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

TL;DR

This paper explores finite type invariants of w-knotted objects, establishing connections to Lie algebras and the Kashiwara-Vergne conjecture, thereby linking knot theory, algebra, and the study of knotted surfaces.

Contribution

It introduces a framework relating w-knotted objects and their invariants to Lie algebras and the Kashiwara-Vergne problem, providing new insights into their algebraic structure.

Findings

Finite type invariants of w-foams relate to solutions of the Kashiwara-Vergne conjecture.

Homomorphic universal finite type invariant of w-foams corresponds to Kashiwara-Vergne solutions.

Connections between w-knotted objects, Lie algebras, and Drinfel'd associators are established.

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.