentially the same as a solution of the
Kashiwara-Vergne conjecture (KV), thus giving a topological explanation
to the work of Alekseev-Torossian work on KV and Drinfel'd associators.
The true value of w-knots, though, is likely to emerge later, for we
expect them to serve as a {w}armup example for the study of virtual
knots. We expect v-knotted objects to provide the global context whose
associated graded structure will be the Etingof-Kazhdan theory of
quantization of Lie bialgebras.