Finite type invariants of w-knotted objects II: tangles, foams and the Kashiwara–Vergne problem

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 base set of knots. But then one imposes a new relation beyond the ordinary collection of Reidemeister moves, called the “overcrossings commute” relation, making w-knotted objects a bit weaker once again. Satoh (J. Knot Theory Ramif. 9-4:531–542, 2000) 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 R⁴. 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 and Vergne (Invent. Math. 47:249–272, 1978) conjecture and much of the Alekseev and Torossian (Ann. Math. 175:415–463, 2012) work on Drinfel’d associators and Kashiwara–Vergne can be re-interpreted as a study of w-foams.