DoubleTree.pdf: last updated Fri, 22 Nov 2013 13:54:23 +0100.

first edition: Not yet.

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 solutionVof 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 forVin terms ofΦ", along with an independent topological proof that the said formula indeed works - that the equations satisfied byVfollow from the equations satisfied byΦ.

**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.