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The Double Tree Construction

Joint with Zsuzsanna Dancso

DoubleTree.pdf: last updated Sun, 04 Dec 2022 06:41:01 -0500.
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 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. Zsuzsi'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.