Abstract. This is the first in a series of papers studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.). These are classes of knotted objects which are wider, but weaker than their "usual" counterparts.The group of w-braids was studied (under the name "welded braids") by Fenn, Rimanyi and Rourke [FRR] and was shown to be isomorphic to the McCool group [Mc] of "basisconjugating" automorphisms of a free group F n -the smallest subgroup of Aut(F n ) that contains both braids and permutations. Brendle and Hatcher [BH], in work that traces back to Goldsmith [Gol], have shown this group to be a group of movies of flying rings in R 3 . Satoh [Sa] studied several classes of w-knotted objects (under the name "weakly-virtual") and has shown them to be closely related to certain classes of knotted surfaces in R 4 . So w-knotted objects are algebraically and topologically interesting.In this article we study finite type invariants of w-brainds and w-knots. Following Berceanu and Papadima [BP], we construct homomorphic universal finite type invariants of w-braids. We find that the universal finite type invariant of w-knots is essentially the Alexander polynomial.Much as the spaces A of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, we find that the spaces A w 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, and in later papers of this series we re-interpret Alekseev-Torossian's [AT] work on Drinfel'd associators and the Kashiwara-Vergne problem as a study of w-knotted trivalent graphs.

Abstract. This is the second in a series of papers dedicated to studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.). These are classes of knotted objects that are wider but weaker than their "usual" counterparts. To get (say) w-knots from usual knots (or u-knots), one has to allow non-planar "virtual" knot diagrams, hence enlarging the the base set of knots. But then one imposes a new relation beyond the ordinary collection of Reidemeister moves, called the "overcrossings commute" relation, making wknotted objects a bit weaker once again. Satoh [Sa] studied several classes of w-knotted objects (under the name "weakly-virtual") and has shown them to be closely related to certain classes of knotted surfaces in R 4 . In this article we study finite type invariants of w-tangles and w-trivalent graphs (also referred to as w-tangled foams). Much as the spaces A of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, the spaces A w 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 universal finite type invariant of w-foams is essentially the same as a solution of the Kashiwara-Vergne [KV] conjecture and much of the Alekseev-Torossian [AT] work on Drinfel'd associators and Kashiwara-Vergne can be re-interpreted as a study of w-foams.

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