Finite Type Invariants of w-Knotted Objects II: Tangles, Foams and the Kashiwara-Vergne Problem

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 w-knotted objects a bit weaker once again. Satoh 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 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 universal finite type invariant of w-foams is essentially the same as a solution of the Kashiwara-Vergne conjecture and much of the Alekseev-Torossian work on Drinfel'd associators and Kashiwara-Vergne can be re-interpreted as a study of w-foams.

Figures (22)

• Figure 1: An algebraic structure ${\mathcal{O}}$ with 4 kinds of objects and one binary, 3 unary and two 0-nary operations (the constants $1$ and $\sigma$).
• Figure 2: $V \in {\mathit v\!D}_{3,3}$ is a $v$-tangle diagram. $V$ is the result of applying the circuit algebra operation $D: C_{2,2} \times C_{2,2} \times C_{2,2} \to C_{3,3}$, given by the wiring diagram shown, acting on two negative crossings and one positive crossing. In other words $V=D(\undercrossing, \undercrossing, \overcrossing)$. The skeleton of $V$ is given by $\varsigma(V)=D(\virtualcrossing, \virtualcrossing, \virtualcrossing)$, which is equal in ${\mathcal{S}}$ to the diagram shown here. Note that we usually suppress the circuit algebra numbering of boundary points. Note also that the apparent "virtual crossings" of $V$ are not virtual crossings but merely part of the circuit algebra structure, see Warning \ref{['warn:virtualxings']}. The same is true for the crossings appearing in the skeleton $\varsigma(V)$.
• Figure 3: The relations ("Reidemeister moves") R$1^{\!s}$, R2 and R3 define $v$-tangles, adding OC to these defines $w$-tangles. VR1, VR2, VR3 and M are not necessary as the circuit algebra presentation eliminates the need for "virtual crossings" as generators. R1 is not imposed for framing reasons, and not imposing UC breaks the symmetry between over and under crossings in ${\mathit w\!T}$.
• Figure 4: Relations for v-arrow diagrams on tangle skeletons. Skeleta parts that are not connected can lie on separate skeleton components; and the dotted arrow that remains in the same position means "all other arrows remain the same throughout".
• Figure 5: Relations for w-arrow diagrams on tangle skeletons.

Theorems & Definitions (86)

• Example 2.1
• Proposition 2.2
• Proposition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Proposition 2.7
• proof
• Example 2.8
• Definition 2.9