Finite Type Invariants of w-Knotted Objects II: from Associators to Solutions of The Kashiwara-vergne Problem

Abstract. In this paper we utilize a certain "double tree construction" to show that every "expansion", namely "universal finite type invariant (UFTI)" of parenthesized braids extends uniquely first to an expansion/UFTI of knotted trivalent graphs (a well known result), and then on to an expansion/UFTI of w-knotted objects, namely to knottings of "2-dimensional foams" and various associated objects in four-dimensioanl space.

In algebraic language, an expansion for parenthesized braids is the same as a "Drinfel'd associator" Φ, and an expansion for w-knotted objects is the same as a solution V of the Kashiwara-Vergne problem [KV] as reformulated by Alekseev and Torossian [AT]. Hence our result amounts to a topological re-interpretation of the result of Alkeseev-Enriquez-Torossian [AET] that "there is a formula for V in terms of Φ", along with an independent topological proof that the said formula indeed works - that the equations satisfied by V follow from the equations satisfied by Φ.

The paper. DoubleTree.pdf, DoubleTree.zip. Dror's version.

References.

[AT]

A. Alekseev and C. Torossian, The Kashiwara-Vergne conjecture and Drinfeld's associators, Annals of Mathematics 175 (2012) 415-463, arXiv:0802.4300.

[AET]

A. Alekseev, B. Enriquez, and C. Torossian, Drinfeld's associators, braid groups and an explicit solution of the Kashiwara-Vergne equations, Publications Mathématiques de L'IHÉS, 112-1 (2010) 143-189, arXiv:0903.4067.

[KV]

M. Kashiwara and M. Vergne, The Campbell-Hausdorff Formula and Invariant Hyperfunctions, Invent. Math. 47 (1978) 249-272.